Dynamics, Processes and Characterization in Classical and ...

Dynamics, Processes and Characterization in Classical and QuantumOptics

by

Omar Gamel

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

c© Copyright 2013 by Omar Gamel

Abstract

Dynamics, Processes and Characterization in Classical and Quantum Optics

Omar Gamel

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2013

We pursue topics in optics that follow three major themes; time averaged dynamics with the asso-

ciated Effective Hamiltonian theory, quantification and transformation of polarization, and periodicity

within quantum circuits.

Within the first theme, we develop a technique for finding the dynamical evolution in time of a time

averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian,

as well as decoherence terms that sometimes manifest in a Lindblad-like form. We also apply the theory

to examples of the AC Stark Shift and three level Raman Transitions.

In the theme of polarization, the most general physical transformation on the polarization state has

been represented as an ensemble of Jones matrix transformations, equivalent to a completely positive

map on the polarization matrix. This has been directly assumed without proof by most authors. We

follow a novel approach to derive this expression from simple physical principles, basic coherence optics

and the matrix theory of positive maps.

Addressing polarization measurement, we first establish the equivalence of classical polarization and

quantum purity, based on the identical mathematical structure of the Poincare and Bloch spheres. We

analyze and compare various measures of polarization / purity for general dimensionality proposed in

the literature, with a focus on the three dimensional case.

In pursuit of the final theme of periodic quantum circuits, we introduce a procedure that synthesizes

the circuit for a simple monoperiodic function that is one-to-one within a single period, of a given period

p. Applying this procedure, we synthesize these circuits for p up to five bits. We conjecture that such a

circuit will need at most n Toffoli gates, where p is an n-bit number.

Moreover, we apply our circuit synthesis to compiled versions of Shor’s algorithm, showing that it

can create more efficient circuits than ones previously proposed. We provide some new compiled circuits

for experimentalists to use in the near future. A layer of “classical compilation” is pointed out as a

method to further simplify circuits. Periodic and compiled circuits are expected to be helpful in creating

experimental milestones, and for the purposes of validation.

ii

Dedication

To my parents Randa and Ehab, whose love and encouragement have shaped my life.

iii

Acknowledgements

I firstly acknowledge my doctoral supervisor, Prof. Daniel James, whose mentorship and guidance have

been invaluable. Working with him has provided me with a much deeper appreciation for physics,

and the entire scientific process. His support has made the research journey an edifying and enjoyable

one. Our memorable discussions ranged from the idiosyncrasies of the culture behind science to the

underlying meaning behind mystifying quantum phenomena. There is much that I have learned from

him; on science, and on life.

I thank my past and present group members, Rene, Asma, Agata, Nicolas, Arnab, Christian, Kero

and Jaspreet who have been great colleagues and friends.

I am also indebted to my supervisory committee members Prof. John Sipe and Prof. Aephraim Stein-

berg, whose thought provoking input and penetrating questions have taught me a great deal throughout

my career as a graduate student. Prof. Man-Duen Choi at the Mathematics Department has given me

much of his time and knowledge through fruitful discussions. Prof. Amr Helmy has graciously agreed to

sit on my departmental defense committee on short notice. I also thank Professors Hoi-Kwong Lo and

Joseph Thywissen for their questions as members of my final defense committee. I extend my apprecia-

tion to Prof. Duncan O’Dell from McMaster University who provided thorough and enriching feedback

on the thesis in his capacity as external examiner for the final defense.

I also extend my thanks to the Natural Sciences and Engineering Research Council of Canada

(NSERC), and the Ontario Ministry of Training, Colleges, and Universities for funding this work. The

University of Toronto and the Department of Physics have provided the best of learning environments. I

also thank Helen Iyer and Krystyna Biel for making the administrative side of my PhD career a breeze.

I thank Abdullah, Guillaume, Sergei, and Alagappan for being pleasant officemates during my gradu-

ate career. I also thank my friend and colleague Ramy El-Ganainy, now Professor of physics at Michigan

Tech, for our engaging and stimulating discussions over the years.

The deepest gratitude goes to my parents Randa and Ehab, who have always been there for me. At

every step in my life, they have provided me with motivation, a nurturing environment, and instilled

within me the value of learning. My debt to them can never be repaid. I thank my younger brother

Mohammed for the encouragement he has given me through life. His curiosity and questions growing

up have fueled my love of learning, and teaching.

Most importantly, a heartfelt appreciation to my wife, life partner, and best friend Marwa. Her

enduring love and companionship transform difficult times to tranquil moments. She has always given

me her unwavering support, and had confidence in me even when I did not.

و بر لهل دمالحينالمالع

iv

Contents

1 Introduction 1

1.1 Time Averaging Quantum Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Polarization and Completely Positive Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Periodicity in Quantum Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Time Averaged Dynamics and the Effective Hamiltonian 5

2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Evolution Equation of Average Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 The Unitary Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.2 The Inverse Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 Approximation up to Second Order . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.4 The Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Harmonic Time Dependent Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 AC Stark Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.2 Three Level Raman Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Completely Positive Maps and Polarization 20

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Complex Analytic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Definition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 Non-linearity of Complex Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Classical Polarization States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.1 Simple Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.2 General Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Positive Linear Map Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

v

4 Measures of Purity and Degree of Polarization 29

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Polarization of Beams and Purity of Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 Classical Polarization States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.2 Quantum Two Level system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.3 Polarization and Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Measures of Purity for N Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.1 Standard Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.2 Von Neumann Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.3 Polarization Purity for N=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.4 Barakat Heirarchy Measures of Purity . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.5 EDPW Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.6 SSKF Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 Comparing Purity Measures for Three Dimensions . . . . . . . . . . . . . . . . . . . . . . 40

4.4.1 Graphical Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4.3 Relationship between SSKF Purity Πsskf and EDPW Purity Πedpw . . . . . . . . 42

4.5 Relation to Entanglement Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Synthesizing Quantum Circuits for Simple Periodic Functions 46

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3 Circuit Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.4 Required Resources for Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Compiled Quantum Factoring Algorithms 54

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.2 Shor’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.2.1 Classical Probabilistic Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.2.2 Quantum Order Finding Subroutine . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.3 Compiled Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.3.1 Implementations of Compiled Shor’s Algorithm . . . . . . . . . . . . . . . . . . . 57

6.3.2 The Compilation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.3.3 Classical Compilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.4 General N and Allowable Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.5 Illustrative Example of Compiled Period Finding . . . . . . . . . . . . . . . . . . . . . . . 63

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7 Conclusions and Outlook 68

Appendices 71

vi

A Addendum to Chapter 2: Time Averaged Dynamics 72

A.1 Derivation of Fk and Lk Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

B Addendum to Chapter 4: Measures of Purity 74

B.1 Gell-Mann Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B.2 Depolarizing Channels as a Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B.3 Partial Derivatives and Agreement of Purity Measures . . . . . . . . . . . . . . . . . . . . 75

C Addendum to Chapter 5: Circuits for Simple Periodic Functions 77

Bibliography 83

vii

List of Tables

4.1 The purity of some states we introduced as well as the pure and maximally mixed states, as

evaluated by four different measures of purity: Πsskf , Πedpw, Πb and Πv. The eigenvalue

spectrum λ1, λ2, λ3 of each state in curly braces. The columns are ordered from purest

to most mixed according to the Πsskf measure. . . . . . . . . . . . . . . . . . . . . . . . . 41

5.1 S3 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 S5 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3 S7 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.4 S9 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.5 S11 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.6 For each period p, we write the period in base 2 ([p]2) and provide its bit-length n. The

informative columns are the number of Toffoli gates (NT ) and CNOT gates (NCN ) needed

to synthesize the Sp circuit. We also include the quantum cost Q = NCN + 6NT . . . . . . 52

6.1 The binary truth table for y = f2,15(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58


6.3 The decimal value table for f4,21(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.4 The period r of fa,21(x) for all a coprime to 21. . . . . . . . . . . . . . . . . . . . . . . . . 59


6.6 The truth table for the partially compiled f4,21(x), with three input qubits and two output

qubits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.7 The fully compiled truth table for y = f4,21(x) with two input and two output qubits.

Identical to S3 table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.8 The period r of fa,33(x) for all a coprime to 33. . . . . . . . . . . . . . . . . . . . . . . . . 62


6.10 For each semiprime N = pq with p and q distinct odd primes, the table lists the value of

the Carmichael function λ(N) ≡ lcm(p− 1, q − 1), and the allowable periods r, given by

all the factors of λ(N). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.11 The probabilities Pp(k) of finding |k〉 after a measurement on the input register of |φp〉.The column index is k and the row index is p. . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.12 The rough separability index S for various values of period p. The values of S are cal-

culated from the probability distributions in table 6.11 by summing the squares of the

entries for each row. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

C.1 S13 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

viii

C.2 S15 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

C.3 S17 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

C.4 S19 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

C.5 S21 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

C.6 S23 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

C.7 S25 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

C.8 S27 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

C.9 S29 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

C.10 S31 truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

ix

List of Figures

2.1 The evolution of the real part of the off diagonal entry for the density matrix; i.e. 12 〈σx〉.

Interaction strength set at b = 0.3. Time in units of the characteristic time 1∆ . Both axes

dimensionless. Since the relative phase of the exact and averaged evolution just depends

on initial conditions and the zero of time, initial conditions were artificially modified in

the figure so the two curves start in phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Illustration of the three level Raman Transitions . . . . . . . . . . . . . . . . . . . . . . . 15

4.1 Classes of cross section of the eight dimensional space in which the generalized Bloch

vectors live, based on a figure by Kimura [124]. In each diagram, the shaded region

represents the allowable states, while the outer circle is a cross section of the enclosing

hypersphere. The pure states are where the shaded region touches the outer circle. Points

A, B, C, D are specific states we examine. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Values of purity measures for various eigenvalues of a three dimensional density matrix.

Each graph has fixed λ1, with λ2 against the horizontal axis, and λ3 ≡ 1 − λ1 − λ2. In

each graph, the upper solid curve in black is Barakat’s last measure Πb, the upper dashed

blue curve is the SSKF purity Πsskf , the lower dashed brown curve is the von Neumann

purity Πv, and the lower solid green curve is the EDPW purity Πedpw. . . . . . . . . . . . 40

5.1 S3 quantum circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49





6.1 The circuit for y = f2,15(x). Gate count: NT = 1 and NCN = 7. This is slightly cheaper in

terms of gates than the analogous arrangement in ref. [74], which used 2 controlled-swap

gates (roughly as hard as a Toffoli) and 2 CNOT gates. . . . . . . . . . . . . . . . . . . . 58

6.2 The fully compiled circuit for y = f2,15(x). Gate count: NT = 0, NCN= 2. . . . . . . . . 58

6.3 The circuit for y = f4,15(x). Gate count: NT=0, NCN = 2. . . . . . . . . . . . . . . . . . 59

6.4 The fully compiled circuit for y = f4,15(x). Gate count: NT=0, NCN = 1. . . . . . . . . . 59

6.5 The circuit for y = f4,21(x) with three input qubits, and five output qubits. Gate count:

NT=2, NCN = 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.6 The circuit for the partially compiled f4,21(x), with three input qubits and two output

qubits. Gate count: NT=2, NCN = 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

x

6.7 The fully compiled circuit for y = f4,21(x) with an additional layer of classical compilation.

Gate count: NT=1, NCN = 3. Identical to S3 circuit in fig. 5.1. . . . . . . . . . . . . . . 61

6.8 The circuit table for y = f4,33(x). Gate count: NT=3, NCN = 7. . . . . . . . . . . . . . . 62

C.1 S13 quantum circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78










xi

Chapter 1

Introduction

Humankind’s fascination with light is as old as humanity itself. Since the dawn of civilization, the

ancients began to manipulate light and study its properties through polished quartz crystals. Yet optics

today has come a long way from the days of Euclid of Alexandria’s treatise on the geometry of vision

[1, 2]. As it blossomed, optics has left us with much to behold; Alhazen’s camera obscura, Newton’s

corpuscles and prisms, Young’s double slit experiment [3, 4, 5] and much vivid imagery.

As optics matured, countless subfields were added, such as the wave theory of light, polarization,

electromagnetics, culminating in the discovery of light as a quantum phenomenon. Optics today is

inseparable from quantum theory, which had its birth and chief applications wherever light is involved.

Quantum theory eventually gave rise to quantum computing, and with it, a new window to understanding

the physical universe. In our age, light has proven itself a far more fascinating and varied phenomenon

than the ancients could have imagined.

As a tribute to this diversity of scientific content, this thesis will cover three major themes within

quantum and classical optics. Firstly, we address time averaged quantum dynamics in chapter 2. We

study how a quantum system, optical or otherwise, evolves if we are only interested in the evolution of

its moving average, rather than the instantaneous state. This leads us to a general theory of Effective

Hamiltonians, which we then apply to well known quantum optical systems.

Our second theme is polarization, and its presence in classical light. In chapter 3 we discuss transfor-

mations on polarization and their relationship to completely positive maps, demonstrating an insightful

and novel derivation of a known operation. Chapter 4 furthers our exploration to an analysis of differ-

ent measures of polarization, which we show is identical to quantum purity. We analyze the different

measures in the literature, shedding light on the properties of each measure.

The third and final theme relates to periodicity in quantum circuits. Chapter 5 presents the simple

problem of creating the quantum circuit for the simple periodic functions. We develop a method for

circuit synthesis, and apply it to the problem at hand. We generate many simple periodic functions, and

observe some techniques useful for general circuit synthesis. Chapter 6 takes the periodic circuit theme

to compiled versions of Shor’s algorithm [6], particularly the modular exponentiation subroutine. We

apply our circuit synthesis methods to compile some basic circuits, and arrive at more efficient results

than in the literature.

In the remainder of this introductory chapter, we explain each theme in more detail, and reference

the major sources in the literature.

1

Chapter 1. Introduction 2

1.1 Time Averaging Quantum Evolution

In many interesting quantum systems with oscillatory Hamiltonians, there are often multiple frequen-

cies of different orders of magnitude. Sometimes an approximation is made, that the high frequency

component may be ignored for the time scale of interest. For example, in a two level atomic system,

the rotating wave approximation is often applied [7, 8, 9]. This approximation simply discards the high

frequency component of the interaction Hamiltonian, i.e. the counter rotating component.

One may ask if it is justified to simply discard this component of the Hamiltonian. Will it not have

any effect at all? In fact, the AC Stark shift [10], the Lamb shift [11], and the Bloch-Siegert shift [12]

result from high frequency oscillatory terms. In each case, the high frequency terms end up modifying

the effective parameters in other terms of the Hamiltonian. In the coarse grained time scale, the high

frequency term is discarded, but modifies the “Effective Hamiltonian”.

The theory of Effective Hamiltonians has many contributors. Cohen-Tannoudji [13] in his classic

textbook discusses the subject. Shore approaches the topic from the adiabatic approximation [14],

as do Gerry and Knight [15]. Effective Hamiltonians have been developed for ion traps and cavities

[16, 17, 18, 19]. Averaged quantum dynamics have also been investigated in stochastic systems [20],

electron systems [21], and open quantum systems [22]. James and Jerke have also derived a form for

the Effective Hamiltonian [23], which we re-derive in a more comprehensive manner in chapter 2. The

contents of chapter 2 have been modified from ref. [24] written by the author.

In deriving the Effective Hamiltonian, the process of discarding some frequencies results in lost

information, and some decoherence. Given the right form of the Hamiltonian, this will lead to an

equation of evolution similar in form to Lindblad’s master equation for open system dynamics [25]. This

link between time averaged systems and open systems is interesting, but not surprising, since averaging

in time will discard information in a similar manner to tracing over the environment in an open quantum

system.

1.2 Polarization and Completely Positive Maps

Supposing we have a classical beam of light, its polarization state can be described in multiple ways. One

common description, in use since the 19th century, relies on the Stokes parameters [26]. The four Stokes

parameters were introduced to specify the beam’s polarization state [27]. They have enough degrees of

freedom to represent a beam in any possible polarization state. If instead of a general beam, we have a

monochromatic fully polarized beam, a Jones vector is commonly used to represent the state [28].

The theory of statistical optics, which treats the electromagnetic field stochastically, has proven a

useful tool over the years. Formalisms by Wiener [29] and Wolf [30, 31] developed the field of statistical

optics, and defined the state through what was known as the coherency matrix. The modern name for

the latter, and the one we use, is the polarization matrix. Results by Van Cittert and Zernike established

the electric field in a light beam can be treated as a random variable [32, 33].

Optical systems acting on an electromagnetic beam will transform its state. The mathematical

machinery for the transformation will obviously depend on the method used to represent the state.

A Stokes vector description of the state requires a Mueller matrix [34] for the transformation, and a

polarization matrix state requires a Jones matrix transformation [28].

Most authors assume on physical grounds that the most general transformation on the polarization


matrix can be represented as an ensemble of Jones matrix transformations [35]. From this assumption,

they convert the Jones matrix ensemble to a Mueller matrix, to find the most general form of the latter

[36, 37, 38]. Recently, Simon et al. [39, 40] have addressed this problem based on properties of matrices

derived from the Mueller matrices, and the structure of the electric field as a tensor product of vectors

from two Hilbert spaces.

In chapter 3, we prove this ensemble assumption based on some more fundamental features of the

physical system, the concept of a complex analytic signal [9, 31], and a brief foray into the theory of

positive and completely positive maps in C* algebras [41, 42, 43]. Chapter 3 is based on ref. [44] by the

author.

In chapter 4, we continue in the same vein, and address the well known mathematical duality between

a two dimensional classical polarization state, and a quantum two level system. The Stokes vector is

analogous to the Bloch vector [45], and the Poincare sphere which geometrically represents the former

[46] is identical to the Bloch sphere, representing the latter. The Pancharatnam phase in classical optics

[47] is the Berry phase in quantum systems [48].

Furthering this analogy, we observe that the point at the origin of the Poincare sphere represents

a completely unpolarized beam of light, while a point on the sphere’s surface represents a completely

polarized beam. The origin of the Bloch sphere is a maximally mixed state while a surface point is a

pure state. This points to a clear analogy between classical degree of polarization and level of quantum

purity. The mathematics behind measuring of the two quantities should, it seems, be identical.

This analogy between polarization and purity has been discussed by some authors [49], and led others

to postulate measures of purity in higher dimensions [50]. And yet, the common measures of classical

degree of polarization and quantum mechanical purity differ. We address this discrepancy in chapter 4

by surveying a range of measures polarization and purity in the literature, and then analyzing them in

a fair level of detail.

We assess the standard purity (so called because it is what is usually thought of when one men-

tions purity in quantum information), von Neumann purity [51], and measures of higher dimensional

polarization due to Barakat [52], Friberg et. al. [53], and Wolf et al. [54]. Our analysis adds to

and organizes much of the discussion on measures of higher dimensional polarization in the literature

[55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. Further, we briefly discuss the relation of purity measures to

entanglement measures. The work in chapter 4 is based on ref. [65] by the author.

1.3 Periodicity in Quantum Circuits

Experimental capabilities in quantum computing technology are currently quite limited. While theorists

develop elaborate and useful algorithms such as Shor’s quantum factoring algorithm [6], experimentalists

are not yet able to fully execute these procedures. To circumvent this problem, experimental groups

have resorted to simplified, or compiled, versions of the important algorithms to serve as milestones for

advancing technology.

Furthermore, the traditional method to verify that a quantum device performs its desired function,

i.e. characterization / validation of the device, has been quantum tomography [66, 67]. However, despite

some advances in the field, full quantum state tomography beyond more than one or two dozen qubits

appears to be intractable [68]. Tomography requires a number of measurements that is exponential in

the number of qubits, limiting its scalability. Therefore, it is of interest to create new methods to validate


quantum devices, and compiled algorithms along with related quantum circuits may be the answer.

In particular, due to its importance in quantum computing, Shor’s algorithm is the ideal candidate

for this. The algorithm makes use of an order finding subroutine that involves a modular exponentiation

operation which leads to a periodic output. The quantum Fourier transform can then be used to extract

the period of the function, from which desired factors can be deduced. The modular exponentiation

operation is the bottleneck of the algorithm, needing the most quantum gates to implement and the

most time to execute. Executing the general modular exponentiation operation is not possible with

currently realizable technology.

For this reason, there has been much theoretical work on compiled circuits for the order-finding

subroutine in Shor’s algorithm and associated modular exponentiation [69, 70, 71, 72, 73]. Additionally,

experimental groups have often demonstrated compiled versions of Shor’s algorithm, whether through

optical techniques [74, 75, 76, 77], nuclear magnetic resonance (NMR) [78, 79], or solid state implemen-

tations [80]. Some experimental groups may realize an uncompiled Shor’s algorithm in the near future

[81], though for small inputs.

Given the above, we pursue two paths. The first focuses on periodicity of the quantum circuit,

which as we saw above is central to Shor’s algorithm. In fact, it is the size of the period rather than

number factored which determines the difficulty of the factorization [82]. Quantum parallelism gives

quantum circuits a significant advantage over their classical counterparts when analyzing properties of

periodicity. This has led to the development of a large number of quantum computing algorithms that

exploit periodic properties. This ubiquity of periodicity motivates us to better understand periodic

functions within quantum computing.

To be precise, it is interesting to study how a simple quantum circuit can be synthesized to implement

a function of a desired period. In addition, given the limited capacity of current experimental systems,

it is of interest to find the minimal number of gates needed for a simple function of a given period.

Creating circuits for simple periodic functions will provide the milestones we seek for experimentalists,

as well possibly provide a route to validation of quantum devices.

In chapter 5, we address the problem of synthesizing a quantum circuit for a simple monoperiodic

function of a given period. In doing so, we only use the CNOT and Toffoli quantum gates [83], as well

as some single qubit gate. The circuits for the simplest periodic functions for up to five-bit periods are

synthesized in the chapter and associated appendix. We use the results from this synthesis to conjecture

an upper limit to the required number of Toffoli gates as a function of the period. The work in this

chapter is based on ref. [84] by the author. Other authors have also addressed algorithms for synthesis

of quantum circuits via Toffoli gates in the context of reversible computing theory [85, 86, 87, 88].

Finally, chapter 6 deals directly with the compilation of the modular exponentiation operation within

Shor’s algorithm. We demonstrate the process of constructing the compiled modular exponentiation

circuits for some semiprime numbers, using the circuit synthesis methods we have established. Examples

of compiled circuits used by experimentalists are provided, as well new ones introduced for future use.

We also point out what may be termed a “classical layer” of compilation that reduces the number of

qubits needed in the compiled circuit. The chapter concludes with a simplified procedure to validate the

function of such a compiled circuit, including its handling of noise and entanglement.

Chapter 2

Time Averaged Dynamics and the

Effective Hamiltonian

2.1 Background

The density operator is the mathematical object that carries all measurable statistical information about

a quantum system, and therefore, completely characterizes the state, whether pure or mixed. However,

in reality, the physical perception of any quantum system takes place over a finite time interval, rather

than instantaneously. Thus understanding the behaviour and evolution of the density matrix convolved

with some time averaging function is essential for a complete understanding of quantum dynamics, as

observed in any realistic circumstances.

Moreover, often in practical applications, such as light-matter interaction systems, the Hamiltonian

often has two distinct parts - one that oscillates at a high frequency and one at a much lower frequency.

If one were observing the quantum systems with a time resolution which is too slow to discern the high

frequency effect, one might idly suppose that the high frequency component of the Hamiltonian should

not play much of a role and can simply be ignored. This is for example is what happens in the rotating

wave approximation for a two level atomic system [7, 8, 9].

And yet effects such as the AC Stark shift [10], the Lamb shift [11], and the Bloch-Siegert shift [12]

can be ascribed to just such high frequency terms in the Hamiltonian. So the high frequency terms

cannot be totally ignored after all. The purpose of the work in this chapter is to formalize this idea, and

find the “Effective Hamiltonian” that goes beyond simply discarding high frequencies, and includes their

overall effects. We formalize and examine the validity of Effective Hamiltonian theory in some detail.

The chapter is organized as follows. In section 2.2, we derive a general formula for the evolution of

the time averaged density matrix, finding the Effective Hamiltonian and decoherence terms. In section

2.3 we apply this theory to the class of Hamiltonians with harmonic disturbances, we derive an equation

of evolution in a form similar to Lindblad’s open system dynamics [25]. The formula for the Effective

Hamiltonian found confirms some previous results [23], and new decoherence terms are found to result

from the averaging process.

Finally, in section 2.4 we test this theory on known physical systems, the AC Stark Shift, and the three

level Raman Transitions, finding a new decoherence effect in the latter, which is potentially realizable

through experiment.

5

Chapter 2. Time Averaged Dynamics and the Effective Hamiltonian 6

Similar work was completed previously by James in the context of Effective Hamiltonian Theory

[23], and by other authors such us Cohen-Tannoudji [13], Shore [14], Gerry and Knight [15]. Effective

Hamiltonians were also used for various systems, such as ion traps and cavities [16, 17, 18, 19]. Averaged

quantum dynamics have also been investigated in stochastic systems [20], electron systems [21], and open

quantum systems [22].

The idea of systems evolving in multiple time scales is ubiquitous in physics. For example, separating

the time evolution to different time scales resembles the well-known method of multiple scales [141]. This

method treats fast and slow scales of time as different independent variables, in order to get rid of secular

terms in the perturbative solution, and find the effective long term periodic behaviour. However, the

method we employed in this chapter is different qualitatively, as there need not be any secular terms in

the solution. We use a time convolution approach rather than creating additional independent variables.

Further, observing behaviour of a physical system at different time or distance scales is reminiscent

of the renormalization group in quantum field theory and condensed matter physics [142]. The Langevin

equation used to describe Brownian motion is stochastic differential equation that describes the evolution

of a system on a macroscopic scale, removing the microscopic effects [93].

The contents of this chapter have been modified from ref. [24] written by the author.

2.2 Evolution Equation of Average Density Matrix

2.2.1 The Unitary Evolution

We start by defining the time-average of an operator O(t):

O(t) ≡∫ ∞−∞

f(t− t′)O(t′)dt′, (2.1)

where the averaging kernel f(·) is positive, real-valued, and has unit area. Since this is a convolution, it

is equivalent to a multiplication in frequency space, and the Fourier transform of f(t) can be seen as a

frequency filter.

In particular, suppose a quantum system has density operator ρ(t), we are interested in the time-

averaged density operator ρ(t). A density matrix must be Hermitian, unit trace, and positive. Since

f(t − t′) is real and ρ is Hermitian, ρ(t) must be Hermitian. The trace of the time-averaged density

matrix is given by

Tr[ρ(t)] = Tr[ ∫ ∞−∞

f(t− t′)ρ(t′)dt′]

=

∫ ∞−∞

f(t− t′) Tr[ρ(t′)]dt′

=

∫ ∞−∞

f(t− t′)dt′ = 1, (2.2)

where in the third line we made use of Tr[ρ] = 1, and the final equality is since f(·) has unit area.

Moreover, since ρ(t′) is a positive matrix for all t′ and f(t − t′) is always positive, the time-averaged

matrix ρ(t), when seen as the limit of a Riemann sum, is a positive linear combination of positive matrices

and is therefore itself a positive matrix.

That is, the Hermiticity, unit trace, and positivity of ρ(t) imply the same properties for ρ(t), meaning


it too is a density matrix. This averaged density operator ρ(t) can be interpreted in multiple ways. One

may interpret the averaging process as a purely theoretical construct that results from a coarse-grained

time resolution. That is, we treat interactions as instantaneous, but choose to focus on certain frequencies

in the interaction. Alternatively, averaging may be interpreted as physically representing the ”perceived”

density matrix through any physical interactions with an apparatus that take a finite nonzero time, with

the averaging kernel f(t− t′) representing the strength of this interaction in the time window involved.

By assumption, ρ(t) represents the evolution of a closed system in the Schrodinger picture, and

therefore it evolves as per the Von Neumann equation:

i~∂ρ

∂t= [H, ρ], (2.3)

where H is the Hamiltonian of the system. This equation implies ρ(t) is given by the unitary evolution

equation:

ρ(t) = U(t, t0)ρ(t0)U(t, t0)† (2.4)

where U(t, t0) is the familiar time-ordered evolution operator which satisfies the Schrodinger equation:

i~∂U(t, t0)

∂t= H(t)U(t, t0). (2.5)

Given the generic equations above, we wish to find an equation analogous to eq. (2.3) that involves

ρ(t) on both sides of the equation rather than ρ(t). That is, we wish to find a differential equation of

evolution for the time-averaged density operator that does not directly involve the instantaneous density

matrix. This equation we seek is in fact a Markovian equation of evolution, meaning the time evolution

of ρ(t) only depends on the value of ρ(t) at the current instant, not the past [100].

However, eq. (2.1) that defines the time average involves a time integral over a finite period, and is

manifestly not instantaneous. This implies that the Markovian equation we seek will be an approximation

at best, and that the exact equation of evolution of ρ(t) will be an integro-differential equation [99]. As

we will later see, making the Markovian approximation may lead to issues with the positivity of the

evolution equation [97].

Nonetheless, finding the Markovian approximation is a useful exercise, and often used in the theory

of quantum operations [103]. Optical systems, for example, are generally treated as Markovian, which

classically is a consequence of Huygen’s principle [94].

We begin by turning our attention to eq. (2.5). To obtain a series expansion for U(t, t0), we adopt

the standard approach of replacing H(t) by λH(t), where λ is a dimensionless expansion parameter.

One can imagine λ gradually being increased from 0 to 1, representing the Hamiltonian being “gradually

turned on”. When λ = 1 we have the standard solution in eq. (2.4). This suggests that we can write

U(t, t0) as a Born series in powers of λ, that is

U(t, t0) ≡∞∑n=0

λnUn(t, t0). (2.6)

We assume that this series converges asymptotically 1. Substituting eq. (2.6) into eq. (2.5) and matching

1The convergence of the (more general) Born series is discussed in page 710 of ref. [117], 7th edition. Asymptoticconvergence is discussed at length in ref. [98].


the coefficients of like powers of λ, we obtain a recursion relation for Un(t, t0):

∂U0(t, t0)

∂t= 0,

i~∂Un(t, t0)

∂t= H(t)Un−1(t, t0) n ≥ 1. (2.7)

Given that for λ→ 0 we must have U(t, t0) = I (the identity operator), then U0(t, t0) = I. Substituting

recursively into eq. (2.7) one obtains Un(t, t0) (for n ≥ 1) as integrals in time of H.

We now apply the time averaging to eq. (2.4), giving us

ρ(t) = U(t, t0)ρ(t0)U(t, t0)†. (2.8)

Denoting ρ0 ≡ ρ(t0), we see that above we have an expression for ρ in terms of ρ0. In what follows,

we seek an inverse expression, which allows us to write ρ0 in terms of ρ. We then express the time

derivative ∂ρ(t)∂t in terms of ρ0, and use our multiple expressions to get rid of ρ0 and create an equation

only involving ρ and its time derivative. This last equation will serve as the evolution equation we seek

for ρ.

On substituting from eq. (2.6) into eq. (2.8), one finds an equation of evolution for ρ(t) in terms of

Un and powers of λ. Suppressing for the moment the explicit dependence on t and t0, we have

ρ =

∞∑m,n

λm+nUmρ0U†n

=

∞∑k

λk( k∑j=0

Uk−jρ0U†j

)≡∞∑k

λkEk[ρ0]

≡ E [ρ0], (2.9)

where Ek[ρ0] are linear maps acting on ρ0, and defined as the bracketed term in the second line. The

expression E [ρ0] is the linear map that acts on ρ0 to give ρ. We also note that from the definition, E0 = I,the identity linear map.

2.2.2 The Inverse Transformation

To proceed, we would like to invert this transformation2 to find F [ρ] = E−1[ρ] that operates on ρ to give

ρ0. That is

ρ0 ≡ F [ρ]

≡∑k

λkFk[ρ], (2.10)

where we have defined Fk[ρ] as the different order terms when one expands F [ρ] in terms of powers of λ.

Naıvely one might invert the time evolution using the unitarity of U(t, t0), i.e. by swapping t and t0 one

2Depending on the averaging kernel, the transform E may not be one-to-one and the inverse may not be unique.However, deconvolution techniques can be used to find a pseudoinverse that applies for the domain of interest [95].


should obtain the inverse. However this approach is invalidated by the time averaging, since unitarity

no longer holds. Instead, we use the fact that F and E together give the identity transformation:

F[E [ρ]

]≡ I[ρ]. (2.11)

Thus expanding both F and E in terms of powers of λ, we find

∞∑m,n=0

λm+nFm[En[ρ]

]= λ0I[ρ], (2.12)

which implies∞∑k=0

λk( k∑j=0

Fj[Ek−j [ρ]

])= λ0I[ρ]. (2.13)

Comparing coefficients of powers λ, will allow us to find the relationship between the Fj superoperators

and the Ej . This is done in the appendix in eq. (A.1), and a recursion relation is derived in eq. (A.2).

From this we find that

F0 = E0 = I

F1 = −E1F2 = −E2 + E1

[E1]. (2.14)

The recursion relation in the appendix can be used to find higher order terms.

2.2.3 Approximation up to Second Order

Next, we find the effective evolution equation of ρ in terms of different orders of λ, so we can examine

the lowest order terms and see what they tell us about the Effective Hamiltonian. That is, we find

an equation that relates the time derivative ∂ρ(t)∂t to the matrix ρ(t) at the same instant time - this

is a Markovian Approximation. Differentiating eq. (2.9) with respect to time (denoted by a dot), and

substituting from eq. (2.10) we have

i~∂ρ(t)

∂t= i~E [ρ(t0)] = i~E

[F [ρ(t)]

]=

∞∑k

λk k∑j=0

i~Ej[Fk−j [ρ(t)]

]. (2.15)

This can be written as

i~∂ρ(t)

∂t=

∞∑k

λkLk[ρ(t)], (2.16)

where the map Lk is defined by the expression in the large curly braces in eq. (2.15). Equation (2.16)

is precisely the equation of evolution of ρ(t) we sought. However, we need to express the terms Lk in

explicit terms of Uj and H. To this end, we first do the same for Ek, Ek and Fk in appendix eqs. (A.3 -

A.5). Then we find, up to second order,

L0[ρ] = i~E0[F0[ρ]

]= 0, (2.17)


L1[ρ] = i~E1[F0[ρ]

]+ i~E0

[F1[ρ]

]= Hρ− ρH, (2.18)

L2[ρ] = HU1ρ−H U1ρ+HρU†1 −HρU†1

− ρU†1H + ρU†1 H − U1ρH + U1ρH. (2.19)

Equation (2.16) together with eqs. (2.17 - 2.19) form the principal result of this section. A more detailed

derivation of the above as well as third order terms are included in appendix eqs. (A.7) and (A.8).

A certain ambiguity of notation in eq. (2.19) must be explained. Terms where the averaging overline

passes over a ρ in the middle, such as the term U1ρH, are meant to have the average apply to the U1

and the H after the matrix/operator multiplication, not to the ρ. More explicitly

U1ρH =

∫ ∞−∞

f(t− t′)U1(t′, t0)ρ(t)H(t′)dt′.

So the time parameter t′ being averaged is only present in U1 and H, not ρ.

To interpret the expressions above, we note that the L1 term is just the commutator term in the

von Neumann equation 2.3, with the averaged Hamiltonian replacing the instantaneous one. This is

expected, since intuitively first order effect of time averaging would be to average the Hamiltonian.

Note that the total expression for all Lk is anti-Hermitian, as we would expect, since the LHS of eq.

(2.16) is an imaginary number multiplied by a Hermitian operator, making it anti-Hermitian. We also

take notice of a familiar pattern occurring in second and higher order terms, of “average of the product

minus product of the averages”, in a manner reminiscent of the definition of statistical covariances.

These difference terms serve as corrections to the first order approximation.

2.2.4 The Effective Hamiltonian

Using eqs. (2.16 - 2.19) in the previous section, and setting the order parameter λ to unity, we have the

following dynamical equation, up to second order,

i~∂ρ

∂t= [H, ρ] +Aρ− ρA† +D2[ρ], (2.20)

where A is defined as

A ≡ HU1 −H U1, (2.21)

A† = −U1H + U1H, (2.22)

and D2[ρ] contains the second order decoherence terms (i.e. all the terms in L2 where ρ is “sandwiched”

between two operators) given by

D2[ρ] ≡ HρU1 + U1ρH −HρU1 − U1ρH. (2.23)


In the above, we have made use of the fact that U†1 = −U1. Turning to the terms involving A in eq.

(2.20), we express them as the sum of a commutator and an anticommutator:

Aρ− ρA† =[A+A†

2, ρ]

+A−A†

2, ρ. (2.24)

The first argument of the commutator is the Hermitian part of A, and the first argument of the anti-

commutator is the anti-Hermitian part of A. This is expected for the Hermiticy to be consistent in both

sides of the equation. Finally, we can write our general dynamical equation in its most insightful form:

i~∂ρ

∂t=[Heff , ρ

]+A−A†

2, ρ

+D2[ρ], (2.25)

where Heff , is the Effective Hamiltonian, given by

Heff = H +1

2(A+A†). (2.26)

We can interpret eq. (2.25) as a type of Lindblad equation [25, 103, 105, 106], with a commutator term

that leads to unitary evolution, and other terms that lead to decoherence.

The expression for the Effective Hamiltonian in eq. (2.26) confirms previous results by James and

Jerke in ref. [23]. But the derivation in this reference simply discarded anti-Hermitian terms to satisfy the

Hermiticity requirement for the Effective Hamiltonian. The derivation provided above, however, treats

this problem with more rigor, and arrives at the detailed solution including the decoherence component

in Lindblad form.

It turns out the decoherence terms are in general important, and yield an evolution closer to the

exact case. Thus eq. (2.25) is the more accurate expression which yields interesting results.

The derivation in this section relied on the Schrodinger picture, however it is valid in the Interaction

picture as well. In the latter picture however, the interpretation of the density matrix changes somewhat.

2.3 Harmonic Time Dependent Hamiltonian

We will now apply our general result to a class of harmonic Hamiltonians of the form

H(t) = H0 +

N∑n=1

hnexp(−iωnt) + h†nexp(iωnt), (2.27)

where H0 is independent of time. This is representative of a wide class of problems, particularly in the

interaction picture Hamiltonian. In this case, the first term in the expansion for U1 is

U1(t) =(t− t0)

i~H0 + V1(t)− V1(t0), (2.28)

where

V1(t) =

N∑n=1

1

~ωn

hnexp(−iωnt)− h†nexp(iωnt)

. (2.29)

Let us assume the averaging kernel f(·) is an ideal low pass filter in frequency space, and an even

function in time, and therefore has a real Fourier transform. We further assume that the frequencies


ωn are sufficiently high that they are filtered out, but sufficiently close to each other so that terms

oscillating at difference frequencies (ωn − ωm) will pass the filter unchanged. Thus, we can make the

following assumptions:

exp±iωnt = 0,

exp±i(ωn + ωm)t = 0,

exp±i(ωn − ωm)t = exp±i(ωn − ωm)t. (2.30)

This implies that

H(t) = H0, V1(t) = 0,

U1(t) =(t− t0)

i~H0 − V1(t0). (2.31)

Using the Harmonic Hamiltonian eq. (2.27) along with eq. (2.30) and our results in eq. (2.25) and eq.

(2.23), we obtain the expression

i~∂ρ

∂t=[H0, ρ

]+

N∑n,m

[(h†mhn~ωn

− hnh†m

~ωm)ρ− ρ

(h†mhn~ωm

− hnh†m

~ωn)

+h†mρhn + hnρh

†m

~ωm− h†mρhn + hnρh

†m

~ωn

]ei(ωm−ωn)t

=[Heff , ρ

]+ N∑n,m

1

~ω−nmh†m, hnei(ωm−ωn)t, ρ

−N∑n,m

2h†mρhn + hnρh

†m

~ω−nmei(ωm−ωn)t, (2.32)

where1

ω±nm=

1

2

( 1

ωn± 1

ωm

), (2.33)

and

Heff = H0 +

N∑n,m

1

~ω+nm

[h†m, hn]ei(ωm−ωn)t. (2.34)

We can then formulate eq. (2.32) in a form very similar to the Lindblad Master equation:

i~∂ρ

∂t=[Heff , ρ

]+

N∑n,m

1

~ω−nm

(L†mLn, ρ − 2LnρL

†m + LnL†m, ρ − 2L†mρLn

), (2.35)

where

Lm = hme−iωmt. (2.36)

Despite the similarity, the above is not exactly a Lindblad master equation since it violates positivity.

To see this, we divide both sides of eq. (2.35) by the imaginary factor i, and treat the expression i/ω−nm as

the entry in the (n,m) position in a square matrix, and find that this matrix is imaginary and Hermitian.

Its nonzero eigenvalues will come in pairs that differ by a sign, ±λ1,±λ2, ... etc. The negative eigenvalues

imply the evolution equation above does not guarantee positivity of evolution, which is a requirement


for Lindblad evolution. This is a common problem in the theory of open quantum systems that arises

from the application of a Markovian approximation to a non-Markovian system [97]. However, usually

the decoherence terms are small in magnitude and are not large enough to destroy the positivity of the

density matrix as it evolves. There are also methods to measure the Markovianity of the system to judge

the validity of the approximation [96].

The issue above with positivity can also be seen as the consequence of an averaging kernel that is not

always positive. The filtering we assumed in eq. (2.30) is equivalent to multiplication by a rect function

in the frequency domain, which in turn is equivalent to convolution with a sinc function in the time

domain. That is, for our example, the time averaging kernel f(·) defined in eq. (2.1) is a sinc function,

which is not always positive, meaning ρ is not guaranteed to be a positive matrix.

Despite this difficulty, eq. (2.35) still preserves Hermiticity and trace, which can be checked in a

straight forward manner. Also note that the Lindblad operators Lm are time dependent in this case.

The Effective Hamiltonian obtained here is exactly the same as the one in ref. [23]. In addition,

we have found the expression for the decoherence terms. If we have only one frequency present in

the Hamiltonian, the decoherence terms vanish since 1/ω−mm = 0, and the Effective Hamiltonian alone

perfectly describes the evolution to second order. In more general cases, the decoherence terms must be

considered.

The appearance of decoherence Lindblad-like terms above suggests the average system may be

thought of as an open system, even though the underlying “real system” is closed, and evolves uni-

tarily. A heuristic argument for this interpretation follows; when one averages quantities, one is of

course throwing away information, and thus increasing the entropy of the system. For example, a low-

pass frequency filtering removes information about high-frequency processes. Naturally, this increasing

entropy will lead to decoherence terms in the evolution equations for the average.

One point we have neglected so far is the truncation of our series at second order. We can obtain a

sufficiency condition for the legitimacy of this approximation simply by considering the ratio of higher

order terms of the expansion. We accomplish this by assuming that ηωn

<< 1 ∀n, where η is the largest

eigenvalue of H. This ensures higher order terms are progressively smaller, and thus may be safely

discarded.

2.4 Examples

In this section, we revisit two examples cited in ref. [23], namely the AC Stark Shift and three level

Raman Transitions. Note that for both examples, we work in the interaction picture [104], starting with

the interaction Hamiltonian. Inclusion of the decoherence terms given in eq. (2.35) will illustrate their

importance and elucidate the validity of the Effective Hamiltonian model.

2.4.1 AC Stark Shift

We begin with an example of the AC Stark shift, also known as the Autler-Townes effect [10]. The

idea behind this effect is that an applied AC field will shift the energy levels of the atom. Similar shifts

have been observed in the Zeeman effect when radio-frequency fields are applied perpendicular to the

magnetic field, which changes the effective Lande g-factor [92].

Consider a two level atom interacting with an external harmonic force. If we use the interaction


picture, the Interaction Hamiltonian is given by

HAC(t) =~Ω

2|2〉〈1| exp(−i∆t) + |1〉〈2| exp(i∆t), (2.37)

where Ω is the Rabi frequency (i.e. the frequency of oscillation between the levels of the atom) and ∆ is

the detuning, (the difference between the harmonic force / laser frequency and the transition frequency

between the two levels of the atom). It is known that the AC Stark Shift results in an effective shift of

Ω2/∆ in the resonance frequency between the two levels.

We use eq. (2.34) to evaluate the Effective Hamiltonian, Heff , for this AC Stark shift system. We

find

Heff = −~Ω2

4∆(|2〉〈2| − |1〉〈1|). (2.38)

The leading coefficient is precisely the expected shift. Applying eq. (2.32) to HAC(t) we find that all

decoherence terms vanish, since there is only one harmonic operator, h1 = ~Ω2 |2〉〈1|, and therefore 1/ω−11

vanishes by definition. So in this case, the Effective Hamiltonian is time independent.

To show the Effective Hamiltonian Theory in action, we compare both the exact and time-averaged

evolution of the density matrix for a specific numerical example. The exact evolution is given by

i~∂ρ(t)

∂t= [HAC(t), ρ(t)], (2.39)

and time averaged evolution via the Effective Hamiltonian follows

i~∂ρ(t)

∂t= [Heff , ρ(t)]. (2.40)

−50 0 50 100 150 200−0.5

0

0.5

Time

Rea

l par

t of C

oher

ence

Time Evolution of Coherence, b = 0.3

Figure 2.1: The evolution of the real part of the off diagonal entry for the density matrix; i.e. 12 〈σx〉.

Interaction strength set at b = 0.3. Time in units of the characteristic time 1∆ . Both axes dimensionless.

Since the relative phase of the exact and averaged evolution just depends on initial conditions and thezero of time, initial conditions were artificially modified in the figure so the two curves start in phase.

We define the dimensionless ratio b ≡ Ω∆ , which represents the strength of the applied AC field, and

measure time in units of the characteristic time 1∆ . Then we start with an arbitrary density matrix,

and plot the real part of the off diagonals - i.e. the real part of the coherences between the populations


in the two levels, which is also equal to half the expectation value of the Pauli σx operator, i.e. 12 〈σx〉.

Figure 2.1 shows the plot for b = 0.3.

From the close correspondence between the two curves, it is clear this method is an effective first

order approximation to the frequency and amplitude of oscillation. As expected, the time averaged

evolution viz the Effective Hamiltonian resembles the exact evolution without the superimposed high

frequency components.

However, Observing the two downward sloping sections, we note that the blue curve and the black

curve do not exactly have the same major period, and will clearly go out of phase in a few cycles. This is

because the interaction strength value of b = 0.3 we chose is in the borderline regime where the Effective

Hamiltonian starts to lose its accuracy. For smaller values of b, the two curves are even closer, and for

larger values they diverge more rapidly.

So we see that Effective Hamiltonian theory yields some insight in the example of the AC Stark Shift.

2.4.2 Three Level Raman Transitions

Suppose we have a three level system, interacting through two time-harmonic terms, as in fig. (2.2). For

example, imagine we have two lasers interacting with a three level atom. We start with the Interaction

Hamiltonian given by

H(t) =~Ω1

2|3〉〈1|e−iω1t +

~Ω2

2|3〉〈2|e−iω2t + h.a., (2.41)

where h.a. represents the Hermitian adjoint of previous terms. Applying our formula eq. (2.34), we once

Figure 2.2: Illustration of the three level Raman Transitions

again get the Effective Hamiltonian

Heff =− ~Ω21

4ω1

(|3〉〈3| − |1〉〈1|

)− ~Ω2

2

4ω2

(|3〉〈3| − |2〉〈2|

)+

~Ω1Ω2

4ω+12

(|1〉〈2|ei(ω1−ω2)t − |2〉〈1|e−i(ω1−ω2)t

). (2.42)

The first two terms represent the AC Stark Shift associated with the two lasers. The final term represents

the effective population transfer between level |1〉 and |2〉, even though the original Hamiltonian had no

direct interaction between these two levels. Interestingly, the Effective Hamiltonian contains no direct

interaction between level |3〉 and the other two levels, despite the presence of such interaction in the

original Hamiltonian. Applying eq. (2.32), the evolution of the density matrix then follows

i~∂ρ

∂t=[Heff , ρ

]+

~Ω1Ω2

4ω−12

[(|2〉〈1|, ρ − 2ρ12|3〉〈3| − 2ρ33|2〉〈1|

)ei(ω1−ω2)t

−(|1〉〈2|, ρ − 2ρ21|3〉〈3| − 2ρ33|1〉〈2|

)e−i(ω1−ω2)t

], (2.43)


where

ρij ≡ 〈i|ρ|j〉. (2.44)

We already assumed that ω1 and ω2 are close in value, which means 1ω−

12

will be small compared to Heff ,

making the decoherence terms relatively small as expected. We now proceed to solve this problem in

some detail by decomposing the density matrix into a generalized Bloch vector. First we write ρ as

ρ =1

3I +

1√3

8∑i=1

riGi, (2.45)

where I is the identity matrix, ri are the components of the generalized Bloch vector, and the Gi are

the Gell-Mann matrices [89], a three dimensional analogue of the Pauli matrices shown in appendix B.1.

We can also write the Gell-Mann matrices in Dirac notation:

G1 = |1〉〈2|+ |2〉〈1| ≡ X, G2 =− i(|1〉〈2| − |2〉〈1|) ≡ Y,

G3 = |1〉〈1| − |2〉〈2| ≡ Z, G4 =|1〉〈3|+ |3〉〈1|,

G5 = −i(|1〉〈3| − |2〉〈3|), G6 =|2〉〈3|+ |3〉〈2|,

G7 = −i(|2〉〈3| − |3〉〈2|), G8 =1√3

(|1〉〈1|+ |2〉〈2| − 2|3〉〈3|) ≡W. (2.46)

The operators X,Y , and Z are the standard Pauli operators for the |1〉, |2〉 subsystem, and W is the

last Gell-Mann matrix. Substituting eq. (2.45) in eq. (2.43), it simplifies to

∂ρ

∂t=

Ω1Ω2

2ω+12

[Z(r1 sin θ + r2 sin θ)− r3

(Y cos θ +X sin θ

)]+

1

4

(Ω21

ω1− Ω2

2

ω2

)(r1Y − r2X

)−√

3Ω1Ω2

2ω−12

[r8

(X sin θ + Y cos θ

)+W

(r1 sin θ + r2 cos θ

)]+ g(...), (2.47)

where

θ ≡ (ω1 − ω2)t, (2.48)

and g is some linear functional that only depends on ri and Gi for i = 4, 5, 6, 7. This implies that the

coherences between level 3 and the other two levels (i.e. the (1,3), (2,3), (3,1), and (3,2) entries of the

density matrix) form a closed subsystem, and only affect each other, evolving independently of the rest of

the density matrix. Keeping this in mind, we ignore this subsystem entirely, and focus on the evolution

of r1, r2, r3, r8, which together form a four dimensional vector. From eq. (2.47), we see that this vector

follows the evolution of the following dynamical system

∂r

∂t= Ar, r =

r1

r2

r3

r8

, (2.49)


where

A =

0 −α −β sin θ −γ sin θ

α 0 −β cos θ −γ cos θ

β sin θ β cos θ 0 0

−γ sin θ −γ cos θ 0 0

. (2.50)

The constants, α, β, γ are given by

α ≡ 1

4

(Ω21

ω1− Ω2

2

ω2

), (2.51)

β ≡ Ω1Ω2

2ω+12

, (2.52)

γ ≡√

3Ω1Ω2

2ω−12

. (2.53)

As we shall see, we can view α and β effectively as driving terms, where γ represents reduction in

frequency due to decoherence. To simplify the matrix A above, we go to the co-rotating frame, and

define

r ≡Mθr, (2.54)

where

Mθ =

cos θ − sin θ 0 0

sin θ cos θ 0 0

0 0 1 0

0 0 0 1

. (2.55)

We can then define the vector d as the first three components of r, and the scalar q as its fourth

component. The vector d can be thought of as the co-rotating Bloch vector of the |1〉, |2〉 subsystem

alone. We also define new vectors γ and Ω as

d ≡

r1

r2

r3

, γ ≡

0

γ

0

, Ω ≡

β

0

α+ ω1 − ω2

. (2.56)

Our equations then reduce to simple vector equations:

d = Ω× d− qγ,

q = −γ · d. (2.57)

This presentation intuitively describes the evolution of the system as the rotation of the Bloch vector

about Ω through the term given by the cross product (as is already well known with the Rabi oscillation

[8]) plus a perturbation given by the terms in γ. Now to solve this, we define the unit vectors eΩ and

eγ as the orthogonal unit vectors along Ω and γ respectively. Together with ep ≡ eΩ×eγ , they give an


orthonormal basis for the three dimensional vector space. Using this basis, we write

d = dΩeΩ + dγeγ + dpep,

Ω = ΩeΩ,

γ = γeγ , (2.58)

substituting this ansatz for d in eq. (2.57) and resolving the equation along each unit vector, we get a

system of coupled differential equations:

dΩ = 0

dγ = −Ωdp − γq

dp = Ωdγ

q = −γdγ , (2.59)

from which we get

dγ = −(Ω2 − γ2)dγ . (2.60)

Solving this simple second order ordinary differential equation, we have

dγ = R cosωt,

dp = RΩ

ωsinωt− γ

Ωq0,

q = −Rγω

sinωt+ q0, (2.61)

where R is some constant oscillation amplitude, and q0 is a constant, both dependent on the initial

conditions. The frequency of oscillation ω given by

ω2 = Ω2 − γ2

= (α+ ω1 − ω2)2 + β2 − γ2. (2.62)

Note that we set the zero of time such that there is no phase angle in the argument of the trigonometric

terms. Then time evolution of the vector d is given by

d(t) = dΩeΩ −γ

Ωqep +R

(cosωteγ +

Ω

ωsinωtep

). (2.63)

Firstly note that when γ vanishes, we have ω = Ω, and d oscillates at the Rabi frequency. When q0 = 0,

this oscillation is around the precession vector Ω, as in the well known result [8]. As γ increases, it

reduces the frequency of precession of d, and changes the oscillation path from a circular one to an

elliptical one. As q0 increases, the centre of this oscillation shifts in the ep direction (perpendicular to

both Ω and γ.

The elliptical nature of the oscillation means the length of the vector is not constant, rather it

oscillates at frequency ω. Note that the length squared of d corresponds to the trace of the square of

the underlying density matrix - i.e. a measure of its purity as discussed in more detail in chapter 4. Let


us define l2 as

l2 ≡ d2Ω + d2

γ + d2p. (2.64)

Taking its time derivative and using eq. (2.59), we find

d(l2)

dt= 2(dΩdΩ + dγ dγ + dpdp)

= −2γqdγ

=γ2

ωR2 sin 2ωt− 2γRq0 cosωt. (2.65)

So in general, the length of the Bloch vector (i.e. the purity of the effective 2 level system) oscillates at

frequency ω, and if q0 vanishes, it oscillates at 2ω. This surprising oscillation in purity may be regarded

once again as a consequence of applying the Markovian approximation to a non-Markovian system. It is

interesting to investigate whether this oscillation in effective purity is testable experimentally to verify

the Effective Hamiltonian theory, as well as the validity of the decoherence terms.

2.5 Summary

We have demonstrated a more rigorous derivation of Effective Hamiltonian Theory and its domain of

applicability. Additionally, it has been shown that additional terms are created resembling Lindblad

evolution for harmonic Hamiltonians - implying that the averaging process introduced a small decoher-

ence factor. Applying this theory to examples such as the AC Stark Shift and Raman Transitions, we

find it introduces some minor corrections.

In the future, applying this theory to other known systems and testing its limitations would inform

us about its usefulness as a tool. In particular, applying it to systems where the newfound decoherence

terms play a larger role would help us better understand their exact role. Additionally, it would be useful

to come up with a general interpretation of the decoherence terms. For example, the averaging process

over a given system can be seen as observing an analogous system-reservoir pair for some hypothetical

reservoir, and then observing the system alone while “tracing out” the reservoir.

Chapter 3

Completely Positive Maps and

Polarization

3.1 Introduction

The polarization state of a classical beam of light can be mathematically represented in a variety of ways.

The first description dates back over a century and a half to the work of Stokes [26], who introduced

the four parameters which bear his name to specify the beam’s polarization state [27]. The Stokes

parameters Sµ, represent, in vector form, the power of the beam in various polarization modes. In the

simple case of a fully polarized monochromatic deterministic beam, a Jones vector is commonly used to

represent the state [28].

Further, powerful models that treat the electromagnetic field stochastically have been developed.

Formalisms by Wiener [29] and Wolf [30, 31, 123] allowed for probabilistic electric fields and the treatment

of statistical optics via the polarization matrix, Φ, formerly known as the coherency matrix [113]. The

Jones vector, or rather the projection operator formed by taking the outer product of the Jones vector

with itself, can be seen as a special case of the polarization matrix. Therefore, we have two rival

mathematical objects that describe classical polarization of a light beam, the Stokes vector Sµ, and the

polarization matrix, Φ.

When an electromagnetic beam passes through an optical system, its state of polarization will, in

general, be transformed. If one uses the Stokes vector to describe the state, then the transformation is

represented via a Mueller matrix [34]. If one represents the state through the polarization matrix, then

transformation is represented via a Jones matrix [28].

It is generally assumed on physical grounds that the most general transformation on the state, in

the polarization matrix formalism, can be represented as an ensemble of Jones matrix transformations

[35], and based on this, the properties of the most general Mueller matrix can be derived [36, 37, 38]

for the Stokes formalism. The proof of this assumption based on rigorous mathematical properties of

the two formalisms has been lacking. Recently, Simon et al. [40] have addressed this problem based on

properties of matrices derived from the Mueller matrices.

In this chapter, we address this problem differently. In section 3.2 we review the concept of a complex

analytic signal, the properties of which will be integral to our argument. We then review the polarization

formalisms above in some detail in sections 3.3 and 3.4. In section 3.5 we show that provided only linear

20

Chapter 3. Completely Positive Maps and Polarization 21

optical effects are allowed, then the most general transformation is indeed an ensemble of Jones matrix

transformations.

We base our main argument on basic properties of positive maps in two dimensions [41, 42] and

simple physical assumptions about the state. The theorem on positive maps we use is similar to Choi’s

theorem for completely positive maps [43] which is popular in quantum information theory [51].

If one however allows nonlinear optical effects, particularly phase conjugation, then a more general

transform is needed. We show the form of this alternative transformation. This chapter is based on ref.

[44] by the author.

3.2 Complex Analytic Signals

3.2.1 Definition and Properties

We begin by reviewing the concept of a complex analytic signal, which is fundamental to our argument.

In what follows, our main reference is Mandel and Wolf, sec. 3.1 [9, 31]. Suppose we have a real-valued

signal x(t), which can be expressed via the Fourier synthesis integral:

x(t) =

∫ ∞−∞

x(ω)e−iωtdω. (3.1)

Since the signal x(t) is real, the Fourier spectrum x(ω) must satisfy x(−ω) = x∗(ω), where the star

denotes the complex conjugate.

We note that the negative frequency components of the spectrum are fully determined by the positive

frequency components. Therefore one can discard the former without loss of information. Thus, we define

the complex analytic signal z(t) as the signal synthesized only from the positive frequency components

of x(ω). That is

z(t) ≡∫ ∞

0

x(ω)e−iωtdω. (3.2)

Alternatively, one may write the full Fourier synthesis equation for z(t) as

z(t) ≡∫ ∞−∞

z(ω)e−iωtdω, (3.3)

where z(ω) ≡ θ(ω)x(ω), and θ(ω) is the Heaviside step function. From the above, it can be shown that

x(t) = 2<[z(t)], (3.4)

where < denotes the real part (and = the imaginary part) of a complex value. For example, for the

simple monochromatic signal x(t) = cos(Ωt), we have x(ω) = 12 [δ(ω+ Ω) + δ(ω−Ω)], z(ω) = 1

2δ(ω−Ω),

and z(t) = 12e−iΩt.

We define y(t) as twice the imaginary part of z(t), that is

y(t) ≡ 2=[z(t)]. (3.5)


Then we can write the complex analytic signal as

z(t) =1

2[x(t) + iy(t)]. (3.6)

One can show that the analytic property of z(t) implies its real and imaginary parts above together

form a Hilbert transform pair [111]:

y(t) = H[x(t)] ≡ 1

πP

∫ ∞−∞

x(t′)

t− t′dt′,

x(t) = −H[y(t)] = − 1

πP

∫ ∞−∞

y(t′)

t− t′dt′, (3.7)

where P denotes the Cauchy principal value, and H is the Hilbert transform defined above. Note that

the negative of a Hilbert transform is also its inverse transform. Considered as a function of complex t,

the analytic signal z(t) is analytic in the lower half of the complex t plane.

The relations in eq. (3.7) are identical, up to a sign, to the Kramers-Kronig relations [109, 110], which

relate the real and imaginary parts of the Fourier transform of a causal response function [108]. The

difference between the case of a complex analytic signal and a causal response function is that the time

and frequency domains have switched roles. In the case of the complex analytic signal, the frequency

domain vanishes for negative arguments, and the time domain obeys the relation in eq. (3.7), whereas

for a causal response function, the time domain vanishes for negative arguments (equivalent to causality)

and the frequency domain obeys the Kramers-Kronig relations.

3.2.2 Non-linearity of Complex Conjugation

Complex analytic signals make dealing with signals more convenient and streamline the mathematics, for

the same reasons one prefers complex exponentials to trigonometric functions. For example, modulation,

phase relationships and derivative properties become easier to deal with. Even though the real part of the

signal is what sets the electric field, the complex part is not simply a convenience, it plays an important

role in determining relative phases, and cross-correlation functions between multiple signals. Changing

the imaginary component of the signal has observable effects.

To illustrate this, suppose we have two complex analytic signals given by zj(t) = 12 [xj(t) + iyj(t)],

(i=1,2), that represent two wide-sense stationary stochastic processes with zero mean. The cross-

correlation function between them is defined as

Γ12(τ) ≡ 〈z∗1(t)z2(t+ τ)〉, (3.8)

where 〈·〉 denotes the ensemble average over different possible realizations.

One can show that Γ12(τ) is itself a complex analytic signal. Note that since the generic complex an-

alytic signal z1(t) contains only positive frequencies, its conjugate z∗1(t) only contains negative frequency

components.

Suppose we have a conjugation device that transforms the stochastic process z1(t) to its complex


conjugate, that is

z1(t)→ z1(t) ≡ z∗1(t)

= x1(t)− iy1(t). (3.9)

We use notation z1(t) for the transformed conjugate function to emphasize that it is just another

complex function for which we can define correlations and a conjugate. We then define a secondary

cross-correlation function:

Γ12(τ) ≡ 〈z∗1(t)z2(t+ τ)〉, (3.10)

= 〈z1(t)z2(t+ τ)〉. (3.11)

While the cross-correlation function Γ12(τ) seems innocuous, using the generalized Wiener-Khintchine

theorem [9, 29, 107], it can actually be shown to be identically zero. The technical reason for this being

that there is no overlap between the spectra of z1(t) and z2(t). That is, they do not share any nonzero

frequencies in their Fourier spectrum; the former only has negative frequencies and the latter only has

positive frequencies.

The same conjugation device will then transform the cross-correlation function as

Γ12(τ)→ Γ12(τ) = 0. (3.12)

So our hypothetical conjugation device would cause the cross-correlation function to always vanish.

Given that the cross-correlation function is linear in z1(t), it seems the conjugation device cannot be

linear in the physical sense. If it were linear, it would only send the zero functions to zero, or else it

would be a trivial device that sends all functions to zero. Since neither is the case, we must conclude it

is not a linear device.

Indeed, known experimental techniques that conjugate phase, such as phase conjugate mirrors [112,

119] are manifestly nonlinear in nature. So if we are restricting ourselves to linear devices, then the

conjugation operation will not be allowed.

Also note that if the operation L acting on electric fields is linear, then by definition it must satisfy

L[αz1(t) + βz2(t)] = αL[z1(t)] + βL[z2(t)], (3.13)

where α and β are arbitrary coefficients. However, the conjugation operation transforms the argument

αz1(t) + βz2(t) to

[αz1(t) + βz2(t)]∗ = α∗z∗1(t) + β∗z∗2(t). (3.14)

Comparing the last two equations, we see that the conjugation operation is not linear (in the sense of

eq. (3.13)). It only satisfies the linearity property if we restrict ourselves to real coefficients α and β.

For example, doubling the input will double the output, but if we have complex coefficients, we see that

the complex conjugation operation is not linear.

Alternatively, one may argue the real and imaginary parts of a signal must form a Hilbert transform


pair for it to be a complex analytic signal. By assumption, z1(t) is a complex analytic signal, and so

y1(t) = H[x1(t)],

x1(t) = −H[y1(t)]. (3.15)

Applying the conjugation operation to z1(t) means flipping the sign of y1(t). However, the conjugated

z1(t) does not describe a complex analytic signal because its imaginary part is not the Hilbert transform

of its real part (there is a sign disparity due to the conjugation), and therefore is not admissible. In other

words, complex conjugation in this context is an unphysical operation, inadmissible by the underlying

formalism of classical stochastic linear optics.

3.3 Classical Polarization States

Consider a classical beam of light propagating in the z direction. The complex electric field values in

the x and y direction are taken to be probabilistic ensembles given by complex analytic signals E1(r, t)

and E2(r, t) respectively, where r is the position vector.

The polarization state of the beam of light is given by the 2×2 polarization matrix Φ(r, t), defined

as

Φij = 〈EiE∗j 〉, i = 1, 2. (3.16)

where position and time dependence have been suppressed. If one thinks of E1 and E2 as random

variables, then Φ is their variance-covariance matrix.

Alternatively, the four element Stokes vector S can be used to represent the polarization state [26].

It is related to Φ by

Sµ = Tr[Φσµ] = Φijσµji, (3.17)

Φ =1

2Sµσ

µ, (3.18)

where σ0 is the identity matrix, and σ1, σ2, and σ3 are the three Pauli matrices σz, σx, and σy respec-

tively. Einstein summation notation has been used, i.e. repeated indices are summed over. Lowercase

Latin letters run from 1 to 2 (corresponding to the two Cartesian components of the transverse field),

while lowercase Greek letters run from 0 to 3. The polarization matrix or Stokes vector contain all the

physical information about the polarization state of the beam [27], and are different ways of mathemat-

ically representing the same information.

3.4 Filters

3.4.1 Simple Filters

When the beam interacts with a linear optical element, generically called a filter, its polarization state

is transformed. The most basic type of filter, which we call a simple filter, linearly transforms the

transverse electric field vector via the well-known Jones matrix [28], denoted T , through simple matrix

multiplication:

E′i = TijEj . (3.19)


This is equivalent to the following unitary transformation on the polarization matrix:

Φ′ = TΦT †. (3.20)

Turning to the Stokes vector, any linear transformation on the state must have the form

S′µ = MµνSν , (3.21)

where Mµν is the 4×4 Mueller matrix [34]. Equations (3.17), (3.18) and (3.20) then imply that the

Mueller matrix corresponding to a simple filter can be expressed as [27, 35, 37]

Mµν =1

2Tr[σµTσνT †] (3.22)

= A(T ⊗ T ∗)A−1

=1

2A(T ⊗ T ∗)A†, (3.23)

where ⊗ is the tensor product, and A is the matrix whose rows are the vectorization of the Identity and

Pauli matrices, given by

A =

1 0 0 1

1 0 0 −1

0 1 1 0

0 i −i 0

. (3.24)

Mueller matrices of this kind are called pure Mueller matrices, or Mueller-Jones matrices [37].

Natural questions that arise at this point are how does one represent the most general type of optical

filter and what are the relevant Jones and Mueller matrices?

3.4.2 General Filters

It has been a generally held axiom, motivated by physical intuition, that the most general kind of optical

filter is an ensemble of simple filters [35, 36, 37]. Therefore, the transformation on the polarization matrix

is then represented by an ensemble of Jones matrices, rather than a single one, through the expression

Φ′ =∑e

peTeΦT†e , (3.25)

where e is the index over the elements of the ensemble, and pe is the probability of realizing the eth

ensemble element (∑e pe = 1).

From eq. (3.22) we find that the Mueller matrix for a general filter is given by

Mµν =∑e

pe1

2Tr[σµTeσ

νT †e ] (3.26)

=∑e

peM(e)µν , (3.27)

where the M(e)µν are pure Mueller matrices, each derived from a single Jones matrix in the ensemble. This

expression tells us that any physically admissible Mueller matrix is given by a convex linear combination


of pure Mueller matrices. That is, the set of all Mueller matrices is the convex hull [114] of pure Mueller

matrices.

At first glance, eq. (3.21) seems to suggest that all matrices Mµν that map the set of physical Stokes

vectors into itself are physical Mueller matrices. This condition of mapping Stokes vectors to Stokes

vectors turns out to be a necessary but not sufficient condition for a physical Mueller matrix [36, 37, 39].

Another necessary condition has to do with the positivity of the polarization matrix. References [39, 40]

show this through the use of matrices derived by rearranging entries of the Mueller matrix, a basis of 16

unitary matrices, and the linear independence of components of the electric field vector. They also make

use of the beam-coherence-polarization matrix [118], which measures coherence between two different

spatial points.

In the next section, we derive eq. (3.25) in a much simpler manner from basic mathematical properties,

making use of an interesting theorem on positive linear maps in C*-algebras [41, 42].

3.5 Positive Linear Map Approach

3.5.1 Axioms

Let F(Φ) denote the operation of a general linear filter on the polarization matrix Φ. Our goal is to

find the form of F(Φ) based on a few intuitive axioms. We assume F(Φ) satisfies the following simple

axioms:

1. It is linear in Φ. That is, for any set of coefficients ci, we have F(∑

i ciΦi

)=∑i ciF(Φi).

2. It is positive. That is, Φ′ = F(Φ) is a positive 2×2 matrix for any positive 2×2 matrix Φ.

3. It is composed of linear operations in the electric field. That is, F must only use operations that

satisfy eq. (3.13).

The validity of the first axiom is based on the superposition principle, assuming of course the different

polarization matrices Φi correspond to independent electric fields (i.e. the electric fields underlying the

polarization matrices being added are mutually uncorrelated) [117]. The second axiom holds true since

the output of F is a physical polarization matrix, and therefore must be positive. The third axiom relies

on the fact that we are only allowing linear optical devices, which ultimately must act in a linear fashion

on the electric field.

We can then make the following proposition:

Proposition. The simplest expression for F(Φ) does not include the conjugate of the input matrix, Φ∗.

In other words, F is holomorphic.

To justify this proposition, we note from the definition of Φ in eq. (3.16) that applying the complex

conjugation operator to Φ is equivalent to applying it to the electric fields. That is, it is equivalent to

Ej → E∗j , j = 1, 2. The electric field of a beam at a given point must be given by the real component of

the complex analytic electric field, expressed as

Ej(t) =1

2[E

(r)j (t) + iE

(i)j (t)], j = 1, 2. (3.28)


Based on our earlier discussion on the nonlinearity of the phase conjugation operation when applied to

complex analytic signals in section 3.2.2, we see that indeed the conjugation operation in the expression

Φ∗ would be nonlinear. Using the third axiom of linearity in the electric field, we see that our proposition

is justified.

3.5.2 Derivation

To derive the form of F , we take the first two axioms. Together, they state that F is a positive linear

map that maps the space of 2 × 2 matrices onto itself. In the theory of C* algebras, the most general

form for such a map is well known [41, 42]. It is given by

F(Φ) =∑i

ViΦV†i +

∑j

WjΦtW †j , (3.29)

where Φt is the transpose of Φ, and Vi, Wj are arbitrary matrix operators of suitable dimension. Recall

that Φ is Hermitian as can be seen from eq. (3.16). Therefore its transpose is equal to its complex

conjugate (Φt = Φ∗). One must mention here that eq. (3.29) is for a positive map, not the more

commonly used completely positive map. There is a subtle difference between the two; a positive map

transforms a positive matrix to a positive matrix, whereas a completely positive map has the additional

requirement that it be positive even when tensored with an identity operation of any dimension [43].

The transpose is the most well known example of an operation that is positive, but not completely

positive [51], hence there is no surprise it appears in eq. (3.29). Note that eq. (3.29) only holds for

dimensionality 2× 2. No analogous expression is known for higher dimensional positive maps [41, 115].

Applying our proposition, which is based on the third axiom, to eq. (3.29), we see that the term

involving the transpose must vanish. This implies that the Wj can no longer be arbitrary, and we must

set Wj = 0 ∀j. Then we have the final form of the most general physical transformation upon the

polarization matrix Φ as

F(Φ) =∑i

ViΦV†i . (3.30)

Interestingly, the expression in eq. (3.30) describes the general form of a completely positive map. How-

ever, we derived it in this particular circumstance without making use of Choi’s well known theorem on

completely positive maps [43], but through constraints of physical linearity applied to the less restrictive

positive map. Dropping the transpose term in the process, we are left with a completely positive map

after all. Note that in the field of quantum information, quantum channels are completely positive trans-

formations that act on the density matrix [51], where the operators Vi are the Kraus operators [101].

This is a foreshadowing of the well known analogies between the quantum and classical cases, addressed

in the following chapter.

A potential point of confusion must be clarified here. We mentioned that we discarded the conjugation

operation (and by extension the transpose) because it is not linear. Yet eq. (3.29) is the expression for a

positive linear transformation. The confusion is resolved when we note that the conjugation operation

was discarded because it is not linear in the electric field E, whereas eq. (3.29) is linear in the polarization

matrix Φ (and not the electric field). So it is linearity in two different senses.

To recap, we have shown that eq. (3.30) describes the most general operation upon a polarization

matrix Φ in the context of linear optics. Comparing eq. (3.30) with eq. (3.25) and setting Vi =√piTi,

we see that, as expected, F(Φ) is of the form of a Jones matrix ensemble.


Alternatively, if one just admits conjugation operations through a specific nonlinear apparatus, then

eq. (3.29) is the most general operation on the polarization matrix Φ. If we allow any nonlinear operation,

then even the first axiom no longer holds, and more general expressions must be used.

3.6 Summary

We have proven the the validity of the expression in eq. (3.25) as the most general physical transfor-

mation on a polarization matrix, using some basic mathematics and simple assumptions. This puts

the assumption that an ensemble of Jones matrices is the most general linear optical filter on more

solid ground, and illustrates exactly where the assumption will break down if we relax the linearity

requirement.

We have also given physical reasons why the transpose map is inadmissible, despite its preservation

of positivity, equating it to the unphysical (and nonlinear) conjugate map. This treatment will break

down in higher dimensions, since eq. (3.29) only applies in the case of 2× 2 matrices. Moreover, if one

admits conjugation operations through nonlinear optical devices, we find the more general eq. (3.29) is

the most general transformation.

Chapter 4

Measures of Purity and Degree of

Polarization

4.1 Introduction

In the previous chapter, we discussed different representations and transformation of the polarization

state of a beam of light. In section 3.3, we introduced the polarization matrix and the Stokes vector

as representatives of this state. There exists a well known mathematical similarity between this two

dimensional classical polarization state, and quantum two level systems. The Stokes vector and Poincare

sphere on the one hand [26, 46] are exactly analogous to the Bloch vector and Bloch sphere on the other

[45]. The Pancharatnam phase in classical optics systems [47] corresponds to the Berry phase in quantum

systems [48].

The point at the origin of the Poincare sphere represents a completely unpolarized beam of light,

while any point on the surface of the sphere represents a completely polarized beam. In the case of the

Bloch sphere, the origin represents the maximally mixed state while any point on the surface represents

a pure state. This suggests an additional analogy between the two systems, that classical polarization

is analogous to quantum purity, and measures of the two quantities should therefore be identical. This

analogy between polarization and purity has been discussed by some authors [49]. Some authors have

suggested the use of Bell’s measure, commonly used in tests of quantum non-locality, to quantify classical

optical coherence (polarization) [64]. Others have suggested a particular measure of polarization in higher

dimensions [50]. However despite this analogy, the most widely used measurements of classical degree

of polarization and quantum mechanical purity are different.

In this chapter, we start in section 4.2 by discussing in more detail the analogy between the classical

and quantum cases, demonstrating that classical degree of polarization and quantum mechanical purity

should be identical quantities. Section 4.3 introduces many existing measures of purity and polarization

in generic N dimensions, with particular attention to the three dimensional case. In doing so, we come

across measures for quantum purity from quantum mechanics, namely the standard purity and von

Neumann purity [51]. We also analyze measures of polarization in three dimensions due to Barakat [52],

Friberg et. al. [53], and Wolf et al. [54].

We then proceed in section 4.4 to compare these measures analytically and numerically, giving them

physical interpretations where possible. Our analysis adds to and clarifies much of the discussion on

29

Chapter 4. Measures of Purity and Degree of Polarization 30

measures of higher dimensional polarization in the literature [55, 56, 57, 58, 59, 60, 61, 62], classifying

the measures and analyzing their relationship.

In section 4.5, we point out that the entanglement of a bipartite pure state can be thought of as the

purity of a subsystem once the other subsystem is traced out. This suggests that using unconventional

measures of purity, we can create new and interesting measures of entanglement. The work in this

chapter is based on ref. [65] by the author.

4.2 Polarization of Beams and Purity of Qubits

4.2.1 Classical Polarization States

Given a classical beam of light we explained in section 3.3 how the state is represented by the polarization

matrix:

Φij = 〈EiE∗j 〉, i = 1, 2, (4.1)

and the four element Stokes vector:

Sµ = Tr[Φσµ]. (4.2)

The inverse relationship to eq. (4.2) is given by

Φ =1

2Sµσ

µ

=1

2

[S0 + S1 S2 − iS3

S2 + iS3 S0 − S1

]. (4.3)

Equation (4.1) implies that Φ is a positive and Hermitian matrix. The positivity of Φ implies

det(Φ) ≥ 0, which when applied to eq. (4.3) implies the following condition on the Stokes parameters

[117]:

S21 + S2

2 + S23 ≤ S2

0 . (4.4)

The inequality in eq. (4.4) shows that the three dimensional vector with coordinates (S1, S2, S3), which

we call the 3D Stokes vector, lies inside a sphere of radius S0. This is the well known Poincare sphere.

Note that S0 = Tr(Φ) represents the total power of the beam.

The degree of polarization, P (2) of a two dimensional polarization matrix Φ is derived by writing

Φ as the unique sum of two polarization matrices, one completely unpolarized (i.e. a multiple of the

identity matrix), and one completely polarized (i.e. a rank 1 matrix) [31]. The degree of polarization is

then the ratio of “power” contained in the completely polarized matrix to the total power. It is given by

P (2) =

√1− 4 det(Φ)

Tr[Φ]2. (4.5)

Using eq. (4.3) to write eq. (4.5) in terms of the Stokes parameters, we find that the degree of polarization


is

P (2) =

√S2

1 + S22 + S2

3

S0. (4.6)

That is, the degree of polarization is simply the length of the 3D Stokes vector divided by the radius of

the Poincare sphere. Put differently, it is relative distance of our state from the origin of the Poincare

sphere. This makes intuitive sense and is a natural measurement, since the origin (where P (2) = 0)

represents the completely unpolarized state, and the surface of the sphere (where P (2) = 1) represents

the set of completely polarized states, and the value of P 2 for other states is “linear” in the distance

metric within the Poincare sphere.

4.2.2 Quantum Two Level system

The most general quantum state is expressed in terms of the density matrix ρ, which contains all the

statistically observable information of the state. This matrix is positive, Hermitian, and of unit trace. In

the case of a quantum two level system, it is of dimension 2× 2. We can write ρ as a linear combination

of the identity and Pauli matrices as follows [51]:

ρ =1

2(I + r1σ1 + r2σ2 + r3σ3), (4.7)

where ~r = (r1, r2, r3) is the well-known Bloch vector [45]. Note that eq. (4.7) is in fact identical to eq.

(4.3). Moreover, since ρ is positive, we can also show that

r21 + r2

2 + r23 ≤ 1. (4.8)

Therefore, the Bloch vector also lies within a (unit) sphere, known as the Bloch sphere. It is clear that

in the two dimensional case, the quantum density matrix ρ is analogous to the classical polarization

matrix Φ, and that the Bloch sphere is analogous to the Poincare sphere.

The only mathematical difference between the two cases is one of scaling. To simplify the mathematics

and make the link to the quantum case obvious, we set the density matrix ρ to be the unit trace scaling

of the polarization matrix Φ. That is

ρ = Φ/Tr [Φ]. (4.9)

So ρ is just the power-normalized version of Φ. In the rest of this chapter, we will only use ρ, keeping

in mind that it applies for both quantum and classical cases.

As a side note, we note that despite the identical mathematical formalism, the Bloch and Poincare

spheres differ in the underlying physical interpretation. If for example we take the quantum two level

system to be the ± 12 spin states of a spin 1

2 particle, then points on the Bloch sphere represent actual

directions of spin in three dimensional space. In other words, each point on the Bloch sphere is an

eigenstate of some (spin) angular momentum operator.

The Poincare sphere however does not have such a simple directional analogy. Its north pole rep-

resents right handed circularly polarized light, its south pole stands for left handed circularly polarized

light, and its equatorial plane gives the linearly polarized states. Since the underlying classical beam of

light is assumed be transverse, there is no longitudinal component. The relationship between the polar-


ization states on the Poincare sphere and three dimensional space is set by the direction of transverse

propagation of the underlying light beam.

The photon being a spin 1 particle, can theoretically have a spin of 1, 0, or -1. However, the zero

spin case represents longitudinal waves and is disallowed. Therefore we only have the 1 and −1 spin

states, that represent right and left circularly polarized light respectively. We can therefore conclude

that contrary to the Bloch sphere, there are only two points on the Poincare sphere that represent

eigenstates of some spin angular momentum operator, the north and south poles, representing right and

left circularly polarized light respectively.

4.2.3 Polarization and Purity

The origin of the Bloch sphere is the maximally mixed state whereas states on the surface of the sphere

are pure states. Comparing this with the Poincare sphere, where the origin is a completely unpolarized

state and the surface contains completely polarized states, this suggests a direct analogue between

quantum purity and and classical degree of polarization.

However, the common measure of classical degree of polarization in eq. (4.6) when expressed as a

function of ρ is given by the expression√

1− 4 det(ρ), while the common measure of purity in quantum

applications is given by Tr[ρ2]. Despite the clear physical analogy, there is a discrepancy in the measures

used. This motivates us to analyze these and other measures of quantum purity and classical degrees of

polarization that have been proposed in the literature.

In the following sections, we will go through several such measures, and probe some of their properties

and relationships, to find which one is appropriate in what situation.

4.3 Measures of Purity for N Dimensions

We proceed to introduce various measures of purity / polarization that have been suggested in the

quantum mechanics and classical optics literature. Since purity is a property intrinsic to the density

operator and invariant of the basis used, it should be invariant under unitary transformations. Therefore,

one can always choose the basis where the density matrix is diagonal, and therefore, purity should be

expressible as a function of the eigenvalues of ρ alone, which we write as λ1 ≥ λ2 ≥ ... ≥ λN , for an N

dimensional system.

We use the symbol Π to denote the various measures of purity, with the appropriate subscript. If we

write the purity as function of the eigenvalues, denoted by Π(λ1, ..., λN ), then we require that it be a

real-valued function that is scaled such that it takes values between 0 and 1. It should take value 1 for a

pure state and 0 for the maximally mixed state; that is, Π(1, 0, ..., 0) = 1 and Π(1/N, 1/N, ..., 1/N) = 0,

respectively.

4.3.1 Standard Purity

In quantum information science, the common measure of purity for a quantum state ρ in an N dimen-

sional system is given by Tr[ρ2] [51]. It takes a maximum value of 1 for a pure state, and minimum value

of 1N for the maximally mixed state. This purity is sometimes scaled linearly so it varies between 0 and


1, giving the following expression, which we call standard purity:

Πs(ρ) ≡ N Tr[ρ2]− 1

N − 1. (4.10)

In terms of eigenvalues, the standard purity is given by

Πs(λ1, ..., λN ) =N∑Ni=1 λ

2i − 1

N − 1. (4.11)

4.3.2 Von Neumann Purity

Shannon entropy is used in classical systems to quantify uncertainty about a random variable [143]. Von

Neumann entropy generalizes this to quantum systems, and is given by

S(ρ) ≡ −Tr[ρ log2(ρ)]. (4.12)

This measure quantifies the departure of a state from a pure state, i.e. its “mixedness” [51]. Note that

the entropy of entanglement (a popular measure of entanglement for bipartite pure states) is defined

to be the von Neumann entropy of one of the subsystems when the other subsystem is traced out, as

discussed in section 4.5. This implies that the von Neumann entropy is a good measure of mixedness.

Therefore, one can define another measure of purity, Πv(ρ) ∈ [0, 1], based on the von Neumann entropy:

Πv(ρ) ≡ 1 +Tr[ρ log2(ρ)]

log2N. (4.13)

Note that approximating the logarithm with its Taylor expansion, and ignoring higher order terms, we

find a result that is a linear function of Πs. That is, standard purity can be thought of as dervied from

the Taylor approximation of von Neumann purity.

Expressed as a function of eigenvalues, von Neumann purity is given by

Πv(λ1, ..., λN ) = 1 +1

log2N

N∑i=1

λi log2(λi). (4.14)

If an eigenvalue λk = 0, we take λk log2(λk) = 0, since limx→0+ x log(x) = 0.

4.3.3 Polarization Purity for N=2

For the simple case of the two dimensional system, we have already seen that the classical degree of

polarization is given by eq. (4.5). Therefore the two dimensional polarization purity as a function of ρ is

P (2)(ρ) ≡√

1− 4 det(ρ). (4.15)

In terms of the two eigenvalues of ρ, this can be written as

P (2)(λ1, λ2) =√

1− 4λ1λ2

= λ1 − λ2, (4.16)


where in the last equality, we used the fact that 1 = (λ1 + λ2)2. Finally, using eq. (4.6) and expressing

the Stokes parameters Si in terms of the Bloch vector elements ri = Si/S0, i=1,2,3, we have

P (2)(~r) =√r21 + r2

2 + r23 = |~r|. (4.17)

So we are left with three equivalent expressions for the polarization purity in two dimensions. Suppose

we wish to generalize this measure of purity to N ≥ 3 dimensions. In principle there are an infinite

number of ways to do this. However, only a handful of them have physical significance. In the following

subsections, we discuss three possible generalizations, each one follows from one of the three equations

(4.15), (4.16), and (4.17). Although the three expressions for purity are identical in two dimensions,

their respective extensions to higher dimensions differ, each forming its own measure.

In our generalizations, we pay particular attention to the N = 3 case, since it corresponds to classical

polarization in three dimensions, a problem which has led to much debate in the literature [50, 54].

The idea of a three dimensional polarization is simple in principle. Rather than dealing with a beam

propagating in one direction with polarization defined in the two dimensional transverse plane, one deals

with an arbitrary electric field distribution in three dimensions. We may, for example, have classical

light that contains longitudinal components and breaks the transversality condition. However, it is not

clear what degree of polarization in this case means physically, leading to differing points of view.

4.3.4 Barakat Heirarchy Measures of Purity

In the case of an N ×N density matrix ρ, Barakat has introduced a hierarchy of N − 1 purity measures

[52]. These measures are defined by first writing out the characteristic polynomial equation of ρ:

det (ρ− λI) = λN − C1λN−1 + C2λ

N−2 − ...+ (−1)NCN = 0. (4.18)

The roots of this polynomial equation are by definition the eigenvalues of ρ. Each coefficient Ck is the

sum of all possible unique products of k eigenvalues of ρ. That is

Ck =∑

1≤i1<...<ik≤N

k∏j=1

λij . (4.19)

For example, if N = 3, then

C1 = λ1 + λ2 + λ3 = 1,

C2 = λ1λ2 + λ1λ3 + λ2λ3,

C3 = λ1λ2λ3 = det (ρ). (4.20)

In fact C1 = 1 and CN = det(ρ) both hold for any N . Moreover, it can be shown that each Ck is

expressible in terms of Tr (ρm), for m = 2, ...k, or alternatively in terms of the first k Casimir invariants

of ρ under the rotation group [125]. For example, for any N , we have [124]

C2 = [1− Tr(ρ2)]/2, (4.21)

C3 = [1− 3 Tr(ρ2) + 2 Tr(ρ3)]/6. (4.22)


Therefore, the Ck are invariant under change of coordinates. If ρ is a pure state (i.e. has rank 1), then

all the Ck are zero, except for C1 which is always unity. If ρ is the maximally mixed state (all eigenvalues

are 1N ), then Ck =

(Nk

)1Nk where

(Nk

)is the binomial coefficient.

With this in mind and noting that Ck coefficients themselves can be thought of as a measure of

purity, Barakat then defines a hierarchy of measures of polarization given by B(N)k (ρ) for k = 2, ...N .

Requiring that B(2)2 (ρ) collapse to P (2)(ρ) in eq. (4.15), one defines

B(N)k (ρ) ≡

√1−

(N

k

)−1

NkCk, k = 2, ..., N, (4.23)

The measure B(N)k (ρ) takes the value zero for all k in the maximally mixed (i.e. the fully unpolarized)

state, and takes the value 1 for all k when ρ is a pure (fully polarized) state. To get a feel for these

measures, let us explore and simplify them using eq. (4.22) for some specific values of N and k. For

N = 2, we have

B(2)2 (ρ) =

√1− 4 det(ρ) = P (2)(ρ), (4.24)

as we required. For N = 3 we have

B(3)2 (ρ) =

√[3 Tr(ρ2)− 1

]/2, (4.25)

B(3)3 (ρ) =

√1− 27 det(ρ)

=√

1− 27λ1λ2λ3. (4.26)

For general N we find

B(N)2 (ρ) =

√N Tr(ρ2)− 1

N − 1=√

Πs(ρ), (4.27)

B(N)N (ρ) =

√1−NN det(ρ)

=√

1−NNλ1λ2...λN . (4.28)

Note the interesting relationship in eq. (4.27) where B(N)2 is simply the square root of the standard

measure of purity Πs. However, B(N)N is unique among the measures we have so far, therefore we define

Barakat’s last measure of purity as Πb, given by

Πb(ρ) ≡ B(N)N (ρ). (4.29)

We add Πb to our collection of measures which will be compared to other measures later in this chapter.

However, it must be mentioned that Πb has a serious shortcoming, in that Πb = 1 if any eigenvalue is

zero. For example, it cannot distinguish between a pure state with eigenvalue spectrum 1, 0, ..., 0, and

a very mixed state with spectrum 1N ,

1N , ...,

1N , 0. That is why Barakat measures are most effective

when different levels of the hierarchy are used together.

4.3.5 EDPW Purity

Another measurement of purity is one proposed by Ellis, Dogariu, Ponomarenko, and Wolf [54, 56]. It

was presented as a measure of three dimensional polarization, however it has the same form for any


dimension. The basic idea is measuring the total power in the fully polarized component. That is, one

splits the 3× 3 density matrix into a unique positive linear combination of the identity matrix, a rank 2

matrix with degenerate eigenvalues, and a rank 1 matrix. To illustrate, suppose U is the unitary matrix

that diagonalizes ρ, as per

U†ρU =

λ1 0 0

0 λ2 0

0 0 λ3

. (4.30)

Then ρ can be written as

ρ = (λ1 − λ2)U

1 0 0

0 0 0

0 0 0

U†︸︷︷︸

fully polarized /maximally pure

rank 1

+(λ2 − λ3)U

1 0 0

0 1 0

0 0 0

U†︸︷︷︸partially polarized

rank 2

+λ3 U

1 0 0

0 1 0

0 0 1

U†︸︷︷︸

unpolarized /maximally mixed

rank 3

. (4.31)

Each of the coefficients (λ1 − λ2), (λ2 − λ3), and λ3 is positive, and the decomposition in eq. (4.31) is

unique. The EDPW purity, denoted Πedpw, is defined to be the ratio of the power in the fully polarized

rank 1 matrix to the total power. That is, it is the ratio of the coefficient of the rank 1 matrix in the

decomposition above to the sum of the eigenvalues, which is simply unity. Therefore, it is given by the

simple expression

Πedpw(λ1, ..., λN ) = λ1 − λ2. (4.32)

This is identical to eq. (4.16) in two dimensions. In fact, this particular measure has the same form for

any N ≥ 2, it is always the difference between the largest two eigenvalues. This can be seen by observing

that eq. (4.31) can be extended to any dimensionality without altering the first coefficient.

The advantage of this measure is that it is physically meaningful. It is the fraction of the power that

is completely polarized, and will be left unchanged if acted on by passive linear elements. That is, if

we use some (hypothetical) three dimensional polarizers with the correct alignment, this fully polarized

component is the only one that will remain unchanged.

However, when one considers the rank 2 component of the ρ matrix, one sees that this component is

not fully polarized, but neither is it fully unpolarized. This suggests that it must have some intermediate

nonzero polarization of its own, and should make a contribution to the overall polarization / purity of the

density matrix. Since Πedpw ignores the rank 2 component completely, it is not suitable as a measure

of overall polarization, but is rather suited for measuring only the component of a field that is fully

polarized. We clarify this in section 4.4.2 with illustrative examples.

4.3.6 SSKF Purity

The measure of purity due to Setala, Shevchenko, Kaivola, and Friberg [53] starts by writing the 3× 3

density matrix ρ as a linear combination of some basis matrices in an expression similar to eq. (4.7). We

write ρ as

ρ =1

3I +

1√3

8∑i=1

riGi, (4.33)


where I is the identity matrix, and Gi, i = 1 − 8 are the popular three dimensional analogue of the

Pauli matrices, the Gell-Mann matrices [89], shown in section B.1 of the appendix. We have modified

the coefficients from those in ref. [53] to ease calculation. The eight coefficients ri together form a

(generalized) Bloch vector ~r. One can use eq. (4.33) together with the orthogonality and tracelessness

of Gell-Mann matrices to show that

Tr [ρ2] =1

3+

2

3|~r|2. (4.34)

The density matrix property Tr [ρ2] ≤ 1 together with eq. (4.34) imply that∑8i=1 r

2i = |~r|2 ≤ 1. That

is, the Bloch vector ~r lies inside an eight dimensional hypersphere of unit radius. We then define the

SSKF purity, denoted Πsskf , in a manner analogous to eq. (4.6) and eq. (4.17) as the length of the Bloch

vector, i.e. the radial distance from the origin in this hypersphere. It is given by

Πsskf (~r) ≡[ 8∑i=1

r2i

] 12

= |~r|. (4.35)

We may alternatively call this measure the radial purity, emphasizing that it gives the length of a radial

Bloch vector in a hypersphere. Equivalently, one can also solve eq. (4.34) for |~r| to write the SSKF

purity as a direct function of ρ:

Πsskf (ρ) =√[

3 Tr(ρ2)− 1]/2, (4.36)

or as a function of the eigenvalues:

Πsskf (λ1, λ2, λ3) =

√1

2

[3(λ2

1 + λ22 + λ2

3)− 1]. (4.37)

Note that eq. (4.36) above is identical to eq. (4.25), and therefore Πsskf (ρ) ≡ B(3)2 (ρ) for N = 3. To

generalize this to general dimensionality N , we write

ρ =1

NI +

1√N

N2−1∑i=1

riQi, (4.38)

where Qi are traceless operators that form a basis for SU(N), and satisfy the orthogonality relation

Tr [QiQj ] = (N − 1)δij . The Bloch vector ~r has N2 − 1 entries ri. Squaring eq. (4.38) and taking the

trace we find

Tr [ρ2] =1

N+N − 1

N|~r|2. (4.39)

From this, we can conclude that for any dimensionality N

Πsskf (ρ) ≡ |~r| (4.40)

=

√N Tr[ρ2]− 1

N − 1. (4.41)

= B(N)2 (ρ) (4.42)

=√

Πs(ρ). (4.43)

So we find that the SSKF / radial purity is equivalent to Barakat’s second measure and the square

root of the standard measure. This is a very interesting result since each of these measures was ostensibly


derived in a different manner. It shows that we really have fewer measures than may initially seem.

Reverting back to the N = 3 case, the picture that has formed seems a natural generalization of

the familiar Poincare/Bloch sphere. The centre of the eight dimensional hypersphere still represents the

totally unpolarized (or maximally mixed) state, and the states on the surface of the hypersphere are

totally polarized (pure).

There is one essential difference however, that undermines the elegance and simplicity of this picture.

In the case N = 2, all states in or on the Bloch sphere represent valid physical states and a positive

density matrix. However, in dimensionality N = 3 or higher, it has been be shown that the physical

constraint of positivity on the density matrix restricts the set of valid states to an irregular convex region

that is a proper subset of the enclosing hypersphere. This physical region touches the surface of the

enclosing hypersphere only in some places (where the fully polarized states lie) [125, 124]. That is, many

states within the hypersphere and on its surface are unphysical since they would create density matrices

that are not positive.

Figure 4.1 shows all possible two dimensional cross sections of the eight dimensional hypersphere.

The shaded regions represent physical polarization/density matrices. Note that in fact most of the

volume inside the hypersphere will be composed of disallowed unphysical states.

Figure 4.1: Classes of cross section of the eight dimensional space in which the generalized Bloch vectorslive, based on a figure by Kimura [124]. In each diagram, the shaded region represents the allowablestates, while the outer circle is a cross section of the enclosing hypersphere. The pure states are wherethe shaded region touches the outer circle. Points A, B, C, D are specific states we examine.

A state lying on the surface of the hypersphere is a necessary but insufficient condition for it to

represent a pure state, for it is unphysical if it is not on the border of the allowable region. States

anywhere on the border of allowable region must have at least one zero eigenvalue. A generic allowable

state does not necessarily lie on a straight line between the maximally mixed state and a pure state

as in the case of the three dimensional Bloch sphere. This must have been the case, since a positive

3× 3 matrix in general cannot just be written as a linear combination of the identity matrix (maximally

mixed) and a rank 1 matrix (pure), there is generally a rank 2 component as shown in eq. (4.31).


To illustrate these features, let us examine the states represented by points A,B,C and D in fig.

(4.1). These four points are given by the following Bloch vectors:

~rA = (0, 0, 0, 0, 0, 0, 0, 1),

~rB = (0, 0, 0, 0, 0, 0, 0,−1),

~rC = (0, 0, 0, 0, 0, 0, 0, 1/2),

~rD = (0, 0,√

3/8, 0, 0, 0, 0, 1/8). (4.44)

Using eq. (4.33), we can construct the corresponding density matrices, and we find

ρA =

2/3 0 0

0 2/3 0

0 0 −1/3

, ρB =

0 0 0

0 0 0

0 0 1

,

ρC =

1/2 0 0

0 1/2 0

0 0 0

, ρD =

1/2 0 0

0 1/4 0

0 0 1/4

. (4.45)

We see that ρA is not positive, and therefore unphysical, despite lying on the unit hypersphere, since it

is not in or on the border of the allowable region. This illustrates the breakdown of the analogy with

the two dimensional Bloch sphere addressed above. The matrix ρB however is positive and therefore

physical. Given that it is physical, we can see that it must be a pure state since it lies on the surface of

hypersphere, and indeed it is. The matrix ρC is also physical, and has a single zero eigenvalue, which is

expected since it is at the boundary of allowable states. If it were to move slightly outside the boundary,

the zero eigenvalue would become negative and therefore unphysical. The state given by ρD is a typical

state inside the allowable region.

One may suggest that these properties are a result of an artificial asymmetry of the Gell-Mann

matrices (in particular G3 and G8), and may be avoided if we opt for a different basis set of matrices

for SU(3). However, this is not true, and the qualitative properties illustrated above are intrinsic to the

N = 3 case, and still hold even if one exchanges the Gell-Mann matrices for a different basis set with

the same basic properties of Hermiticity, tracelessnesss, and orthogonality.

To see this, note the following basis-independent property: the surface of the hypersphere is seven

dimensional, and pure states only form a three dimensional surface. This implies that, independent of

the choice of the basis, the pure states form a very small part of the surface of the generalized Bloch

hypersphere. Most states on the hypersphere surface will be analogous to ρA above, that is they will be

unphysical due to violation of positivity.

For general N , the enclosing hypersphere is of dimension N2−1, and its surface of dimension N2−2.

The space of pure states is only of dimension N . Only in the case N = 2 do we have the dimensions of

the surface and of the pure state space coinciding, giving us the simple properties of the conventional

Bloch sphere.

Yet despite the loss of the simple geometry of a filled hypersphere, the SSKF / radial purity still,

in some sense, quantifies the distance of the state from the maximally mixed state. Furthermore,

if we suppose that the system of interest involves depolarizing channels, a popular type of quantum

noise channel [90], we find that Πsskf satisfies an intuitive depolarization criterion, making it the most


convenient and logical purity measure. We discuss this in more detail in appendix B.2.

4.4 Comparing Purity Measures for Three Dimensions

4.4.1 Graphical Comparison

Thus far, we have discussed five contending measures of purity for N = 3 dimensions: the standard

purity Πs, the von Neumann purity Πv, Barakat’s last measure Πb, the EDPW purity Πedpw, and the

SSKF purity measure Πsskf . Since the standard purity Πs is just the square of the SSKF purity Πsskf ,

we ignore the former and only include the latter in our comparison.

To compare the four remaining measures, we set N = 3, and recall that the purity will only be a

function of the eigenvalues λ1, λ2, and λ3 = 1 − λ1 − λ2, leaving us with only two degrees of freedom.

In figure 4.2, we plot the various measures of purity against λ2, for some fixed values of λ1. Note the

interesting point in the second graph where all the purity measures are zero, this is the maximally mixed

state (λ1 = λ2 = λ3 = 13 ).

0.2 0.4 0.6 0.8Λ2

0.2

0.4

0.6

0.8

1.0

PurityΛ1=0.2

0.1 0.2 0.3 0.4 0.5 0.6Λ2

0.2

0.4

0.6

0.8

1.0

PurityΛ1=0.33

0.1 0.2 0.3 0.4 0.5 0.6Λ2

0.2

0.4

0.6

0.8

1.0

PurityΛ1=0.4

0.1 0.2 0.3 0.4 0.5Λ2

0.2

0.4

0.6

0.8

1.0

PurityΛ1=0.5

Figure 4.2: Values of purity measures for various eigenvalues of a three dimensional density matrix.Each graph has fixed λ1, with λ2 against the horizontal axis, and λ3 ≡ 1− λ1 − λ2. In each graph, theupper solid curve in black is Barakat’s last measure Πb, the upper dashed blue curve is the SSKF purityΠsskf , the lower dashed brown curve is the von Neumann purity Πv, and the lower solid green curve isthe EDPW purity Πedpw.

Examining figure 4.2, we first note that in the graphs above, Πb, Πsskf and Πv behave similarly. That

is, comparing the purity of any two states within the same graph, these three measures agree which state

is more pure. The values of these measures all increase together, decrease together, and have extrema


at the same eigenvalues. Derivatives of their three curves always have the same sign. In appendix B.3,

we show that this behaviour of these three measures will be seen for any dimensionality N , provided we

only vary two eigenvalues and not more.

We also observe that Πedpw often, but not always, yields results opposite to those of all the other

measures. That is, it sometimes disagrees with other measures in deciding which of two states is purer.

This suggests that it is measuring something entirely different, and can be better understood through

examples.

4.4.2 Illustrative Examples

Recall state C with eigenvalue spectrum 1/2, 1/2, 0 and state D with spectrum 1/2, 1/4, 1/4, with

their respective density matrices ρC and ρD defined in eq. (4.45).

Suppose we wish to use one of our measures of purity to find which density matrix ρC or ρD, is more

pure. If for example we use the measure of purity Πsskf , we find that state ρC has higher purity than

state D. If we use Πedpw we find the opposite, state D is higher in purity. To get a more comprehensive

idea, we also introduce state E with eigenvalue spectrum 3/4, 1/8, 1/8 and state F with spectrum

2/3, 1/6, 1/6. In table 4.1, we evaluate the measures Πsskf , Πedpw, Πb and Πv for all of these states.

Pure State E State F State C State D Max. Mixed

1, 0, 0 34, 18, 18 2

3, 16, 16 1

2, 12, 0 1

2, 14, 14 1

3, 13, 13

Πsskf 1 0.625 0.5 0.5 0.25 0

Πedpw 1 0.625 0.5 0 0.25 0

Πb 1 0.827 0.707 1 0.395 0

Πv 1 0.330 0.210 0.369 0.054 0

Table 4.1: The purity of some states we introduced as well as the pure and maximally mixed states,as evaluated by four different measures of purity: Πsskf , Πedpw, Πb and Πv. The eigenvalue spectrumλ1, λ2, λ3 of each state in curly braces. The columns are ordered from purest to most mixed accordingto the Πsskf measure.

The most striking feature of table 4.1 is that no two of the measures agree on the ordering of the states

from purest to most mixed. To help us reason more clearly, we note that in general, the more mixed

a state is, the closer all the eigenvalues are to each other, with the extreme case being the maximally

mixed state where all eigenvalues are equal (to 1/N). The purer a state is the more a small number of

eigenvalues should stand out, with pure states having a single eigenvalue equal to unity and as far as

possible from the rest, which are all zero.

Restricting ourselves to the states C and D, we can now reason that each of the density matrices ρC

and ρD have two identical eigenvalues (a “mixed” property), but in ρC , the third distinct eigenvalue is

further away from the identical two than in ρD (that is, |0− 1/2| > |1/2− 1/4|), therefore ρC should be

more pure. We can also reason that since both states have one eigenvalue of 1/2, then they are equal

in this respect, and the other two eigenvalues should be the deciding factor in which state is more pure.

The remaining eigenvalues for matrix ρD are 1/4, 1/4, these are identical (more mixed), and for matrix

ρC are 1/2, 0, these are as different as possible. So we expect that ρC must have higher overall purity.

Therefore it seems Πsskf is more suited for the general idea of overall purity than Πedpw.

However, suppose instead of overall purity, we are interested in the component of the density matrix

that is fully polarized (i.e. the component that can be acted upon by a hypothetical three dimensional


polarizer and remain unchanged). We see that we can write

ρD =1

4

1 0 0

0 0 0

0 0 0

+1

4I. (4.46)

That is, ρD has a nontrivial fully polarized component, the magnitude of which will be given by

Πedpw(ρD) = 1/4. The matrix ρC however cannot be decomposed in this way, has no fully polar-

ized component, and therefore Πedpw(ρC) = 0. We conclude that the choice of a more suitable measure

of purity depends on what one is interested in measuring, though Πsskf seems more suitable for general

purposes. Since the standard purity Πs is simply the square of the latter, it can be used as quick and

simple way to measure purity, and its ubiquity in quantum information seems justified.

4.4.3 Relationship between SSKF Purity Πsskf and EDPW Purity Πedpw

We have already discussed the properties, strengths and weaknesses of Πsskf and Πedpw. It is of interest

to find a simple relationship between the two measures with aid of a pair of suitably defined variables.

The following analysis is similar to results by Sheppard [61]. In the N = 3 case there are only two

degrees of freedom in setting the eigenvalues (since they must sum to unity). We define x and y as

x ≡ Πedpw = λ1 − λ2,

y ≡ 3(λ1 + λ2 − 2/3) = 1− 3λ3. (4.47)

Physically, x is the EDPW purity, i.e. the fraction of the power that is in the fully polarized component.

and y can be thought of as the fraction of power that is not in the completely unpolarized / mixed

component. In other words, x represents the power in the rank 1 component of the density matrix, while

y represents combined power in the rank 1 and rank 2 components, i.e. the power not in the rank 3

component. Both x and y vary between 0 and 1, with the condition that y ≥ x, which can be seen from

y − x = 1− λ1 + λ2 − 3λ3

= 2λ2 − 2λ3 ≥ 0, (4.48)

where in the second equality we used 1 = λ1 + λ2 + λ3, and the last inequality we noted that λ2 ≥ λ3.

Then we can express the eigenvalues in terms of x and y:

λ1 =1

3+

1

2(y

3+ x), (4.49)

λ2 =1

3+

1

2(y

3− x), (4.50)

λ3 =1

3− y

3. (4.51)

We can make use of these expressions to show that λ21 + λ2

2 + λ23 = (2 + 3x2 + y2)/6. Using this result

along with eq. (4.37), Πsskf is expressible as

Πsskf =1

2

√3x2 + y2. (4.52)


This tells us that Πsskf includes the purity from x (i.e. Πedpw) plus an additional component from y.

Note that for a given x, the minimum value y can take is x, in which case we see from eq. (4.37) that

Πsskf = x = Πedpw. This is expected, since for y to equal x, this means there is no power in the rank 2

component, and all the polarized power is in the rank 1 component, so both measures agree.

4.5 Relation to Entanglement Measures

Erwin Schrodinger first pointed out the uniquely quantum phenomenon of entanglement in his semi-

nal 1935 paper with Max Born [116]. At that time, entanglement was poorly understood, and subject

to paradoxes, such as the famous Einstein-Podolsky-Rosen thought experiment [128], which was sub-

sequently analyzed by John Bell, leading him to his famous inequalities [129]. Though much of the

same fundamental mystery of quantum entanglement remains today, we have at least a large array of

potential tools to measure it [127]. In this section, we propose yet another potential approach with

which to measure entanglement, namely measuring the purity of a subsystem through our various purity

measures.

The entanglement of a bipartite system is directly related to the purity of a subsystem once the other

subsystem has been traced out. For example, say we have a bipartite system of two qubits, A and B,

given by the Bell state

|Φ〉AB =|00〉+ |11〉√

2. (4.53)

This system is maximally entangled. If we define ρA as the improper density matrix of the first qubit

once the second one has been traced, we get

ρA ≡ TrB [|Φ〉〈Φ|]

=

(1/2 0

0 1/2

). (4.54)

Note that ρA is maximally mixed. If we had traced out system A and kept ρB it would have been

identical. Moreover, if |Ψ〉AB where a separable state, then ρA would have been a pure rank 1 matrix. So,

we see that maximal entanglement leads to maximal mixedness in the subsystem, and no entanglement

leads to a pure subsystem state. This argument suggests that the mixedness (one minus the purity) of

a subsystem is a good measure of entanglement of the whole system.

A common measure of entanglement for a bipartite system is entropy of entanglement E [130]. It is

defined as the von Neumann entropy S (i.e. a measure of mixedness) of a subsystem once the other has

been partially traced out. That is, it can be written as

E(|Ψ〉) = 1−Πv[TrB(|Ψ〉〈Ψ|)]. (4.55)

What if we replace Πv in eq. (4.55) with another measure of purity, say Πsskf or Πb? This would give

rise to another class of entanglement measures with different properties, which could possibly be more

relevant for some applications.

For example we consider a bipartite system with N = 3, i.e. a system of two qutrits. Such a

system has been studied by some authors, and even geometric descriptions developed to help visualize


its entanglement [126]. Say we had the following two states:

|ΨC〉 =|00〉+ |11〉√

2, (4.56)

|ΨD〉 =1√2|00〉+

1

2|11〉+

1

2|22〉

=

√2− 1

2|00〉+

√3

2

( |00〉+ |11〉+ |22〉√3

). (4.57)

If we partially trace out one subsystem in each of these two states, we are left with the density matrices

ρC and ρD given by eq. (4.45) in the previous section. Then we can revisit the discussion in section 4.4.2

regarding which measures of purity are more suitable, Πsskf or Πedpw. The question of which is more

pure, ρC or ρD, can be asked differently: which is less entangled, |ΨC〉 or |ΨD〉?In this case, we can further appreciate why Πedpw favoured ρD as more pure, because it favours |ΨC〉

as more entangled. Note that |ΨC〉 is a two dimensional Bell state in a three dimensional system, and

one can think of it in a very specific sense as more entangled than |ΨD〉 since it is equal to a Bell state

(even if it is of a lower dimension). One may also reason that |ΨC〉 should have higher entanglement

since it has no separable component, whereas |ΨD〉 can be written as a combination of the separable

state |00〉, and the maximally entangled qutrit state as shown above. Note that the latter is a linear

combination of non-orthogonal states, and therefore the squares of the coefficients√

2−12 and

√3

2 do not

add to unity.

However, we can conclude |ΨD〉 is more entangled than |ΨC〉 (and therefore ρC purer than ρD), since

both have the same coefficient for the |00〉 state, yet |ΨD〉 has both a |11〉 and |22〉 component while

|ΨC〉 only has a |11〉 with no entanglement in the third level whatsoever. This definition of entanglement

is the one of interest for practical purposes.

All of this of course assumes the bipartite state |Ψ〉AB , is pure. In general, this state can itself be

mixed and is written as a density matrix ρAB , which complicates the situation, and gives rise to the large

and growing number of entanglement measures in the literature [127]. Furthermore, it is not immediately

clear how this idea of using purity to measure entanglement may be generalized to measures of tripartite

entanglement between three systems, or multipartite entanglement where an arbitrary number of systems

is involved. Most likely, it would involve a type of average of the purities of each each subsystem

individually, obtained by tracing out all the other systems. This may open the door to new interesting

measures of entanglement applicable in different situations.

Entanglement can also be related to purity / polarization in other subtle ways. For example, the

Schmidt decomposition can be used to decompose entangled states to a unique positive sum of separable

states [121, 122]. This has been exploited by some authors to create a measure of polarization based on

the Schmidt decomposition [63].

4.6 Summary

We showed that quantifying quantum purity for an N level system is equivalent mathematically to

quantifying the degree of classical polarization in N dimensions. Then we described and analyzed

different measures of purity, finding interesting properties and strength and weaknesses of each.

In the more common case of measuring overall purity, the SSKF / radial purity Πsskf seems the


strongest option, since it is most consistent with depolarizing channels, commonly used quantum noise

channels. The measure is also motivated through a simple geometric analogy of a generalized Bloch

sphere, which still provides insights despite some limitations in higher dimensions. The standard purity

Πs in particular is the simplest to use and the most common in the quantum information literature. It

turns out to be just the square of Πsskf , and therefore inherits its validity as a measure of purity.

Barakat’s last measure of purity Πb is easy to compute, but becomes useless as soon as one of the

eigenvalues approaches zero. The von Neumann purity Πv is also interesting due to its connection to

entropy, but it has few useful properties.

If instead we are interested only in the component that is fully polarized, then the EDPW purity

Πedpw is a more suitable measure. It will yield the strength of only the fully polarized part, discarding

other components. It can also be shown that for N = 3, Πsskf and Πedpw are related in a simple manner

once we add a variable to represent the second degree of freedom.

Moreover, there is a direct relationship between the entanglement of a pure bipartite state and the

purity of one of its subsystems once the other subsystem has been traced out. This can be used to give

insight into measures of entanglement, and possibly create new entanglement measures based on various

measures of purity.

Chapter 5

Synthesizing Quantum Circuits for

Simple Periodic Functions

5.1 Introduction

As quantum computers improve and grow in size beyond a dozen or so qubits, they face two daunting

problems. Firstly, since complete tomographic reconstruction of the quantum state becomes increasingly

intractable [68, 67], how might these devices be characterized and their performance validated? Secondly,

and by no means distinctly, is the problem of finding meaningful milestones for device development along

the long road to true large scale devices capable of tackling useful problems.

Shor’s factoring algorithm [6], nearly 20 years since it was discovered, remains arguably the most

promising and compelling application of quantum computing. It allows one to factor large numbers

in polynomial time, undermining the most common cryptographic schemes in use today, such as RSA

cryptography [131]. Highly simplified versions of this algorithm have been implemented using both NMR

techniques [78] and linear optical implementations [74, 75, 76], providing just such a milestone for two

of the possible technologies under consideration as quantum computers.

Shor’s algorithm makes use of the periodicity of the modular exponential function, whose period can

be evaluated efficiently due to the inherent massive parallelism of quantum computing. The quantum

Fourier transform can then be used to extract the period of the function, from which desired factors

can be deduced. Periodicity is also of particular interest, since it has been shown that it is the size

of the associated period rather than the number being factored which determines the difficulty of the

factorization [82]. This motivates us to better understand periodic functions as an independent object of

study. In particular, it is of interest to study how a simple quantum circuit can be created to implement

functions of a given period, and the resources such a circuit requires. Moreover, given the limited capacity

of currently realizable experimental systems, it is of interest to find the minimal number of gates needed

to create the circuit for a function of a given period.

Synthesizing these simple circuits will provide interesting milestones as experimental technology ad-

vances. Furthermore, periodic functions are relatively straightforward to verify, via a quantum Fourier

transform [102]. Therefore, they may play a role in validating new quantum devices as they are intro-

duced. That is, we use a new device or technology to implement a periodic quantum circuit, and then

use the quantum Fourier transform to check that the circuit indeed produces a periodic superposition

46

Chapter 5. Synthesizing Quantum Circuits for Simple Periodic Functions 47

state when it should, to ensure that the new device functions correctly.

There has been much literature published on the synthesis of quantum circuits in the context of

reversible computing theory, using Toffoli gates [85, 86, 87, 88]. However, the algorithms they presented

are fairly general, whereas we intend to focus on simple periodic functions.

In this chapter, we investigate the process of creating a quantum circuit for a simple periodic function

of a given period p, using only the basic quantum gates: CNOT and Toffoli gates [83]. Note that Toffoli

gates along with local unitary gates, form a universal set of gates [83]. We begin in section 5.2 by defining

the types of periodic functions and the class of simple periodic function we choose to investigate. In

section 5.3 we explain the process of synthesizing some circuits of the simple periodic functions, while

giving some illustrative examples for some values of period p. More circuits are provided in appendix

C. Finally, in section 5.4 we list the required resources (gates) found in synthesizing the circuits for up

to 5-bit periods p. In doing so, we conjecture an upper limit to the required number of Toffoli gates in

such a circuit for any period p. The work in this chapter is based on ref. [84] by the author.

5.2 Periodic Functions

We define a periodic injective function as any function Fp,n satisfying the following properties:

1. It is a binary function with n input bits and m output bits. The input x is an integer satisfying

0 ≤ x < 2n, and output y satisfies 0 ≤ y < 2m.

2. It has a period p, i.e. Fp,n(x) = Fp,n(x− p) for x ≥ p.

3. It is one-to-one (injective) within a single period, i.e. Fp,n(x1) = Fp,n(x2) for 0 ≤ x1, x2 < p

implies x1 = x2.

For example, looking at continuous functions instead of discrete ones for the moment, we see that

tan has period π and is injective within a single period. While sin and cos both have period 2π and are

not injective within a period.

Writing the input variable x in binary notation, we have x = xn...x2x1, where the xi denote each of

the n input bits. For example, if x has the value 13 in decimal notation, then in binary notation it is

x = 1101, with individual bits x4 = x3 = x1 = 1 and x2 = 0. Similarly, if y is the output, we write it in

binary notation as y = ym...y2y1 and the yj denote each of the m output bits. Note that m need only be

large enough so the number p can be represented in m bits, i.e. m = dlog2(p)e, where dwe is the ceiling

of w (defined as the smallest integer which is not smaller than w; for example d3.142e = 4, d5e = 5).

We further define a monoperiodic function Gp as a periodic injective function with period p, and a

number of input bits enough to contain just one complete period p. That is, n = m = dlog2(p)e. So we

write

Gp ≡ Fp,dlog2(p)e. (5.1)

Note that there are many different monoperiodic functions Gp for a given p. All of them have the

same basic structure, and are related to each other through a simple isomorphism on integers in the

output space (i.e. a straight forward relabeling of output states). For example, in the period 3 case,

suppose we define V3(x) ≡ x mod 3, and W3 is defined such that W3(0) = W3(3) = 1,W3(1) = 0, and

W3(2) = 2, where the argument satisfies 0 ≤ x < 4. Then both V3 and W3 qualify as monoperiodic


functions of period 3, and they can be related via the isomorphism t defined as t[0] = 1, t[1] = 0, t[2] = 2,

and t[3] = 3, satisfying W3(x) = t[V3(x)]. This example isomorphism t happens to be self-inverse,

but that need not always be the case. The fact that they can be related via isomorphisms means all

monoperiodic functions of a given period are linked by a type of equivalence relation.

However, despite this mathematical equivalence relation between all monoperiodic functions of a given

period p, there is some practical difference between them. Suppose we wish to synthesize a quantum

circuit for each of these functions using only the basic quantum gates, CNOT and Toffoli gates. It turns

out that some of them will be easier to construct than others, requiring fewer gates.

In what follows, we seek to synthesize the quantum circuit for the monoperiodic function of period

p that is easiest to implement. We define the easiest function as one with the smallest gate count.

Toffoli gates are the most demanding to implement experimentally. If one implements a Toffoli gate

via CNOT gates on a system of qubits, it takes six of the latter to implement, in addition to some

comparatively cheap single qubit gates [132, 133, 139]. It takes fewer resources if we allow qutrits in

an optical implementation [134] or use vibrational modes in an ion trap to store additional information

[135], however for our purposes we will count a Toffoli gate as equivalent to six CNOT gates.

There are many methods to quantify the cost of synthesizing a quantum circuit [88]. We define NT

and NCN as the number of Toffoli and CNOT gates for a given circuit, respectively. We will focus

on Toffoli gates since they are the main drain on resources, and minimizing their number is of most

interest. We will also pay attention to the quantum cost, defined as Q ≡ NCN + 6NT . The quantum

cost is roughly equivalent to the number of elementary two-qubit (CNOT) gates needed to implement a

circuit.

Suppose we implement the most efficient quantum circuit for all possible monoperiodic functions Gp.

We define a simple periodic function Sp as one which minimizes the Toffoli gate count NT among all

monoperiodic functions Gp. Again, Sp may or may not be unique, as there may be multiple functions

with this minimum Toffoli gate count.

To summarize, there are many periodic injective functions with period p, denoted Fn,p. A subset

of them, monoperiodic functions denoted Gp, has only one complete period in the domain of its input

qubits. A smaller subset of these, simple periodic functions denoted Sp, minimizes the Toffoli gate count

during the circuit synthesis. It is this final subset we are interested in.

5.3 Circuit Synthesis

We now address the task of synthesizing the quantum circuit for a function Sp, for a given p. In fact,

it is by synthesizing the circuit for a monoperiodic function Gp while trying to minimize the number of

Toffoli gates that Sp can be found. We conjecture that the process used below to synthesize the circuits

minimizes the number of Toffoli gates. A rigorous proof will be addressed in a future work.

Note that if p is even, then the quantum circuit for Sp is simply the circuit of S p2

synthesized between

the input and output bits xi and yi for (i = 2, ..., n), with an additional CNOT gate that copies x1 to

y1. With this in mind, we are only interested in odd p, since they trivially generalize to even p.

In figures 5.1, 5.2, and 5.3 below, we construct the circuits for the first few odd periods S3, S5 and

S7 respectively. The truth table for these circuits are included in tables 5.1, 5.2 and 5.3. The circuits are

constructed by inspection, and trial and error, making use of some general patterns and principles. Each

circuit can be seen as two processes. The first is copying a linear combination of the input qubits to each


x2 • • x2x1 • x1

|0〉 • y2

|0〉 • y1

Figure 5.1: S3 quantum circuit.

x2 x1 y2 y1

0 0 0 00 1 0 11 0 1 01 1 0 0

Table 5.1: S3 truth table.

output qubit via CNOT gates. This process must be used to create linearly independent combinations

of the input bits in the output bits, which serve as the canvas on which the second process will operate.

The linear independence insures the condition of injectivity within a single period is satisfied.

The second process is the application of cascades of Toffoli gates to modify the results of the first

process by flipping some entries in the truth table. The first Toffoli gate of the cascade uses two suitable

control bits. The Toffoli gate at each subsequent level of the cascade uses as one of its control bits the

target bit of the previous Toffoli gate. It is this Toffoli cascade process where most of the creativity

lies, since it is what actually creates the desired periodicity. The first process alone cannot create odd

periodicity. Note that the two processes may be made to occur in tandem. Also note that all output

qubits yi are initialized to the state |0〉.If one thinks of the truth tables below, the first Toffoli gate applied modifies a number of entries

equal to a quarter of the total length of the column (i.e. 2n−2). Each consecutive Toffoli gate in the

cascade flips half the number of entries of its target qubit as in the previous level of the cascade. For

example, if n = 3 (as is the case for S5 and S7), then the first Toffoli gate will flip 23−2 = 2 entries, the

second gate in the cascade will flip 1 entry. If n = 4, then the first Toffoli gate will flip 24−2 = 4 entries,

the second gate in the cascade will flip 2 entries, and the third gate will flip 1 entry. For n input qubits,

a Toffoli cascade will have n− 1 gates, with the final gate flipping only 1 entry in the truth table.

Where possible, the output of a Toffoli cascade is copied over to other qubits via CNOT gates, to

avoid need for an identical cascade. This copying may take place in the middle of, or at the end of

the cascade. Circuits for higher p values may in addition have separate Toffoli gates that continue from

where a copy of the cascade was made but in a “different direction”, or are not part of a cascade at all.

Examples of this are provided in appendix C.

To discuss the synthesis in more detail, we consider S3, which implies n = 2 bits each for the input

and output registers. We see in figure 5.1 that it requires 1 Toffoli gate, and 3 CNOT gates. The process

is as follows, we set y1 = x1 ⊕ x2, where ⊕ denotes addition modulo 2 (i.e. the XOR operation). Then

we use a Toffoli gate with y2 as its target, which we then copy onto y1. We use standard notation for

CNOT and Toffoli gates, with a black circle indicating the control qubit, and a large circle with a plus

sign inside indicates the target qubit. The small white circles we see in some circuits (such as fig. 5.2)

are inverted control qubits, in the sense that the target bit is modified if the inverted control bit has

value 0, and is unchanged if it has value 1.

The underlined entries in the truth table shown in table 5.1 denote the entries flipped by the action

of a Toffoli gate, whether directly or indirectly (where the result of the Toffoli is copied by a CNOT to

another bit). Each underline is an entry flip, so an even number of underlines leaves the entry unchanged.

The single horizontal line amid the truth table demarcates where the first period ends and the second

period begins. The values of the output bits under this line must repeat the values at the top of the

table.


x3 • • x3x2 • x2x1 • x1

|0〉 y3

|0〉 • y2

|0〉 • y1


x3 • • x3x2 • • x2x1 • • x1

|0〉 y3

|0〉 • y2

|0〉 y1


x3 x2 x1 y3 y2 y1

0 0 0 0 0 00 0 1 0 0 10 1 0 0 1 00 1 1 0 1 11 0 0 1 1 11 0 1 0 0 01 1 0 0 0 11 1 1 0 1 0


x3 x2 x1 y3 y2 y1

0 0 0 0 0 00 0 1 0 0 10 1 0 0 1 00 1 1 0 1 11 0 0 1 0 01 0 1 1 0 11 1 0 1 1 11 1 1 0 0 0


Similarly, the circuit and truth table for S5 are shown in fig. 5.2 and table 5.2 respectively. The

circuit requires 2 Toffoli gates, and 3 CNOT gates. It involves setting y1 = x1 ⊕ x3 and y2 = x2, then

using a cascade of two Toffoli gates to flip (2+1 =) 3 entries in the truth table. The modified entries are

once again marked with an underline. Note that the only nonzero entry in the y3 column of the truth

table was not created by using CNOT additions from the xi, as y3 is not the target of any CNOT gates.

Rather the value of y3 was ’synthesized’ through a cascade of Toffoli gates. This same idea emerges in

many circuits as can be seen in the appendix.

The circuit and truth table for S7 are shown in fig. 5.3 and table 5.3 respectively. The circuit requires

2 Toffoli gates, and 4 CNOT gates. Here we set y1 = x1 and y2 = x2, then we use a cascade of two Toffoli

gates to flip 3 entries in the truth table, followed by a CNOT which duplicates one of these modified

entries to another output bit, bringing the total number of flipped entries to 4, which are marked with

an underline in the table. Finally, x3 is added to y3. Note that this final copying step was not done

earlier in the process to facilitate the copying of the result of the second Toffoli gate from y3 to another

bit.

The circuit S9 requires 3 Toffoli gates and 4 CNOT gates to construct, as shown in fig. 5.4. In a

process very similar to the S5 circuit, we set y1 = x1⊕x4, y2 = x2, and y3 = x3. Then a cascade of three

Toffoli gates is used to flip (4 + 2 + 1=) 7 entries in the truth table shown in table 5.4. Once again, the

value of y4 is synthesized solely using Toffoli gates, with no CNOT gates acting on that bit.

The circuit S11 requires 4 Toffoli gates and 5 CNOT gates to construct in fig. 5.5. Here the

construction of the circuit is more complicated. We start by setting y1 = x1 ⊕ x4 and y2 = x2. We then

follow it by a cascade of three Toffoli gates which flips 7 entries in table 5.5. We then add x3 ⊕ x4 to

y3, which again we chose to do after the cascade of Toffoli gates. Finally, we add a fourth Toffoli gate,

independent of the initial Toffoli cascade, to synthesize the contents of the y4 column in the truth table.

Overall, nine entries in the truth table have been modified by Toffoli gates. An entry with a double

underline means its bit value was flipped twice, and therefore it was unchanged.

In appendix C we include the circuits and truth tables for Sp, for all odd p up to p = 31, i.e. up to p


a 5-bit number. The process of circuit construction is the same as the ones above, and we include some

explanations and insights into any techniques used.

x4 • • x4x3 • x3

x2 • x2x1 • x1

|0〉 y4

|0〉 • y3

|0〉 • y2

|0〉 • y1


x4 • • • • x4x3 • x3x2 • • x2x1 • • x1

|0〉 y4

|0〉 • y3

|0〉 y2

|0〉 y1


x4 x3 x2 x1 y4 y3 y2 y1

0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 1

0 0 1 0 0 0 1 0

0 0 1 1 0 0 1 1

0 1 0 0 0 1 0 0

0 1 0 1 0 1 0 1

0 1 1 0 0 1 1 0

0 1 1 1 0 1 1 1

1 0 0 0 1 1 1 1

1 0 0 1 0 0 0 0

1 0 1 0 0 0 0 1

1 0 1 1 0 0 1 0

1 1 0 0 0 0 1 1

1 1 0 1 0 1 0 0

1 1 1 0 0 1 0 1

1 1 1 1 0 1 1 0


x4 x3 x2 x1 y4 y3 y2 y1

0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 1

0 0 1 0 0 0 1 0

0 0 1 1 0 0 1 1

0 1 0 0 0 1 0 0

0 1 0 1 0 1 0 1

0 1 1 0 0 1 1 0

0 1 1 1 0 1 1 1

1 0 0 0 1 1 0 1

1 0 0 1 1 1 1 0

1 0 1 0 1 1 1 1

1 0 1 1 0 0 0 0

1 1 0 0 0 0 0 1

1 1 0 1 0 0 1 0

1 1 1 0 0 0 1 1

1 1 1 1 0 1 0 0


There are interesting patterns in circuits for p where p = 2k± 1 for any integer k. If p = 2k + 1, then

the circuit for Sp follows the pattern in S5, S9, and S17 shown in figures 5.2, 5.4 and C.3. The circuit

will have k + 1 input bits and the same number of output bits. It will start by copying each input bit

xi to the output bit yi for i = 1, ..., k. Then a single CNOT gates adds xk+1 to y1. Finally, a Toffoli

cascade of k gates is implemented, with bits y2, y3 .. yk+1 as the target as each step. In total for such

a circuit NT = k and NCN = k + 1.

If p = 2k − 1, then the circuit for Sp follows the pattern in S7, S15, and S31 shown in figures 5.3,

C.2, and C.10. The circuit will have k input bits and the same number of output bits. It will start by

copying each input bit xi to the output bit yi for i = 1, ..., k − 2. Then a Toffoli cascade of k − 1 gates,

with the first gate having xk and xk−1 as the control bits, and y1 the target. The final target of the

Toffoli cascade will be yk−1, which is then copied by a CNOT to yk. Finally xk and xk−1 are added to

yk and yk−1 respectively. For this circuit NT = k − 1 and NCN = k + 1.


Appendix C discusses other interesting patterns and relationships between the circuits, which may

be exploited in the future to help create an optimal circuit synthesis algorithm.

5.4 Required Resources for Synthesis

One can follow the process illustrated in the previous section and expounded upon in the appendix to

synthesize these circuits for arbitrary odd numbers. We have continued this process for larger circuits

for period p up to 5 bits. The required resources for each period have been summarized in table 5.6,

which provides the number of Toffoli and CNOT gates needed to synthesize the circuit of the function

Sp. Note that [p]2 is the period p expressed in base 2.

Period p [p]2 n NT NCN Q

3 11 2 1 3 9

5 101 3 2 3 15

7 111 3 2 4 16

9 1001 4 3 4 22

11 1011 4 4 5 29

13 1101 4 3 6 24

15 1111 4 3 5 23

17 10001 5 4 5 29

19 10011 5 5 6 36

21 10101 5 5 6 36

23 10111 5 5 7 37

25 11001 5 4 8 32

27 11011 5 5 7 37

29 11101 5 4 7 31

31 11111 5 4 6 30

Table 5.6: For each period p, we write the period in base 2 ([p]2) and provide its bit-length n. Theinformative columns are the number of Toffoli gates (NT ) and CNOT gates (NCN ) needed to synthesizethe Sp circuit. We also include the quantum cost Q = NCN + 6NT .

Given the above information, one can conjecture the following result of this chapter:

Conjecture. To synthesize the circuit for a simple periodic function Sp, where p is an n-bit number,

one needs at most n Toffoli gates.

More precisely, let [c]2 be the binary string equal to [p]2 with the last bit truncated (since it is always

1, because p is odd). Then, we conjecture that for a given p, if the respective [c]2 contains the substring

01, which we call type A, then exactly n Toffoli gates are needed for a simple periodic function. If the

substring 01 does not occur in [c]2, which we call type B, then exactly n− 1 Toffoli gates are needed.

For a given bit-length n, there are 2n−2 odd periods p where p is an n-bit number. Of these 2n−2

possible odd periods, n− 1 will be of type B, and the rest of type A. As an example, for p = 23, we have

[p]2 = 10111, then [c]2 = 1011, which does contain the substring 01, i.e. is type A, therefore NT = n = 5

Toffoli gates are needed. In the case p = 25, then [p]2 = 11001, and [c]2 = 1100, which does not contain

the substring 01, therefore it is type B and NT = n− 1 = 4 Toffoli gates are needed.

The requirement that a certain substring be present in the binary representation of the period may

seem a strange condition. However, its predictive power in the above examples seems to stem from the


binary structure and recurrence of powers of 2 in the truth tables, particularly in the input columns. An

exact analysis of the conjecture will be addressed in a future work.

5.5 Summary

We have defined an interesting class of simple periodic functions, with the intention that studying them

yield some insights into periodicity in quantum circuits. We have demonstrated a custom procedure for

circuit synthesis of Sp, a simple periodic function with period p, using only CNOT and Toffoli gates.

We have provided examples of these circuits for many values of p. The procedure is immediately

scalable to exactly construct circuits for periods p of special forms p = 2k ± 1. For other p values, the

circuit synthesis procedure can be scaled on an ad-hoc basis. By analyzing the required resources for

the synthesized circuits, we conjecture that for p an n-bit number, one needs at most n Toffoli gates to

construct Sp. These simple periodic circuits may serve as stepping stones for experimental procedures

as technology improves.

This leaves us with many interesting questions to address in subsequent work, such as the need for

a scalable procedure for more general forms of p, and a proof for the conjecture above. A proof that

a simple periodic function Sp is the ‘simplest’ possible (i.e. truly minimize the Toffoli gate count) is

also of interest. Moreover, the periodic properties of these functions and their behaviour under Fourier

transforms should be analyzed. Finally, we may ask how would one generalize the problem at hand to

more complicated periodic functions with more than just one complete period?

Chapter 6

Compiled Quantum Factoring

Algorithms

6.1 Introduction

Theory remains very much ahead of experiment in quantum computing. While complicated quantum

algorithms are developed and studied, experimental capabilities are not advanced enough to implement

them. As experimental techniques progress on the road to a fully functional quantum computer, there

is a great need for small scale algorithms and circuit verification techniques to verify the functionality

of new technology.

Traditionally, tomography has been used to characterize and validate quantum computing technology,

and test the functionality of new devices [66, 67]. However, despite advances in compressed sensing and

other enhancements, full quantum state tomography beyond more than one or two dozen qubits appears

to be intractable due to the amount of measurements required (which grows exponentially with the

number of qubits) [68]. Therefore, it is of interest to create new methods of verifying that new quantum

devices actually execute their desired function.

To this end, it is natural to use well known algorithms, such as Shor’s factoring algorithm [6], for

verification and characterization of new technology. However, the complete version of Shor’s algorithm

uses an order finding subroutine that involves a modular exponentiation operation, which is the bot-

tleneck of the algorithm, needing the most quantum gates to implement and the most time to execute.

Creating circuits for a general modular exponentiation operation is not possible with currently realizable

technology for any nontrivial parameters.

Therefore, there has been much theoretical work on “compiled circuits” for the order-finding subrou-

tine in Shor’s algorithm and associated modular exponentiation operation [69, 70, 71, 72, 73]. Compila-

tion in this sense uses known information about the solution to simplify the circuit from its complicated

general form to a more manageable size. Additionally, experimental demonstrations of compiled circuits

have been carried out using various technologies [74, 75, 76, 77, 78, 79, 80]. Some experimental groups

may realize an uncompiled Shor’s algorithm in the near future [81], though for relatively small numbers.

In this chapter we will use compiled versions of the modular exponentiation operation for the purposes

of validation. We demonstrate some techniques to construct the simple compiled circuits, including a

“classical layer” of compilation that reduces the number of qubits needed. It remains to be seen if one

54

Chapter 6. Compiled Quantum Factoring Algorithms 55

can generalize these compilation and validation techniques beyond a dozen or so qubits, to the regime

where tomography has significant limitations.

In section 6.2, we provide a brief overview of Shor’s algorithm and the steps it involves, distilling

it to the modular exponentiation operation. Section 6.3 demonstrates the process of constructing the

compiled modular exponentiation circuits for some semiprimes, using circuit synthesis methods similar

to those in the previous chapter. In section 6.4 we briefly discuss the relationship between allowable

periods in the modular exponentiation subroutine for each number to be factored. Section 6.5 analyzes

a specific example of compiled modular exponentiation and order-finding, taking into account the effects

of noise and entanglement in the compiled circuits. The content of this chapter is based on ref. [138] by

the author.

6.2 Shor’s Algorithm

6.2.1 Classical Probabilistic Steps

Shor’s factoring algorithm is held as one of the most promising and useful applications of quantum

computing. It allows one to factor large numbers in polynomial time, undermining the most common

cryptographic scheme in use today, RSA cryptography [131]. The algorithm is based on an order-finding

subroutine, which in turn makes heavy use of the modular exponentiation operation [51, 120]. The

modular exponentiation is the bottleneck of the algorithm, needing the resources in terms of quantum

gates and computation time.

The purpose of Shor’s algorithm is to factorize a number N . It is assumed N is odd, and that it is

not a power of a prime number. The case N = pk for some prime p and integer k can be checked by

calculating k√N (the kth root of N) for all integer k ≤ log3N , which is an efficient procedure [51, 120].

Since N is not the power of a prime, it can be written as the product of two coprime integers.

Shor’s algorithm non-deterministically finds a nontrivial factor of N in the following manner. First,

a number a < N is chosen arbitrarily. The greatest common divisor of a and N , denoted gcd(a,N), is

found using the Euclidean algorithm [91]. If gcd(a,N) 6= 1, then gcd(a,N) is a nontrivial factor of N,

and no further work is needed. In case a and N are coprime, i.e. gcd(a,N) = 1, the algorithm uses the

order finding subroutine explained in the next subsection to find the order of a modulo N . Defining the

function fa,N (x) as

fa,N (x) ≡ ax mod (N), (6.1)

the order of a modulo N is defined as the smallest positive integer x such that fa,N (x) = 1. We denote

the order r. The order r is also the period of the function fa,N (x), that is fa,N (x) = fa,N (x + r). For

Shor’s algorithm to succeed, either r has to be even, or a has to be a square of an integer, so that either

way ar/2 is an integer.

If r is found to be odd, and a is not a square, then the algorithm trial has failed and a new random

number a above must be attempted. If r is even, we compute ar/2 mod (N). If ar/2 ≡ −1 mod (N),

once again the algorithm has failed and a new random number a must be tried. We note that ar/2 6= 1

mod (N), because if ar/2 was congruent to 1 modulo N then r would not be the smallest power that

satisfies the order condition.

Supposing the algorithm succeeded, we have that ar/2 6= −1 mod (N) and ar/2 6= 1 mod (N), yet

ar = 1 mod (N). These statements imply that N - (ar/2 + 1) and N - (ar/2 − 1), yet N |(ar − 1) =


(ar/2 + 1)(ar/2 − 1). We have used standard number theory notation, where b|c means b is a factor of c,

and b - c means b is not a factor of c.

Since N divides the product of (ar/2 + 1) and (ar/2 − 1), but not each of them alone, then the

nontrivial factors of N must be split between the two numbers. To find the factors, we simply use the

Euclidean algorithm to calculate gcd(ar/2 ± 1, N). With this, Shor’s algorithm concludes.

Note that the processes involved in Shor’s algorithm can all be executed efficiently (in polynomial

time) on a classical computer, with the exception of order finding. It is a bottleneck for classical com-

puting, since it requires a large number of evaluations of the underlying function; quantum parallelism

removes this problem and fewer evaluations are needed, leading to an efficient computation.

However, despite this, the order finding subroutine comes with its own limitations. Order finding

itself is broken down to two steps; modular exponentiation and a quantum Fourier transform. The

modular exponentiation is what requires the most resources. The quantum Fourier transform requires

few resources, particularly when implemented semi-classically [102]. So Shor’s algorithm while efficient,

is still very difficult to apply practically given today’s limitations in quantum technology. To see this,

we discuss the order finding subroutine separately below.

6.2.2 Quantum Order Finding Subroutine

To find the order of the function fa,N (x) = ax mod (N) defined in eq. (6.1), Shor’s algorithm makes

heavy use of the modular exponentiation operation given by

U(a) : |x〉|0〉 → |x〉|axmod(N)〉. (6.2)

Clearly the structure of the circuit that implements this operation will depend on the random number

a chosen. The quantum state is initialized at

1√M

M∑x=0

|x〉|0〉, (6.3)

where M = 2m is the smallest power of 2 that satisfies M ≥ N2. We then apply the modular exponen-

tiation operation in eq. (6.2) to our initial state, yielding

1√MU(a)

M∑x=0

|x〉|0〉 =1√M

M∑x=0

|x〉|ax(modN)〉. (6.4)

Applying the inverse quantum Fourier transform will yield a quantum state which when measured, will

with a high probability lead us to the order r of the function fa,N (x) = ax mod (N).

Since the modular exponentiation operation U(a) depends on the arbitrary number a, the general

circuit implementing it must have a as a variable input, making it very complicated. In other words, it

is very difficult to create a single circuit that will execute the modular exponentiation in eq. (6.2) for

arbitrary values of x and a. For this reason, the modular exponentiation is by far the most resource

intensive part of Shor’s algorithm, and is currently the biggest practical obstacle to implementing the

algorithm for useful values of N [73]. Though some experimental groups may soon realize an uncompiled

Shor’s algorithm [81], they will still be limited to small numbers for the foreseeable future.

To handle this difficulty while still implementing most aspects of Shor’s algorithm, researchers have


resorted to ’compiled’ versions of the algorithm. Practically, this means that knowing N in advance, a

specific value of the number a is chosen that is known to yield a successful result within the algorithm.

Then the circuit U(a) for the modular exponentiation for the specific value of a is constructed and used

for the compiled algorithm. This will be much simpler than the generic algorithm, since we no longer

need to construct a modular exponentiation circuit that works for any value of a, but for a single value.

Of course, this roundabout process does not really factor the number N , in fact it assumes previous

knowledge of the solution. This has led some authors to suggest that “to even call such a procedure

compilation is an abuse of language” [82], since compilers cannot know the solution of the problem

beforehand.

Nonetheless, the so called compilation process is of interest as a proof of concept of aspects of the

algorithm, and is often the best that can be done with current technology. It is of interest to explore

the process of compiling circuits to create simple examples that provide milestones for experimentalists

as technology advances.

6.3 Compiled Circuits

In this section, we discuss some examples of compiled versions of Shor’s algorithm that have been studied

by researchers in the field, and then propose and explain the additional classical layer of compilation.

6.3.1 Implementations of Compiled Shor’s Algorithm

Much theoretical work has been completed on compiled circuits for the order-finding subroutine in Shor’s

algorithm. Beckman et. al. published one of the first theoretical works on the topic [69], discussing

a possible implementation through ion traps. The focus of that work was minimizing the number of

memory qubits and operations needed by using known properties about the number N to be factored.

In particular, N = 15 was used as an example, and general methods developed for any N .

Similar work in developing theoretical techniqes to compile the modular exponentiation subroutine

was carried out by Vedral [70], Van Meter and Itoh [71], and most recently by Markov and Saeedi [72, 73].

Additionally, experimental demonstrations of compiled circuits through photonic qubits [74, 75, 77] as

well those with recycled photonic qubits [76], nuclear magnetic resonance [78, 79], and superconducting

qubits [80] have been carried out.

As discussed above, compilation in the sense used in this field does not really simplify the problem at

hand. Its main purpose is to give milestones that closely resemble useful tasks to nascent experimental

devices. This will help sharpen experimental techniques on the way to a full implementation of the

algorithm. Moreover, Shor’s algorithm involves several steps, compilation oversimplifies only one of

them, leaving the other steps intact. Therefore, implementing a compiled version will directly help us

improve the technique involved in these steps.

6.3.2 The Compilation Process

As explained above, compilation hitherto used consists of choosing a specific number a that is known

to factor a given N via Shor’s algorithm, and creating the circuit for the function U(a) defined in eq.

(6.2). We will demonstrate compilation examples for the semiprimes N = 15, 21, and 33. These are

the smallest numbers that are the product of two distinct odd primes. We will use methods of circuit


synthesis discussed in section 5.3. In the process, we show we can use a simple classical function to

reduce the number of qubits and gates needed. One may call this a ’layer of classical compilation’.

N=15

We begin by considering the case N = 15. As in ref. [74], we consider two subcases, a = 2 resulting

in r = 4, and a = 4 with r = 2. Starting with the case a = 2, we have table 6.1 showing the truth

table for y = f2,15(x), for input x between 0 and 3. We use standard binary notation, and the input x

is represented by 2 qubits x2, and x1 and the output y is represented by 4 qubits y4, y3, y2, and y1. For

example, if x = 2, then x2 = 1, and x1 = 0.

x2 x1 y4 y3 y2 y1

0 0 0 0 0 1

0 1 0 0 1 0

1 0 0 1 0 0

1 1 1 0 0 0

Table 6.1: The binary truth table for y = f2,15(x).

The function y = f2,15(x) can be implemented using the circuit in fig. 6.1. In a practice similar to

the previous chapter, the underlined entries in truth table 6.1 are the ones that were modified by the

action of a Toffoli gate, whether directly or indirectly via copying through a CNOT gate. We also reuse

the definition of NT as the number of Toffoli gates in a circuit and NCN as the number of CNOT gates.

x2 • • • x2x1 • • • x1

|0〉 • • • y4

|0〉 y3

|0〉 y2

|0〉 y1

Figure 6.1: The circuit for y = f2,15(x). Gate count: NT = 1 and NCN = 7. This is slightly cheaper interms of gates than the analogous arrangement in ref. [74], which used 2 controlled-swap gates (roughlyas hard as a Toffoli) and 2 CNOT gates.

The authors in ref. [74] define a fully compiled function, which we denote f2,15(x) ≡ log2(f2,15(x)).

For x = 0 to 3, we have f2,15(x) = x. The circuit for this function is much simpler than the one

above, and is given in fig. 6.2 below. Although this circuit is a gross oversimplification of modular

exponentiation, it is still in some sense “compiled”.

x2 • x2x1 • x1

|0〉 y2

|0〉 y1

Figure 6.2: The fully compiled circuit for y = f2,15(x). Gate count: NT = 0, NCN= 2.

The use of the log2 function can be seen as a layer of “classical compilation”. We exploit this idea

later in the chapter to generalize it to other sorts of functions.


Moving to the case a = 4, the truth table for y = f4,15(x), for input x equal to 0 or 1 is in table 6.2.

x1 y3 y2 y1

0 0 0 1

1 1 0 0


The circuit for y = f4,15(x) is given in fig. 6.3 below.

x1 • • x1

|0〉 y3

|0〉 y2

|0〉 y1

Figure 6.3: The circuit for y = f4,15(x). Gate count: NT=0, NCN = 2.

Single qubit gates such as the NOT gate are easy to implement, and we do not factor them into

the gate count. Although this circuit is simple as it is, one can fully compile it further [74] by defining

f4,15(x) ≡ log4(f4,15(x)). This function simply maps 0 and 1 to themselves, and can be implemented

through a single CNOT gate as in fig. 6.4.

x1 • x1

|0〉 y1

Figure 6.4: The fully compiled circuit for y = f4,15(x). Gate count: NT=0, NCN = 1.

The above compilation for N = 15 has been implemented in ref. [74]. We generalize these ideas and

drive them further for larger N . We will construct the general uncompiled circuits, and then compile

them through different intermediate functions to reach simpler circuits.

N=21

Suppose we wish to compile the factorization of N = 21, and that we have chosen a = 4 for this purpose.

To get a better intuitive understanding of the modular exponentiation process, we compute the values

of f4,21(x) = 4x mod (21). The results are in table 6.3 below.

x 0 1 2 3 4 5 6 7

4x mod (21) 1 4 16 1 4 16 1 4

Table 6.3: The decimal value table for f4,21(x).

From the values in the table, we note that f4,21(x) is has period 3. Therefore, modulo 21, the order

of 4 is 3. Similarly, for any a coprime to 21 one can find the corresponding order r modulo 21. We do

so in the table 6.4.

a 2 4 5 8 10 11 13 16 17 19 20

r 6 3 6 2 6 6 2 3 6 6 2

Table 6.4: The period r of fa,21(x) for all a coprime to 21.


Note that all the periods are factors of 6, which is to be expected for N = 21 as will be explained in

section 6.4. For now, we simply note that the largest period is 6, meaning that in general, to implement

U(a) defined in eq. (6.2), one needs at most 3 qubits in the input register. Returning to the function

y = f4,21(x), we include its truth table for input x between 0 and 7 in table 6.5.

x3 x2 x1 y5 y4 y3 y2 y1

0 0 0 0 0 0 0 1

0 0 1 0 0 1 0 0

0 1 0 1 0 0 0 0

0 1 1 0 0 0 0 1

1 0 0 0 0 1 0 0

1 0 1 1 0 0 0 0

1 1 0 0 0 0 0 1

1 1 1 0 0 1 0 0


We make some observations about table 6.5, first that y4 and y2 are always zero, and need not have

any gates. We observe that y3 can almost be written as y3 = x3 ⊕ x2 ⊕ x1, with the exception being

the underlined 0 in the third row, which would have been 1 if the formula was exact. Similarly y1 can

almost be written as y1 = x3 ⊕ x2 ⊕ x1 ⊕ 1, with the exception being the underlined 0 in the sixth row,

which would have been a 1 if the formula was exact.

We also note that y5 has only two nonzero entries, which can potentially be constructed from a

single Toffoli gate. Observe that y5 takes the value 1 when x3 = x1 6= x2, or equivalently when

x3 ⊕ x2 = x2 ⊕ x1 = 1. Therefore, we can construct y5 as the output of a Toffoli gate with the two

control bit values x3 ⊕ x2 and x2 ⊕ x1. Then we can use another Toffoli gate (forming a Toffoli cascade,

discussed in the prior chapter) that has y5 as one control bit and x3 as the other control bit to modify

the entry in the fifth row of y1, the target bit. To flip the entry in the third row for y3, we simply add

the result of the two previous Toffoli gates, that is, we add y5 and the modifier entry of y1, which will

result in a single modifier entry in the third row of y3.

All of the above is shown in the circuit constructed in fig. 6.5.

x3 • • • • • x3x2 • • • • x2x1 • • x1

|0〉 • • y5

|0〉 y4

|0〉 • y3

|0〉 y2

|0〉 • • y1

Figure 6.5: The circuit for y = f4,21(x) with three input qubits, and five output qubits. Gate count:NT=2, NCN = 12.

We have again underlined all entries in the truth table shown in table 6.5 that are flipped by a Toffoli

gate, whether directly or indirectly (i.e. the result of a Toffoli being copied by a CNOT). An entry that

is flipped twice (i.e. unchanged) is doubly underlined. This circuit synthesis procedure is ad-hoc, and

based on taking advantage of some patterns in the function truth table. It is much more efficient than


some alternatives ones. It uses only 2 Toffoli gates and 12 CNOT gates (with a quantum cost of 24),

whereas for example, the circuit for the identical function in ref. [72] uses 8 Toffoli gates and 5 CNOT

gates (quantum cost of 53).

Even so, we note that the circuit above has 5 output qubits to represent the three output values

1, 4, and 16. To partially compile the circuit, we map these three outcomes to the integers 0, 1, and 2

instead, via the operation of a logarithm base 4. That is, we define a function f4,21(x) ≡ log4(f4,21(x)),

which needs only two out put qubits. Its truth table is shown in table 6.6 below. Using our same circuit

synthesis procedure, the circuit for f4,21(x) is shown in fig. 6.6.

x3 • • x3

x2 • • • • x2

x1 • • x1

|0〉 • y2

|0〉 y1

Figure 6.6: The circuit for the partially compiledf4,21(x), with three input qubits and two outputqubits. Gate count: NT=2, NCN = 6.

x3 x2 x1 y2 y1

0 0 0 0 0

0 0 1 0 1

0 1 0 1 0

0 1 1 0 0

1 0 0 0 1

1 0 1 1 0

1 1 0 0 0

1 1 1 0 1

Table 6.6: The truth table for the partially com-piled f4,21(x), with three input qubits and two out-put qubits.

The function log4 can be used through an intermediate classical operation, and simplifies the circuit

to 2 Toffoli gates and 6 CNOT gates (quantum cost of 18), and uses fewer qubits. Again, the logarithm

can be considered a classical layer of compilation that reduces the number of qubits needed.

Since the circuit in fig. 6.6 has period 3, it can be further compiled to the fully compiled function

f4,21(x), whose truth table is in table 6.7 and circuit in fig. 6.7. This fully compiled circuit has only two

input qubits. It can be thought of as the “underlying circuit”. Note that in a fully compiled circuit, the

values of the function output generated by all the possible inputs create one full period of the function,

but not two or more. This is similar to the idea of the simplest periodic function. In fact, the function

f4,21(x) is identical to S3, the simplest periodic function of period 3, introduced in chapter 5. Many

fully compiled circuits for higher N will also fall into the simplest periodic function category.

x2 • • x2x1 • x1

|0〉 • y2

|0〉 • y1

Figure 6.7: The fully compiled circuit for y =f4,21(x) with an additional layer of classical compi-lation. Gate count: NT=1, NCN = 3. Identical toS3 circuit in fig. 5.1.

x2 x1 y2 y1

0 0 0 0

0 1 0 1

1 0 1 0

1 1 0 0

Table 6.7: The fully compiled truth table for y =f4,21(x) with two input and two output qubits.Identical to S3 table 5.1.


N=33

Suppose N = 33, the table 6.8 shows all the possible a coprime to 33, and the order of each modulo 33

(i.e. the period of fa,33(x)).

a 2 4 5 7 8 10 13 14 16 17 19 20 23 25 26 28 29 31 32

r 10 5 10 10 10 2 10 10 5 10 10 10 2 5 10 10 10 5 2

Table 6.8: The period r of fa,33(x) for all a coprime to 33.

We choose a = 4 once again, since it is easy to work with, and its odd period of 5 is admissible in

Shor’s algorithm since 4 is a square. We can construct the circuit for calculating f4,33(x) = 4x mod (33),

which takes on values 1, 4, 16, 31, 25 as x varies from 0 to 4, and then repeats for x ≥ 5. To represent

the input values 0 to 4 we need three qubits in the input register, and to represent the output values up

to 31 we will need five qubits in the output register.

However, we choose instead to construct the compiled circuit, for f4,33(x) = g(f4,33(x)) where g(y) =

(y − 1)/3. This maps the output values to 0, 1, 5, 10, 8. This map works well since g(y) is a simple

differentiable function that maps the output values of f4,33(x) to smaller integer values. Here, g(y)

performs the classical compilation step, and takes the place of the logarithms used in previous examples.

Constructing the circuit for g(f4,33(x)) we have fig. 6.8 below.

x3 • • x3

x2 • • • • x2

x1 • • • • • x1

|0〉 • y4

|0〉 y3

|0〉 y2

|0〉 y1

Figure 6.8: The circuit table for y = f4,33(x). Gate count: NT=3, NCN = 7.

x3 x2 x1 y4 y3 y2 y1

0 0 0 0 0 0 0

0 0 1 0 0 0 1

0 1 0 0 1 0 1

0 1 1 1 1 0 0

1 0 0 1 0 0 0

1 0 1 0 0 0 0

1 1 0 0 0 0 1

1 1 1 0 1 0 1


6.3.3 Classical Compilation

What we have called the classical compilation step works by defining fa,N (x) = g(fa,N (x)) where g(y) is

some differentiable function that maps the output values of fa,N (x) (for all x in the function’s domain)


to smaller integer values. The smaller output values after applying the function g allow us to use fewer

qubits to represent the output register, and thus saving us some resources. We have seen examples

where g(y) is a logarithm, or a linear function. In general, other simple functions can be used, though

for some combinations of a and N this compilation is not possible except through very artificially defined

complicated functions g.

Circuit compilation is only interesting when it involves simple intermediate steps that offer a sub-

stantial simplification of a complex circuit. If the function we use to compile is itself overly complicated,

then we have not gained much. This puts a limitation of the usefulness compilation process as a whole.

6.4 General N and Allowable Periods

Given that the period is what determines the difficulty of factorization [82], and even the size of the

compiled circuit, it is of interest to find which periods of fa,N (x) = ax mod (N) are allowed for a given

N . That is, we wish to find the set of all orders of any a modulo a given N .

Suppose N = pq is fixed, where p and q are distinct primes. From elementary number theory we

know there are ϕ(N) = (p − 1)(q − 1) positive integers less than N that are coprime to N [136]. The

function ϕ(N) is called Euler’s totient function [137]. The group formed by coprime integers modulo N

is of size (p − 1)(q − 1), and so any subgroup must have a size, also known as the group’s order, that

divides (p− 1)(q − 1). In other words, for any number a chosen, the period of fa,N (x) must be a factor

of (p− 1)(q − 1).

In fact, an even stronger condition has been established. The period r of fa,N (x) can be shown to

divide the Carmichael function of N [140], given by

λ(N) ≡ lcm(p− 1, q − 1), (6.5)

where lcm is the least common multiple.

For example, if N = 21, then p = 3 and q = 7, therefore λ(21) = lcm(2, 6) = 6. Table 6.10 gives pairs

of distinct odd primes p, q, and the corresponding N , λ(N), and allowed values of the period r (which

are all the factors of λ(N)) for N < 90.

Table 6.10 is useful for experimentalists who want to construct a compiled factoring circuit that has

a given period/order. For example, if we wish to construct a compiled circuit for factoring some number

N where the period r is 11, then we know the smallest such N is 69.

6.5 Illustrative Example of Compiled Period Finding

Suppose we are able to construct the compiled circuits described in the previous section using some

quantum device. We can run the circuit with the initial state |+〉 in all input registers (generating an

equal superposition), and |0〉 in all the output registers. After running the circuit we can measure the

probability distribution of the output state and compare it to theoretical expectations to benchmark the

device, and judge if it works correctly. In addition, by comparing the measured probability distribution

to the theoretically expected one, we can roughly quantify the noise (assuming a Werner state model),

and entanglement of the state. We demonstrate this by way of an illustrative example.

Suppose we have a “toy” quantum circuit with three qubits in the input register and three in the


p q N λ(N) allowed r3 5 15 4 2, 43 7 21 6 2, 3, 63 11 33 10 2, 5, 105 7 35 12 2, 3, 4, 6, 123 13 39 12 2, 3, 4, 6, 123 17 51 16 2, 4, 8, 165 11 55 20 2, 4, 5, 10, 203 19 57 18 2, 3, 6, 9, 185 13 65 12 2, 3, 4, 6, 123 23 69 22 2, 11, 227 11 77 30 2, 3, 5, 6, 10, 15, 305 17 85 16 2, 4, 8, 163 29 87 28 2, 4, 7, 14, 28

Table 6.10: For each semiprime N = pq with p and q distinct odd primes, the table lists the value ofthe Carmichael function λ(N) ≡ lcm(p− 1, q − 1), and the allowable periods r, given by all the factorsof λ(N).

output register. If we represent the binary numbers in each register as decimal numbers, the value in

each register is an integer from 0 to 7. Suppose further that the circuit is periodic. That is, if we treat

the circuit as a function with this input range, it has a period p, for some p between 1 and 8 inclusive.

Denoting the action of the circuit by the function Fp, and initializing each of the three qubits in the

input register to the state |+〉, then the action of the circuit creates the output state |ψp〉 given by

|ψp〉 ≡ Fp[ 1√

8

7∑j=0

|j〉i|0〉o]

=1√8

7∑j=0

|j〉i|j modp〉o, (6.6)

where i subscript denotes the input register and o subscript denotes the output registers. For example,

if p = 3, we have

|ψ3〉 =1√8

[(|0〉+ |3〉+ |6〉

)i|0〉o +

(|1〉+ |4〉+ |7〉

)i|1〉o +

(|2〉+ |5〉

)i|2〉o

]. (6.7)

As a side note, the state |ψ3〉 is the one produced by the partially compiled circuit for f4,21 in fig. 6.6

and table 6.6, with the subtle difference the here we have defined three qubits in the output register,

and the f4,21 only has two.

In Shor’s algorithm, the final step is applying the quantum Fourier transform (QFT) to the input

register. We follow suit and do the same in our toy model. The QFT is defined as

QFT : |j〉 → 1√N

N−1∑k=0

ωkjN |k〉, (6.8)

where ωN is the Nth root of unity, defined as ωN ≡ e2πi/N . Continuing with our example, we have

N = 23 = 8. Dropping the subscript, we have ω ≡ ω8 = eπi/4 = 1√2(1 + i). Applying the the QFT to


the input register in eq. (6.6), we have

|φp〉 ≡ QFTi|ψp〉 =1

8

7∑j=0

7∑k=0

ωkj |k〉i|j modp〉o. (6.9)

For example, if p = 3, we have

|φ3〉 ≡1√8

[[3|0〉+ 0.293(1−i)|1〉 − i|2〉+ 1.707(1+i)|3〉+ |4〉+ 1.707(1−i)|5〉+ i|6〉+ 0.293(1+i)|7〉

]i|0〉o

+[3|0〉+ 0.414|1〉+ 1|2〉 − 2.414|3〉 − |4〉 − 2.414|5〉+ |6〉+ 0.414|7〉

]i|1〉o +

[2|0〉 − (0.707−0.293i)|1〉

− (1−i)|2〉+ (0.707−1.707i)|3〉+ (0.707+1.707i)|5〉 − (1+i)|6〉 − (0.707+0.293i)|7〉]i|2〉o

]. (6.10)

Then we can define the reduced density matrix ρp as the result when the output register is traced out,

that is

ρp ≡ Tro(|φp〉〈φp|

). (6.11)

From using eq. (6.10) and eq. (6.11) we can calculate ρ3 be

ρ3 = 10−3

344 11+5i 16+16i −11−27i 0 −11−27i 16−16i 11−5i

11−5i 15 27+11i −31−31i −2−5i 22i 8−20i 9−9i16−16i 27−11i 63 −102−42i −16−16i 5+11i −31i 8−20i−11+27i −31+31i −102+42i 235 64+27i 53+53i 5+11i 22i

0 −2+5i −16+16i 64−27i 31 64+27i −16−16i −2−5i−11−27i −22i 5−11i 53−53i 64−27i 235 −102−42i −31−31i16+16i 8+20i 31i 5−11i −16+16i −102+42i 63 27+11i11+5i 9+9i 8+20i −22i −2+5i −31+31i 27−11i 15

(6.12)

If one were to measure the input register in the state |φp〉, the probabilities Pp(k) of finding the state

|k〉 (for 0 ≤ k ≤ 7) are the values along the diagonal of ρp. Computing these probabilities for different

values of p, we have the probability of measuring |k〉 in the input register for each value or p, tabulated

in the table 6.11.

In effect, each row in table 6.11 gives the probability distribution of the resulting ket after measuring

the input register, for a given period p. Practically speaking, this means one can construct the circuit

for the function Fp, apply the QFT, then measure input register. Repeating the entire process many

times, an observed probability distribution, P(k), is obtained and can be compared with the theoretical

distribution, Pp(k), from table 6.11. The comparison will help assess the accuracy and effectiveness of

the quantum information processing device on which it was implemented.

p |0〉 |1〉 |2〉 |3〉 |4〉 |5〉 |6〉 |7〉1 1 0 0 0 0 0 0 02 0.5 0 0 0 0.5 0 0 03 0.344 0.015 0.063 0.235 0.031 0.235 0.063 0.0154 0.25 0 0.25 0 0.25 0 0.25 05 0.219 0.059 0.125 0.19 0.031 0.191 0.125 0.0596 0.188 0.125 0.063 0.125 0.188 0.125 0.063 0.1257 0.156 0.147 0.125 0.103 0.093 0.103 0.125 0.1478 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125

Table 6.11: The probabilities Pp(k) of finding |k〉 after a measurement on the input register of |φp〉.The column index is k and the row index is p.

To make the model more realistic, one can introduce a depolarizing channel model of noise (as in eq.

(B.2)) via the transform

ρp → ρ′p = (1− ε)I8

+ ερp, (6.13)


for some parameter ε. The value ε = 1 results in an unchanged (noiseless) state, and ε = 0 a maximally

mixed (totally noisy) state. Under this transformation due to the depolarizing channel, the probabil-

ity distribution in row p of table 6.11 will be “diluted” by the maximally mixed distribution (which

corresponds to the p = 8 row). That is

Pp(|k〉)→ P ′p(|k〉) =1

8(1− ε) + εPp(|k〉). (6.14)

The transformation in eq. (6.14) follows from eq. (6.13) together with the realization that the probability

distribution Pp(|k〉) consists of the diagonal elements of ρp. The transformed probabilities sum to unity,

since∑7k=0 Pp(|k〉) = 1 implies

∑7k=0 P ′p(|k〉) = 1.

The presence of entanglement between the input and output registers is also of interest. We note

that the state |ψp〉 defined in eq. (6.6) above is completely separable for p = 1 and maximally entangled

for p = 8, with intermediate entanglement for 1 < p < 8. The state |φp〉 will have the same entanglement

level as |ψp〉, since they are related by a local unitary transformation, the QFT.

Suppose we wish to find some rough measure of entanglement of our pre-measurement state using only

the experimentally observed probability distribution, P(k). In this case, we can introduce a new quantity

called the rough separability index, S, defined as the sum of the squares of the observed probabilities:

S =

8∑k=0

P(k)2. (6.15)

We calculate S for the theoretical probability distributions Pp(k) for each period p, and tabulate the

result in table 6.12.

p 1 2 3 4 5 6 7 8S 1 0.5 0.238 0.25 0.16 0.141 0.129 0.125

Table 6.12: The rough separability index S for various values of period p. The values of S are calculatedfrom the probability distributions in table 6.11 by summing the squares of the entries for each row.

We note that S is almost monotonically decreasing when taken as a function of p, with just one

exception at p = 3. And since entanglement monotonically increases with p, we can take 1 − S as a

rough measure of entanglement created by our circuit. This can act as a proxy for the ability of the

constructed circuit to handle entanglement, assuming a simple depolarizing channel noise model.

Using eq. (6.14) and eq. (6.15), we find the value of S is affected by the introduction of a depolarization

channel model of noise by

S → S′ = ε2S +1

64(1− ε)(1 + 15ε). (6.16)

Supposing we know the theoretical (noiseless) value of S based on our knowledge of the period p, and we

measure (noisy) S′ after executing the circuit, we can use eq. (6.16) to find the value of ε, and estimate

the noisiness of our system.

To sum up, in this section we introduced a simple method to test and validate a nascent quantum

technology using some periodic circuit and the QFT. This procedure applied will give us some basic

insight into the operation of our test circuit, how well it functions, as well as its handling of noise and

entanglement.


6.6 Summary

Shor’s algorithm is based on probabilistic classical steps as well as a quantum order finding subroutine.

The latter has a modular exponentiation step, which in its general form is very difficult to implement

with currently realizable quantum technology. Therefore, experimentalists over the last decade or so

have resorted to compiled versions of Shor’s algorithm, that simplify the modular exponentiation step

using known information about the solution.

In this chapter, we have extended the method of compiling the modular exponentiation operation,

and demonstrated a more efficient method of circuit synthesis than previously used. We have provided

the compiled circuits for several semiprimes, and illustrated how the process can be generalized. A

simple layer of “classical compilation” has been discussed as potentially simplifying complicated circuits

further.

An emphasis has been placed on periodicity in the modular exponentiation circuits, and a link to the

simplest periodic functions in chapter 5 has been pointed out. Finally, we have presented a simplified

procedure to validate the function of such a circuit, including its handling of noise and entanglement.

Chapter 7

Conclusions and Outlook

In this thesis, we have approached three major themes within optics; time averaged dynamics and

associated Effective Hamiltonian theory, measures and transformations of polarization, and periodicity

within quantum circuits.

In studying the dynamics of a time averaged quantum system, we have derived a general expression

for the Effective Hamiltonian in a more rigorous manner than previous work. The theory showed

that unlike the rotating wave approximation where high frequency components are simply discarded,

these components actually have a discernible effect on the Effective Hamiltonian. Additionally, it has

been shown that the averaging process creates decoherence terms, that resemble the Lindblad master

equation evolution for harmonic Hamiltonians. This implies that the averaging process introduces a small

decoherence factor. Applying this theory of time averaging to examples such as the AC Stark Shift and

Raman Transitions, we find it confirms previous results, and introduces some minor corrections.

In the future, we would gain more insight into the validity of the theory by applying it to known

systems. In particular, applying it to systems where the newfound decoherence terms play a larger role

would help us better understand their exact effect. Additionally, a general and more physically intuitive

interpretation of the decoherence terms would be a useful construct. For example, it may be possible to

see the time averaging process in a manner analogous to a system-reservoir pair where the latter is traced

out, and the high frequency components somehow play the role of this discarded reservoir. However the

details of such a process are not yet clear.

In our studies of polarization, we have proven the the validity of the expression for the completely

positive map in eq. (3.25) as the most general physical transformation on a two dimensional polarization

matrix. We did so based on mathematics of positive maps and fundamental physical assumptions, rather

than an a priori assumption as is usually the case. We have also made use of complex analytic signals

to explain why the transpose / conjugation map on the polarization matrix is inadmissible. It has been

demonstrated that the conjugation map will be nonlinear in an important sense, rendering it unphysical

within the context of linear optics.

In the process, we emphasized the difference between a positive and completely positive map, and

showed how they are related in two dimensions. It may be interesting in the future to generalize this

particular line of reasoning to higher dimensions, especially given the difficulty that no general form for

positive maps exists in dimensionality three or higher.

We discussed the equivalence between polarization and purity, and provided a detailed description

68

Chapter 7. Conclusions and Outlook 69

and analysis of different measures of these quantities. Despite some agreement between the measures

in some cases, they generally will not agree which of two arbitrary quantum states is more pure. We

find that each measure has its strengths, with some of them being more relevant for most cases of

interest. In the more common case of measuring overall purity, the SSKF / radial purity Πsskf seems

the strongest option, since it is most consistent with depolarizing channels, commonly used quantum

channels simulating noise. The measure is also motivated through a simple geometric analogy of a

generalized Bloch sphere, which still provides insights despite some limitations in higher dimensions.

The standard purity Πs is simply the square of the radial purity, and remains the simplest to use and

most common in quantum information literature.

Barakat’s last measure of purity Πb is easy to compute, but loses usefulness as soon as one of the

eigenvalues approaches zero. The strength of Barakat’s hierarchy of measures lies in using them together,

not in isolation. The von Neumann purity Πv is also interesting due to its connection to entropy, but it

has few other useful properties. If instead we are interested only in the component that is fully polarized,

then the EDPW purity Πedpw is a more suitable measure. It will yield the strength of only the fully

polarized part, discarding other components. It can also be shown that in three dimensions Πsskf and

Πedpw are related in a simple manner once we add a variable to represent an additional second degree

of freedom.

We also observed the direct relationship between the level of entanglement of a pure bipartite state

and the purity of either of its subsystems once the other subsystem has been traced out. This implies

a relationship between measures of entanglement and measures of purity. In particular, we can use any

measure of purity we choose on the subsystems to induce a measure of entanglement on the bipartite

state in the whole space. This can be used to give insight into measures of entanglement, and possibly

create new useful entanglement measures in the future.

Through our final theme of periodic quantum circuits, we have pursued multiple paths. We started

by defining an interesting class of simple periodic functions, in the hope that studying them yield insight

into periodicity in quantum circuits. A procedure for synthesizing quantum circuits more efficient than

many that exist in the literature has been demonstrated. We used this synthesis process to construct

circuits of Sp, a simple monoperiodic function with period p, using only the basic CNOT and Toffoli

gates.

We have provided the constructed circuits for values of p up to five bits long. The synthesis procedure

is also able to easily and exactly construct circuits for periods p of special forms p = 2k ± 1. For other

p values, the circuit synthesis procedure can be scaled on an ad-hoc basis, which however requires more

work. By analyzing the required resources for the synthesized circuits, we conjecture that for p an n-bit

number, one needs at most n Toffoli gates to construct Sp. These simple periodic circuits may serve as

stepping stones for experimental procedures as technology improves.

This leaves many interesting extensions to be addressed in a future work. One needs a scalable

procedure for more general forms of p, transforming our ad-hoc procedure and observations on circuit

patterns into a formal step by step algorithm for circuit synthesis. Developing such an algorithm will

help us address a proof for the conjecture above. Likewise, generalizing the problem at hand to more

complicated periodic functions with more than just one complete period forms an interesting future

research trajectory.

Finally, we dealt with circuits that execute compiled version of Shor’s algorithm. These are important

since the full algorithm involves quantum modular exponentiation, which in its general form is very

Chapter 7. Conclusions and Outlook 70

difficult to implement with current technology. Therefore, experimentalists have routinely made use

of compiled versions of Shor’s algorithm, which simplify the modular exponentiation step using known

information about the solution.

We have improved the method of compiling the modular exponentiation operation, and demonstrated

a more efficient method of circuit synthesis to this end. We have synthesized the compiled circuits for

several semiprimes, and illustrated how the process can be generalized. To potentially compile the circuits

further, we pointed out a simple layer of “classical compilation”. To conclude this line of thought, we

presented a simplified process to validate the function of such a compiled circuit, including its handling

of noise, entanglement, and the measured probability distribution.

This research into quantum circuits is mainly meant to help experimentalists by facilitating validation

of quantum devices, as well as provide milestones for nascent quantum computers.

Appendices

71

Appendix A

Addendum to Chapter 2: Time

Averaged Dynamics

A.1 Derivation of Fk and Lk Terms

The terms derived by comparing the coefficients of each power of λ in eq. (2.13) are as follows:

λ0 : F0

[E0]

= I,

λ1 : F0

[E1]

+ F1

[E0]

= 0,

λ2 : F0

[E2]

+ F1

[E1]

+ F2

[E0]

= 0,

λn : F0

[En]

+ F1

[En−1

]+ F2

[En−2

]= 0. (A.1)

The recursion relation for calculating Fn is given by

Fn = −n−1∑j=0

Fj[En−j [ρ]

]. (A.2)

In what follows, we will calculate Ek and Fk terms for the first few orders. Using eq. (2.9) we get

E0[ρ] = ρ,

E1[ρ] = U1ρ+ ρU†1 ,

E2[ρ] = U2ρ+ U1ρU†1 + ρU†2 ,

E3[ρ] = U3ρ+ U2ρU†1 + U1ρU

†2 + ρU†3 . (A.3)

From eq. (2.10) and eq. (A.2) we have

F0[ρ] = ρ, (A.4)

F1[ρ] = −U1ρ− ρU†1 ,

F2[ρ] = −U2ρ− U1ρU†1 − ρU

†2 + U1

(U1ρ+ ρU†1

)+(U1ρ+ ρU†1

)U†1 . (A.5)

72

Appendix A. Addendum to Chapter 2: Time Averaged Dynamics 73

Applying the differential operator i~ ∂∂t to the Ek, we have

i~E0[ρ] = 0,

i~E1[ρ] = Hρ− ρH,

i~E2[ρ] = HU1ρ+HρU†1 − U1ρH − ρU†1H,

i~E3[ρ] = HU2ρ+HU1ρU†1 − U2ρH +HρU†2 − U1ρU

†1H − ρU

†2H. (A.6)

Making use of the above, the second order term L2 is found to be

L2[ρ] = i~E2[F0[ρ]

]+ i~E1

[F1[ρ]

]+ i~E0

[F2[ρ]

]= HU1ρ+HρU†1 − U1ρH − ρU†1H −H

(U1ρ+ ρU†1

)+(U1ρ+ ρU†1

)H

= HU1ρ−H U1ρ+HρU†1 −HρU†1 − ρU

†1H + ρU†1 H − U1ρH + U1ρH. (A.7)

The third order term L3 is found to be

L3[ρ] = i~E3[F0[ρ]

]+ i~E2

[F1[ρ]

]+ i~E1

[F2[ρ]

]+ i~E0

[F3[ρ]

]= HU2ρ+HU1ρU

†1 − U2ρH +HρU†2 − U1ρU

†1H − ρU

†2H −HU1

(U1ρ+ ρU†1

)−H

(U1ρ+ ρU†1

)U†1 + U1

(U1ρ+ ρU†1

)H +

(U1ρ+ ρU†1

)U†1H −H U2ρ−H U1ρU

†1

−HρU†2 +H U1

(U1ρ+ ρU†1

)+H

(U1ρ+ ρU†1

)U†1 + U2ρH + U1ρU

†1H + ρU†2H

− U1

(U1ρ+ ρU†1

)H −

(U1ρ+ ρU†1

)U†1 H

= HU2ρ−H U2ρ+HρU†2 −HρU†2 − ρU

†2H + ρU†2 H − U2ρH + U2ρH

+HU1ρU†1 −HU1ρU

†1 −HU1ρU

†1 −H U1ρU

†1 + 2H U1ρU

†1 − U1ρU

†1H + U1ρU

†1H

+ U1ρU†1H + U1ρU

†1 H − 2U1ρU

†1 H −HU1 U1ρ+H U1 U1ρ−HρU†1U

†1 +HρU†1 U

†1

+ ρU†1 U†1H − ρU

†1 U†1 H + U1U1ρH − U1 U1ρH. (A.8)

Note that the terms in L2 and L3 follow a distinct pattern. The terms come in groups of four. One

term is an average of a product, another is the product of the averages and has opposite sign. Then

another pair of terms is simply the Hermitian adjoint of the first pair, and has opposite sign.

In L3, some terms have an averaging overline under another averaging overline, and so this notation

must be exlpained. In this case, the outer averaging line only applies to terms outside the inner one.

For example, in the term U1ρU†1H, the outer averaging operator only applies to the U1 and the H, since

the U†1 was already averaged by the inner averaging operator.

Appendix B

Addendum to Chapter 4: Measures

of Purity

B.1 Gell-Mann Matrices

The Gell-Mann matrices (Gi, i = 1, ..., 8) are the most widely used set of generators for the group of

special unitary 3× 3 matrices, SU(3) [89]. They are given by

G1 =

0 1 0

1 0 0

0 0 0

, G2 =

0 −i 0

i 0 0

0 0 0

, G3 =

1 0 0

0 −1 0

0 0 0

,

G4 =

0 0 1

0 0 0

1 0 0

, G5 =

0 0 −i0 0 0

i 0 0

, G6 =

0 0 0

0 0 1

0 1 0

,

G7 =

0 0 0

0 0 −i0 i 0

, G8 =1√3

1 0 0

0 1 0

0 0 −2

. (B.1)

They are all Hermitian, traceless, and satisfy the orthogonality relation Tr [GiGj ] = 2δij , where δij is

the Kronecker delta. However they are not unitary like the Pauli matrices.

B.2 Depolarizing Channels as a Criteria

Consider a depolarizing channel, an important type of quantum noise [51]. It is a transformation which

depolarizes the input quantum state (i.e. replaces it with I/N) with probability 1 − p, and leaves it

unchanged with probability p. The action of this channel on the density matrix is given by the following

superoperator [90]:

E(ρ) = (1− p) IN

+ pρ. (B.2)

74

Appendix B. Addendum to Chapter 4: Measures of Purity 75

Squaring eq. (B.2) and taking the trace, we have

Tr[E(ρ)2] =(1− p)2

N+

2p(1− p)N

+ p2 Tr[ρ2]

=1− p2

N+ p2 Tr[ρ2]. (B.3)

Applying the SSKF polarization measure of purity in eq. (4.41) to E(ρ), we have

Π2sskf (E(ρ)) =

N Tr[E(ρ)2]− 1

N − 1

=Np2 Tr[ρ2] + 1− p2 − 1

N − 1

= p2N Tr[ρ2]− 1

N − 1

= p2Π2sskf (ρ). (B.4)

where in the second line we made use of eq. (B.3). Taking the square root of both sides, we have the

simple result

Πsskf (E(ρ)) = pΠsskf (ρ). (B.5)

We observe that eq. (B.5) has a very simple and intuitive form, showing that the purity simply scales

down by a factor of p after the state passes through the depolarizing channel. This intuitive relationship

does not hold for other measures of purity, even ones whose partial derivatives have the same sign as

Πsskf , shown in the next section B.3. This suggests Πsskf is a measure with more physical meaning,

and more relevant whenever depolarizing channels are in effect, as they often are.

B.3 Partial Derivatives and Agreement of Purity Measures

The graphical comparison in section 4.4.1 shows that the measures Πb, Πsskf and Πv behave similarly;

their derivative always has the same sign, and one is tempted to conclude that they will always agree

which of any two states is more pure. It is of interest to ask if this will aways be true. To gain some

insight into this question, we first examine the signs of the partial derivatives of the various measures.

Assume we have an N dimensional state, with eigenvalues λ1, ..., λN , with the first N −1 eigenvalues

independent, and λN a dependent variable satisfying

λN = 1−N−1∑j=1

λj . (B.6)

Then by eq. (4.11), eq. (4.14), and eq. (4.28), we have

Πs =N [∑λ2j + (1−

∑λj)

2]− 1

N − 1, (B.7)

Πv = 1 +

∑λj log2 λj + (1−

∑λj) log2(1−

∑λj)

log2N, (B.8)

Π2b = 1−NNλ1λ2...λN−1(1−

∑λj), (B.9)

Appendix B. Addendum to Chapter 4: Measures of Purity 76

where all sums over j in this section run from 1 to N−1. We have squared Πb since it does not affect the

sign of the derivative, and makes the calculation more tractable. Also, since Πs = Π2sskf , the properties

we find for Πs will also apply to Πsskf .

Keeping in mind that ∂λN

∂λi= −1 for i = 1, ..., N , we can compute the partial derivatives ∂Πs

∂λi, ∂Πv

∂λi,

and∂Π2

b

∂λias follows:

∂Πs

∂λi=N [2λi − 2(1−

∑λj)]

N − 1

=2N [λi − λN ]

N − 1, (B.10)

∂Πv

∂λi=

1

log2N[log2 λi − log2(1−

∑λj)]

=1

log2N[log2 λi − log2 λN ], (B.11)

∂Π2b

∂λi= −NNλ1...λi−1λi+1...λN−1(1−

∑λj − λi)

= NNλ1...λi−1λi+1...λN−1(λi − λN ). (B.12)

Note that λi − λN will always have the same sign as log2 λi − log2 λN , since λi > λN ⇐⇒ log2 λi >

log2 λN . Therefore the derivatives ∂Πs

∂λiand ∂Πv

∂λimust have the same sign. Moreover, assuming none of

the eigenvalues are zero, we see that ∂Πs

∂λiand

∂Π2b

∂λiboth equal a positive number multiplied by λi − λN ,

and therefore will also have the same sign.

Therefore for any given point in the eigenvalue space, the partial derivatives of the measures Πs,

Πsskf , Πv and Πb will always have the same sign, yielding the similar graphical behaviour exhibited in

figure 4.2.

This implies that the purity measures above will behave similarly so long as we are varying only two

eigenvalues (λi and as a consequence, λN ). If we vary more eigenvalues simultaneously, then in general,

each measure behaves differently. In other words, if we use the aforementioned measures to compare the

purity of two quantum states, they will all agree which state is purer as long as the two states differ in

only two eigenvalues. If the two states differ in three or more eigenvalues, the measures will, in general,

not agree which is purer. This is clearly illustrated in table 4.1.

Appendix C

Addendum to Chapter 5: Circuits

for Simple Periodic Functions

In this appendix, we list the circuits and truth tables for simple periodic functions Sp, for odd p in the

range 13 ≤ p ≤ 31. The circuits in this appendix and their associated truth tables follow similar patterns

to the ones in chapter 5. However, some comments on a few of them are in order.

The case of S19, whose circuit is in fig. C.4 and associated truth table in table C.4, demonstrates

another interesting technique in circuit construction. The circuit starts with a Toffoli cascade on bits

y2, y3, and y4 that flip 8, 6, and 2 entries in the truth table respectively. Then another Toffoli gate

from outside the cascade acts on y4, flipping 8 entries, 2 of which were already flipped, resulting in an

effective flip of (8 − 2 =) 6 entries. Another Toffoli gate then acts within the initial cascade, with the

bit y5 as its target, and flipping half the number in the previous level of the cascade, i.e. half of 6, and

so 3 entries are flipped.

A pure Toffoli cascade can only flip a number of enties that is a power of 2 at each step. The technique

used in the S19 circuit demonstrates that one can interrupt the cascade with an ’independent’ Toffoli

gate to alter the number of entries flipped.

The S21 circuit in fig. C.5 uses the same technique, with the difference that the Toffoli gate that

intervenes in the cascade just adds to the number of flipped entries, so we have (8 + 2 =) 10 entries

flipped in the y4 column, and half that number, 5 entries flipped in the last level of the cascade in the

y5 column.

Interestingly, the circuits for S19 and S21 in figs. C.4 and C.5 respectively are almost identical, and

only differ in the control value of some Toffoli gates (some black-filled circles denoting control bits are

exchanged for white-filled ones, and vice versa).

77

Appendix C. Addendum to Chapter 5: Circuits for Simple Periodic Functions 78

x4 • • x4x3 • • • x3x2 • x2

x1 • x1

|0〉 y4

|0〉 • y3

|0〉 • y2

|0〉 • y1

Figure C.1: S13 quantum circuit.

x4 • • x4x3 • • x3x2 • • x2x1 • • x1

|0〉 y4

|0〉 • y3

|0〉 y2

|0〉 y1


x4 x3 x2 x1 y4 y3 y2 y1

0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 1

0 0 1 0 0 0 1 0

0 0 1 1 0 0 1 1

0 1 0 0 1 1 0 0

0 1 0 1 1 1 0 1

0 1 1 0 1 1 1 0

0 1 1 1 1 1 1 1

1 0 0 0 1 0 0 0

1 0 0 1 1 0 0 1

1 0 1 0 1 0 1 0

1 0 1 1 1 0 1 1

1 1 0 0 0 1 1 1

1 1 0 1 0 0 0 0

1 1 1 0 0 0 0 1

1 1 1 1 0 0 1 0

Table C.1: S13 truth table.

x4 x3 x2 x1 y4 y3 y2 y1

0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 1

0 0 1 0 0 0 1 0

0 0 1 1 0 0 1 1

0 1 0 0 0 1 0 0

0 1 0 1 0 1 0 1

0 1 1 0 0 1 1 0

0 1 1 1 0 1 1 1

1 0 0 0 1 0 0 0

1 0 0 1 1 0 0 1

1 0 1 0 1 0 1 0

1 0 1 1 1 0 1 1

1 1 0 0 1 1 0 1

1 1 0 1 1 1 1 0

1 1 1 0 1 1 1 1

1 1 1 1 0 0 0 0



x5 • • x5x4 • x4

x3 • x3

x2 • x2x1 • x1

|0〉 y5

|0〉 • y4

|0〉 • y3

|0〉 • y2

|0〉 • y1


x5 • • • • x5x4 • x4

x3 • x3x2 • • x2x1 • • x1

|0〉 y5

|0〉 • y4

|0〉 • y3

|0〉 • y2

|0〉 y1


x5 x4 x3 x2 x1 y5 y4 y3 y2 y1

0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 1

0 0 0 1 0 0 0 0 1 0

0 0 0 1 1 0 0 0 1 1

0 0 1 0 0 0 0 1 0 0

0 0 1 0 1 0 0 1 0 1

0 0 1 1 0 0 0 1 1 0

0 0 1 1 1 0 0 1 1 1

0 1 0 0 0 0 1 0 0 0

0 1 0 0 1 0 1 0 0 1

0 1 0 1 0 0 1 0 1 0

0 1 0 1 1 0 1 0 1 1

0 1 1 0 0 0 1 1 0 0

0 1 1 0 1 0 1 1 0 1

0 1 1 1 0 0 1 1 1 0

0 1 1 1 1 0 1 1 1 1

1 0 0 0 0 1 1 1 1 1

1 0 0 0 1 0 0 0 0 0

1 0 0 1 0 0 0 0 0 1

1 0 0 1 1 0 0 0 1 0

1 0 1 0 0 0 0 0 1 1

1 0 1 0 1 0 0 1 0 0

1 0 1 1 0 0 0 1 0 1

1 0 1 1 1 0 0 1 1 0

1 1 0 0 0 0 0 1 1 1

1 1 0 0 1 0 1 0 0 0

1 1 0 1 0 0 1 0 0 1

1 1 0 1 1 0 1 0 1 0

1 1 1 0 0 0 1 0 1 1

1 1 1 0 1 0 1 1 0 0

1 1 1 1 0 0 1 1 0 1

1 1 1 1 1 0 1 1 1 0


x5 x4 x3 x2 x1 y5 y4 y3 y2 y1

0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 1

0 0 0 1 0 0 0 0 1 0

0 0 0 1 1 0 0 0 1 1

0 0 1 0 0 0 0 1 0 0

0 0 1 0 1 0 0 1 0 1

0 0 1 1 0 0 0 1 1 0

0 0 1 1 1 0 0 1 1 1

0 1 0 0 0 0 1 0 0 0

0 1 0 0 1 0 1 0 0 1

0 1 0 1 0 0 1 0 1 0

0 1 0 1 1 0 1 0 1 1

0 1 1 0 0 0 1 1 0 0

0 1 1 0 1 0 1 1 0 1

0 1 1 1 0 0 1 1 1 0

0 1 1 1 1 0 1 1 1 1

1 0 0 0 0 1 1 1 0 1

1 0 0 0 1 1 1 1 1 0

1 0 0 1 0 1 1 1 1 1

1 0 0 1 1 0 0 0 0 0

1 0 1 0 0 0 0 0 0 1

1 0 1 0 1 0 0 0 1 0

1 0 1 1 0 0 0 0 1 1

1 0 1 1 1 0 0 1 0 0

1 1 0 0 0 0 0 1 0 1

1 1 0 0 1 0 0 1 1 0

1 1 0 1 0 0 0 1 1 1

1 1 0 1 1 0 1 0 0 0

1 1 1 0 0 0 1 0 0 1

1 1 1 0 1 0 1 0 1 0

1 1 1 1 0 0 1 0 1 1

1 1 1 1 1 0 1 1 0 0



x5 • • • • x5x4 • x4

x3 • • x3

x2 • x2

x1 • x1

|0〉 y5

|0〉 • y4

|0〉 • y3

|0〉 • y2

|0〉 y1


x5 • • • • x5x4 • • • • x4x3 • • x3x2 • • x2x1 • • x1

|0〉 • y5

|0〉 y4

|0〉 • y3

|0〉 • y2

|0〉 y1


x5 x4 x3 x2 x1 y5 y4 y3 y2 y1

0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 1

0 0 0 1 0 0 0 0 1 0

0 0 0 1 1 0 0 0 1 1

0 0 1 0 0 0 0 1 0 0

0 0 1 0 1 0 0 1 0 1

0 0 1 1 0 0 0 1 1 0

0 0 1 1 1 0 0 1 1 1

0 1 0 0 0 0 1 0 0 0

0 1 0 0 1 0 1 0 0 1

0 1 0 1 0 0 1 0 1 0

0 1 0 1 1 0 1 0 1 1

0 1 1 0 0 0 1 1 0 0

0 1 1 0 1 0 1 1 0 1

0 1 1 1 0 0 1 1 1 0

0 1 1 1 1 0 1 1 1 1

1 0 0 0 0 1 1 0 1 1

1 0 0 0 1 1 1 1 0 0

1 0 0 1 0 1 1 1 0 1

1 0 0 1 1 1 1 1 1 0

1 0 1 0 0 1 1 1 1 1

1 0 1 0 1 0 0 0 0 0

1 0 1 1 0 0 0 0 0 1

1 0 1 1 1 0 0 0 1 0

1 1 0 0 0 0 0 0 1 1

1 1 0 0 1 0 0 1 0 0

1 1 0 1 0 0 0 1 0 1

1 1 0 1 1 0 0 1 1 0

1 1 1 0 0 0 0 1 1 1

1 1 1 0 1 0 1 0 0 0

1 1 1 1 0 0 1 0 0 1

1 1 1 1 1 0 1 0 1 0


x5 x4 x3 x2 x1 y5 y4 y3 y2 y1

0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 1

0 0 0 1 0 0 0 0 1 0

0 0 0 1 1 0 0 0 1 1

0 0 1 0 0 0 0 1 0 0

0 0 1 0 1 0 0 1 0 1

0 0 1 1 0 0 0 1 1 0

0 0 1 1 1 0 0 1 1 1

0 1 0 0 0 1 1 0 0 0

0 1 0 0 1 1 1 0 0 1

0 1 0 1 0 1 1 0 1 0

0 1 0 1 1 1 1 0 1 1

0 1 1 0 0 1 1 1 0 0

0 1 1 0 1 1 1 1 0 1

0 1 1 1 0 1 1 1 1 0

0 1 1 1 1 1 1 1 1 1

1 0 0 0 0 1 0 0 0 1

1 0 0 0 1 1 0 0 1 0

1 0 0 1 0 1 0 0 1 1

1 0 0 1 1 1 0 1 0 0

1 0 1 0 0 1 0 1 0 1

1 0 1 0 1 1 0 1 1 0

1 0 1 1 0 1 0 1 1 1

1 0 1 1 1 0 0 0 0 0

1 1 0 0 0 0 0 0 0 1

1 1 0 0 1 0 0 0 1 0

1 1 0 1 0 0 0 0 1 1

1 1 0 1 1 0 0 1 0 0

1 1 1 0 0 0 0 1 0 1

1 1 1 0 1 0 0 1 1 0

1 1 1 1 0 0 0 1 1 1

1 1 1 1 1 1 1 0 0 0



x5 • • x5x4 • • • x4x3 • • x3x2 • x2

x1 • x1

|0〉 y5

|0〉 • y4

|0〉 • y3

|0〉 • y2

|0〉 • • y1


x5 • • x5x4 • • • x4x3 • • • • x3x2 • • x2x1 • • x1

|0〉 y5

|0〉 • y4

|0〉 y3

|0〉 • y2

|0〉 • • y1


x5 x4 x3 x2 x1 y5 y4 y3 y2 y1

0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 1

0 0 0 1 0 0 0 0 1 0

0 0 0 1 1 0 0 0 1 1

0 0 1 0 0 0 0 1 0 0

0 0 1 0 1 0 0 1 0 1

0 0 1 1 0 0 0 1 1 0

0 0 1 1 1 0 0 1 1 1

0 1 0 0 0 1 1 0 0 0

0 1 0 0 1 1 1 0 0 1

0 1 0 1 0 1 1 0 1 0

0 1 0 1 1 1 1 0 1 1

0 1 1 0 0 1 1 1 0 0

0 1 1 0 1 1 1 1 0 1

0 1 1 1 0 1 1 1 1 0

0 1 1 1 1 1 1 1 1 1

1 0 0 0 0 1 0 0 0 0

1 0 0 0 1 1 0 0 0 1

1 0 0 1 0 1 0 0 1 0

1 0 0 1 1 1 0 0 1 1

1 0 1 0 0 1 0 1 0 0

1 0 1 0 1 1 0 1 0 1

1 0 1 1 0 1 0 1 1 0

1 0 1 1 1 1 0 1 1 1

1 1 0 0 0 0 1 0 1 1

1 1 0 0 1 0 0 0 0 0

1 1 0 1 0 0 0 0 0 1

1 1 0 1 1 0 0 0 1 0

1 1 1 0 0 0 0 0 1 1

1 1 1 0 1 0 0 1 0 0

1 1 1 1 0 0 0 1 0 1

1 1 1 1 1 0 0 1 1 0


x5 x4 x3 x2 x1 y5 y4 y3 y2 y1

0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 1

0 0 0 1 0 0 0 0 1 0

0 0 0 1 1 0 0 0 1 1

0 0 1 0 0 0 1 1 0 0

0 0 1 0 1 0 1 1 0 1

0 0 1 1 0 0 1 1 1 0

0 0 1 1 1 0 1 1 1 1

0 1 0 0 0 1 1 0 0 0

0 1 0 0 1 1 1 0 0 1

0 1 0 1 0 1 1 0 1 0

0 1 0 1 1 1 1 0 1 1

0 1 1 0 0 1 0 1 0 0

0 1 1 0 1 1 0 1 0 1

0 1 1 1 0 1 0 1 1 0

0 1 1 1 1 1 0 1 1 1

1 0 0 0 0 1 0 0 0 0

1 0 0 0 1 1 0 0 0 1

1 0 0 1 0 1 0 0 1 0

1 0 0 1 1 1 0 0 1 1

1 0 1 0 0 1 1 1 0 0

1 0 1 0 1 1 1 1 0 1

1 0 1 1 0 1 1 1 1 0

1 0 1 1 1 1 1 1 1 1

1 1 0 0 0 0 1 0 0 1

1 1 0 0 1 0 1 0 1 0

1 1 0 1 0 0 1 0 1 1

1 1 0 1 1 0 0 0 0 0

1 1 1 0 0 0 0 0 0 1

1 1 1 0 1 0 0 0 1 0

1 1 1 1 0 0 0 0 1 1

1 1 1 1 1 0 1 1 0 0



x5 • • • x5x4 • • x4x3 • • • x3x2 • x2

x1 • x1

|0〉 y5

|0〉 y4

|0〉 • y3

|0〉 • y2

|0〉 • y1


x5 • • x5x4 • • x4x3 • • x3x2 • • x2x1 • • x1

|0〉 y5

|0〉 • y4

|0〉 y3

|0〉 y2

|0〉 y1


x5 x4 x3 x2 x1 y5 y4 y3 y2 y1

0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 1

0 0 0 1 0 0 0 0 1 0

0 0 0 1 1 0 0 0 1 1

0 0 1 0 0 0 1 1 0 0

0 0 1 0 1 0 1 1 0 1

0 0 1 1 0 0 1 1 1 0

0 0 1 1 1 0 1 1 1 1

0 1 0 0 0 1 0 0 0 0

0 1 0 0 1 1 0 0 0 1

0 1 0 1 0 1 0 0 1 0

0 1 0 1 1 1 0 0 1 1

0 1 1 0 0 1 1 1 0 0

0 1 1 0 1 1 1 1 0 1

0 1 1 1 0 1 1 1 1 0

0 1 1 1 1 1 1 1 1 1

1 0 0 0 0 1 1 0 0 0

1 0 0 0 1 1 1 0 0 1

1 0 0 1 0 1 1 0 1 0

1 0 0 1 1 1 1 0 1 1

1 0 1 0 0 1 0 1 0 0

1 0 1 0 1 1 0 1 0 1

1 0 1 1 0 1 0 1 1 0

1 0 1 1 1 1 0 1 1 1

1 1 0 0 0 0 1 0 0 1

1 1 0 0 1 0 1 0 0 0

1 1 0 1 0 0 1 0 1 1

1 1 0 1 1 0 1 0 1 0

1 1 1 0 0 0 0 1 1 1

1 1 1 0 1 0 0 0 0 0

1 1 1 1 0 0 0 0 0 1

1 1 1 1 1 0 0 0 1 0


x5 x4 x3 x2 x1 y5 y4 y3 y2 y1

0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 1

0 0 0 1 0 0 0 0 1 0

0 0 0 1 1 0 0 0 1 1

0 0 1 0 0 0 0 1 0 0

0 0 1 0 1 0 0 1 0 1

0 0 1 1 0 0 0 1 1 0

0 0 1 1 1 0 0 1 1 1

0 1 0 0 0 0 1 0 0 0

0 1 0 0 1 0 1 0 0 1

0 1 0 1 0 0 1 0 1 0

0 1 0 1 1 0 1 0 1 1

0 1 1 0 0 0 1 1 0 0

0 1 1 0 1 0 1 1 0 1

0 1 1 1 0 0 1 1 1 0

0 1 1 1 1 0 1 1 1 1

1 0 0 0 0 1 0 0 0 0

1 0 0 0 1 1 0 0 0 1

1 0 0 1 0 1 0 0 1 0

1 0 0 1 1 1 0 0 1 1

1 0 1 0 0 1 0 1 0 0

1 0 1 0 1 1 0 1 0 1

1 0 1 1 0 1 0 1 1 0

1 0 1 1 1 1 0 1 1 1

1 1 0 0 0 1 1 0 0 1

1 1 0 0 1 1 1 0 1 0

1 1 0 1 0 1 1 0 1 1

1 1 0 1 1 1 1 1 0 0

1 1 1 0 0 1 1 1 0 1

1 1 1 0 1 1 1 1 1 0

1 1 1 1 0 1 1 1 1 1

1 1 1 1 1 0 0 0 0 0


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