1 p 2 (|0i|0i + |1i|1i) Two-qubit entangled state A B 1 measurement (in Z-basis) 0 1 completely...

Markus Müller

Peter-Grünberg-Institut 2, Forschungszentrum Jülich Institute for Quantum Information, RWTH Aachen

www.rwth-aachen.de/mueller-group

Quantifying Spatial Correlations in General Quantum Dynamics

Autumn School on Correlated Electrons: Topology, Entanglement, and Strong Correlations

New J. Phys. 17 (2015) 062001 doi:10.1088/1367-2630/17/6/062001

FAST TRACK COMMUNICATION

Quantifying spatial correlations of general quantum dynamics

Ángel Rivas andMarkusMüllerDepartamento de Física Teórica I, UniversidadComplutense, E-28040Madrid, Spain

E-mail: [email protected]

Keywords: open quantum systems, spatially correlated dynamics, correlation quanti!ers

AbstractUnderstanding the role of correlations in quantum systems is both a fundamental challenge as well asof high practical relevance for the control ofmulti-particle quantum systems.Whereas a lot of researchhas been devoted to study the various types of correlations that can be present in the states of quantumsystems, in this workwe introduce a general and rigorousmethod to quantify the amount ofcorrelations in the dynamics of quantum systems. Using a resource-theoretical approach, weintroduce a suitable quanti!er and characterize the properties of correlated dynamics. Furthermore,we benchmark ourmethod by applying it to the paradigmatic case of two atomsweakly coupled to theelectromagnetic radiation !eld, and illustrate its potential use to detect and assess spatial noisecorrelations in quantumcomputing architectures.

1. Introduction

Quantum systems can display awide variety of dynamical behaviors, in particular depending on how the systemis affected by its coupling to the surrounding environment. One interesting feature which has attractedmuchattention is the presence ofmemory effects (non-Markovianity) in the time evolution. These typically arise forstrong enough coupling between the system and its environment, or when the environment is structured, suchthat the assumptions of thewell-knownweak-coupling limit [1–3] are no longer valid.Whereasmemory effects(or time correlations) can be present in any quantum system exposed to noise, another extremely relevantfeature, whichwewill focus on in this work, are correlations in the dynamics of different parts ofmulti-partitequantum systems. Since different parties of a partition are commonly, though not always, identi!edwithdifferent places in space, without loss of generality wewill in the following refer to these correlations betweendifferent subsystems of a larger system as spatial correlations.

Spatial correlations in the dynamics give rise to awide plethora of interesting phenomena ranging fromsuper-radiance [4] and super-decoherence [5] to sub-radiance [6] and decoherence-free subspaces [7–11].Moreover, clarifying the role of spatial correlations in the performance of a large variety of quantumprocesses,such as e.g. quantum error correction [12–17], photosynthesis and excitation transfer [18–28], dissipative phasetransitions [29–33] and quantummetrology [34] has been and still is an active area of research.

Along the last few years, numerousworks have aimed at quantifying up towhich extent quantumdynamicsdeviates from theMarkovian behavior, see e.g. [35–43].However,much less attention has been paid to developquanti!ers of spatial correlations in the dynamics, although someworks e.g. [44, 45] have addressed this issuefor some speci!cmodels. Thismay be partially due to thewell-known fact that undermany, though not allpractical circumstances, dynamical correlations can be detected by studying the time evolution of correlationfunctions of properly chosen observables A! and B! , acting respectively on the two partiesA andB of interest.For instance, in the context of quantum computing, sophisticatedmethods towitness the correlated character ofquantumdynamics, have been developed and implemented in the laboratory [45]. Indeed, any correlationC ( , )A B A B A B= ! " # $ ! #! #! ! ! ! ! ! detected during the time evolution of an initial product state,

A B! ! != " , witnesses the correlated character of the dynamics. However, note that there exist highlycorrelated dynamics, which cannot be realized by a combination of local processes, which do not generate any

OPEN ACCESS

RECEIVED

19March 2015

REVISED

14April 2015

ACCEPTED FOR PUBLICATION

29April 2015

PUBLISHED

18 June 2015

Content from this workmay be used under theterms of theCreativeCommonsAttribution 3.0licence.

Any further distribution ofthis workmustmaintainattribution to theauthor(s) and the title ofthework, journal citationandDOI.

© 2015 IOPPublishing Ltd andDeutsche PhysikalischeGesellschaft

My background and research areas

Rydberg atomsTrapped ions Trapped Rydberg ions

Rydberg excited Calcium Ions for quantum interactions

Warsaw – 08.03.2012

Innsbruck – Mainz – Nottingham

Igor Lesanovsky

Research Lines & Applications

Quantum Information ProcessingQuantum Gate Design

AMO System EngineeringMicroscopic modelling

SE

Quantum SimulationsQuantum Many-Body Systems

Reservoir Engineering Topological Quantum Matter

QEC and Fault-Tolerant QCQuantum Error Correction Protocols

Topological Quantum CodesDecoders

functions of statistical mechanical models with three-bodyinteractions and external magnetic fields. Section VI is de-voted to conclusions.

II. TOPOLOGICAL COLOR CODES

A. Construction

Let us start by recalling the notion of a topological colorcode in order to see what type of classical spin models weobtain from them with appropriate projections onto factor-ized quantum states and specific lattices.

A TCC, denoted by C, is a quantum stabilizer error cor-rection code constructed with a certain class of two-dimensional lattices called 2-colexes. The word !colex! is acontraction that stands for color complex, where complex isthe mathematical terminology for a rather general lattice. A2-colex, denoted by C2, is a 2D trivalent lattice which hasthree-colorable faces and is embedded in a compact surfaceof arbitrary topology such as a torus of genus g. A trivalentlattice is one for which three edges meet at every vertex. Theproperty of being three-colorable means that the faces !orplaquettes" of the lattice can be colored with these colors insuch a way that neighboring faces never have the same color.We select as colors red !r", green !g", and blue !b". An ex-ample of a 2-colex construction is shown in Fig. 1.

Edges can be colored according to the coloring of thefaces. In particular, we attach red color to the edges thatconnect red faces, and so on and so forth for the blue andgreen edges and faces. When studying higher dimensionalcolexes, it turns out that the coloring of the edges is the keyproperty of D-colexes: all the information about a D-colex isencoded in its 1-skeleton, i.e., the set of edges with its col-oring #11$.

Given a 2-colex !C2", a TCC !C" is constructed by placingone qubit at each vertex of the colored lattice. Let us denoteby V, E, and F the sets of vertices v, edges e, and faces f,respectively, of the given 2-colex. Then, the generators of thestabilizer group, denoted by S, are given by face operatorsonly. For each face f, they come into two types depending onwhether they are constructed with Pauli operators of X or Ztype,

Xf: = !v!f

Xv,

Zf: = !v!f

Zv, !1"

and there are no generators associated to lattice vertices. Forexample, a hexagonal lattice is an instance of a 2-colex, seeFig. 2. The operators for the face f displayed in the figuretake the form Xf=X1X2X3X4X5X6, Zf=Z1Z2Z3Z4Z5Z6.

We notice that for the purpose of this paper, we are deal-ing only with lattices of trivial topology, such as the plane orthe sphere. In these cases, it suffices to require the lattice tohave the following properties: !i" Each vertex has coordina-tion number 3; !ii" each face has even number of edges !ver-tices". These conditions guarantee that the stabilizers Xf andZf pairwise commute and the TCC state is unique for thetrivial topology. Nevertheless, we have kept the general defi-nition of a 2-colex in terms of three-colorability since we arereferring to previous works where nontrivial topologies areconsidered.

A given state %!c&!C is left trivially invariant under theaction of the face operators,

Xf%!c& = %!c&, Zf%!c& = %!c&, ! f ! F . !2"

An erroneous state %!&e is one that violates conditions !2" forsome set of face operators of either type. As the generatoroperators Xf ,Zf!S satisfy that they square to the identityoperator, !Xf"2=1= !Zf"2, !f!F, then an erroneous state isdetected by having a negative eigenvalue with respect tosome set of stabilizer generators: Xf%!&e=!%!&e and/orZf%!&e=!%!&e.

Interestingly enough, it is possible to construct a quantumlattice Hamiltonian Hc such that its ground state is degener-ate and corresponds to the TCC !C", while the erroneousstates are given by the spectrum of excitations of the Hamil-tonian #9$. Such Hamiltonian is constructed out of the gen-erators of the topological stabilizer group S,

Hc = ! 'f!F

!Xf + Zf" . !3"

The ground state of this Hamiltonian exhibits what is calleda topological order #12$, as opposed to a more standard orderbased on a spontaneous symmetry breaking mechanism. Oneof the signatures of that topological order is precisely the

FIG. 1. !Color online" An example of 2-colex. Both edges andfaces are three-colorable and they are colored in such a way thatgreen edges connect green faces !light gray" and so on and so forthfor red !medium gray" and blue !dark gray" edges and faces.

f

!

1 2

345

6

a bc

d

e f

FIG. 2. !Color online" A hexagonal lattice is an instance of2-colex. Numbered vertices belong to the face f. Vertices labeledwith letters correspond to the red string " displayed. " is an openstring, because it has an end point in a red face.

H. BOMBIN AND M. A. MARTIN-DELGADO PHYSICAL REVIEW A 77, 042322 !2008"

042322-2

Neutral atoms

Dynamics of Quantum Systems

Correlated vs Uncorrelated Dynamics

A B

Evolution of a part of a multipartite quantum system may depend on the evolution of the others

“Spatial” Correlations

Correlated Quantum DynamicsExamples:

Any multipartite quantum system with interaction among their parties, or with a common bath

Blatt group (Innsbruck, Austria)

Trapped Ions Biomolecules

Superconducting Qubits

Martinis group (UCSB, California)

Optical Lattices

Ye group (NIST, Colorado)

Fundamental processes and applications, e.g. performance of quantum information processing protocols

Atomic gases

Quantum Computers

and Simulators

Harnessing dynamical correlationsProtection of quantum states

Decoherence-Free Subspaces (DFS), Dynamical decoupling, quantum control

Quantum technologyEntanglement-based magnetometry,Quantum sensing

Outline of this lecture

1) Quantifying correlations in quantum states

2) Dynamics in closed and open quantum systems

3) Quantifying spatial correlations in quantum dynamics

4) Physical application: radiating atoms

5) Experimental noise characterisation in a quantum computer

6) Harnessing spatially correlated noise

System

Environment

First attempt: correlations of observables

Quantifying Correlations in Dynamics

C(XA, YB) = hXA ⌦ YBi � hXAihYBi

S

⇢A ⌦ ⇢B C

XA YB

If Correlated Dynamics

Not enough!

C(XA, YB) 6= 0<latexit sha1_base64="tzFBK1vzMyLvpsOmKSsEplRmH3A=">AAAB8nicbZDNTgIxFIU7/iL+MOrSTSMxwcSQGTTRJcLGJSbyY2Ay6ZQ70NDpjG3HhBBeRFdG3fkovoBvY8FZKHhWX+85Te65QcKZ0o7zZa2srq1vbOa28ts7u3sFe/+gpeJUUmjSmMeyExAFnAloaqY5dBIJJAo4tINRfea3H0EqFos7PU7Ai8hAsJBRos3Itwv1Use/Prv3a6c9AQ+ObxedsjMXXgY3gyLK1PDtz14/pmkEQlNOlOq6TqK9CZGaUQ7TfC9VkBA6IgPoGhQkAuVN5otP8UkYS6yHgOfv39kJiZQaR4HJREQP1aI3G/7ndVMdXnkTJpJUg6AmYrww5VjHeNYf95kEqvnYAKGSmS0xHRJJqDZXypv67mLZZWhVyu55uXJ7UazWskPk0BE6RiXkoktURTeogZqIohQ9ozf0bmnryXqxXn+iK1b25xD9kfXxDX17j1w=</latexit>

Introduce a quantifier to assess the amount of correlations in quantum dynamics in a general way, and independently of

specific underlying situations or models

No need for good guesses of test statesNo need for good guesses of test observables Quantitative measure - not a simple yes / no / I don’t know output

Wishlist


ALICEBOB

1p2(|0i|0i+ |1i|1i)

Example: Bell states (EPR pairs)

Correlations in quantum dynamicsstates

1p2(|0i|0i+ |1i|1i)

Two-qubit entangled state

A B1

measurement(in Z-basis)

0

1

completely random

measurement outcomes(+1 or -1)

50% - 50%

|0i and |1i

16.4 Markus Muller

classical correlations: think of a machine that with probability of 50% prepares both qubitsin |00iAB, and with 50% in |11iAB – the resulting measurement statistics would be the same.But what happens if the qubits of the Bell state (1) are measured in the X-basis instead, i.e.,the observable Pauli matrix X = |0ih1| + |1ih0| is measured? The Bell state can be equallywritten as

��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i

�/p2 denoting the eigenstates of X , X |±i = ± |±i. Thus, the mea-

surement outcomes for measurements in this different basis are also perfectly correlated! Thisfeature is a signature of the entangled nature of this two-qubit state: in fact, there exists nobasis, in which |�+iAB can be written as a product of single-qubit states | 1iA and | 2iB,|�+i 6= | 1iA ⌦ | 2iB – therefore, the two qubits are entangled.How can one then quantify the amount of correlations in a bi-partite quantum state? This canbe done by means of the quantum mutual information, which is a generalization of the Shannonmutual information in the classical case [12], which quantifies the mutual dependence betweentwo random variables. Let us start by considering the density matrix ⇢S describing the jointstate of a system S, which is composed of two parts A and B. The von Neumann entropy S(⇢S)

of the state ⇢S is defined byS(⇢S) := �Tr⇢S log(⇢S) (3)

The density operator ⇢S can be written in terms of its eigenstates | ii, ⇢S =P

i pi | iih i|,with pi � 0 and

Pi pi = 1. Then the expression for the von Neumann entropy reduces to

S(⇢S) = �P

i pi log pi. For the system S in a pure state | i, ⇢S = | i h |, and S(⇢S) = 0.The reduced density operators, associated to parts A and B of the composite system, ⇢S|A and⇢S|B, are obtained by performing the partial trace [12] over the respective complementary parts

⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)

For the Bell state of Eq. (1), the reduced density operators correspond to the fully mixed state

⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)

and thus the von Neumann entropies evaluate to S(⇢S|A) = S(⇢S|B) = log 2.Now, the quantum mutual information of a state ⇢S is given by

I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)

For the Bell state of Eq. (1) the quantum mutual information assumes its maximum value fora two-qubit system, I(⇢S) = 2 log 2, indicating that the Bell states are indeed maximally cor-related quantum states. On the other hand, for any product state, i.e., if ⇢S = ⇢S|A ⌦ ⇢S|B,the quantum mutual information vanishes, which indicates that the subsystems A and B areindependent. In order words, outcomes of local measurements on subsystems A and B are inthe latter case completely uncorrelated, and thus from measuring one subsystem no information

16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


Spatial Correlations in Quantum Dynamics 16.5

about the state of the other subsystem can be inferred. Thus, quantum mutual information, as itsclassical counterpart, indicates by how much knowing about one part of a larger system reducesthe uncertainty about the other part.Finally, how can correlations in quantum states be detected in practice, i.e., in an experiment?As discussed, they reveal themselves in correlations in the measurement statistics of suitablychosen observables: Non-vanishing values for correlation functions such as C(OA,OB) =

hOA⌦OBi�hOAihOBi signal the presence of correlations. For the example of the Bell state ofEq. (1), the choice of, e.g., OA = ZA and OB = ZB is suitable, since hZAi = TrA(ZA⇢S|A) = 0,and similarly for qubit B, whereas hZA ⌦ ZBi = +1. In contrast, for a product state such ase.g. |0iA ⌦ |0iB, the correlator expectation value vanishes, C(ZA, ZB) = 0.

2 Quantum dynamics

2.1 Closed- and open-system quantum dynamics

After this brief discussion about correlations that can be present in quantum states, let us turnour attention to quantum dynamics. General time evolution of a quantum system S, whichcan be coupled to an environment, can be described by quantum operations ES [3]. Here, wewill focus on completely positive and trace-preserving (CPT) maps, often also called Krausmaps, which map valid physical density matrices describing the state of the system S onto otherphysical density matrices

ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)

Here, the set of so-called Kraus operators {Ki} fulfillP

i K†iKi = 1S. Note that this includes

the case of time evolution in closed quantum systems, where ⇢S 7! US ⇢S U†S, i.e., one Kraus

operator corresponds to the unitary time evolution operator US and all other Kraus operatorsvanish.As an example for open-system dynamics let us briefly discuss dephasing dynamics of a singlequbit or spin-1/2 system. This dynamics is present in many physical systems, and it is a limitingfactor in almost all architectures that are being used for quantum processors. Such dynamicscan be generated for instance by fluctuating fields (e.g. magnetic background fields) in the lab.We can thus describe the dephasing process using a single fluctuating variable B(t), referred toin the following as effective magnetic field

HG(t) =1

2B(t)Z. (8)

For simplicity, we assume the random fluctuation in the values of the effective magnetic field toobey a Gaussian distribution P (B), which implies that

⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)




2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)


A B

1p2(|0i|0i+ |1i|1i)

1

0

1

completely random

measurement outcomes(+1 or -1)

50% - 50%

measurement(in Z-basis)

|0i and |1i




2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)


A B

1p2(|0i|0i+ |1i|1i)

1 measurement(Z-basis)

0

1 measurement(Z-basis)

0

0

0

Completely random, but perfectly correlated

+1/-1 outcomes




2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)

<latexit sha1_base64="AcCnB8rKunoCYg6WyzXCLtJl/t4=">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</latexit>

C(ZA, ZB) = hZA ⌦ ZBi � hZAihZBi = 1

1p2(|0i|0i+ |1i|1i) = 1p

2(|+i|+i+ |�i|�i)

Two-qubit entangled state<latexit sha1_base64="d9o0TjgtSUtHy2J6yaWdmNJvuko=">AAAB+3icbVBNS8NAEJ3Ur1q/Uj16WSyCIJRECnoRil48VrC20MSy2W7apZtN2N0oJe1P8eJBQbz6R7z5b9y2OWjrg4HHezPMzAsSzpR2nG+rsLK6tr5R3Cxtbe/s7tnl/XsVp5LQJol5LNsBVpQzQZuaaU7biaQ4CjhtBcPrqd96pFKxWNzpUUL9CPcFCxnB2khduzz2GgP2cOpJLPqcokvUtStO1ZkBLRM3JxXI0ejaX14vJmlEhSYcK9VxnUT7GZaaEU4nJS9VNMFkiPu0Y6jAEVV+Njt9go6N0kNhLE0JjWbq74kMR0qNosB0RlgP1KI3Ff/zOqkOL/yMiSTVVJD5ojDlSMdomgPqMUmJ5iNDMJHM3IrIAEtMtEmrZEJwF19eJq2zqluruu5trVK/yvMowiEcwQm4cA51uIEGNIHAEzzDK7xZY+vFerc+5q0FK585gD+wPn8Am/KTYg==</latexit>

|�+i =

16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


Entangled:<latexit sha1_base64="/fdEhtw1YcN09XeArKoohgNwupQ=">AAACI3icbZBNS8NAEIY3flu/oh69LBZBEEoignoQRC8eK1hbaEKYbCft0s0m7G6EUv0tXvwrXjwoiBcP/he3bRC/Xlh4eWaG2XnjXHBtPO/dmZqemZ2bX1isLC2vrK656xvXOisUwwbLRKZaMWgUXGLDcCOwlSuENBbYjPvno3rzBpXmmbwygxzDFLqSJ5yBsShyj2+DXPOIBwpkVyA9oQGIvAcRv/VKFnG6R4MYzQj6XzByq17NG4v+NX5pqqRUPXJfg07GihSlYQK0bvtebsIhKMOZwLtKUGjMgfWhi21rJaSow+H4xDu6Y0mHJpmyTxo6pt8nhpBqPUhj25mC6enftRH8r9YuTHIUDrnMC4OSTRYlhaAmo6O8aIcrZEYMrAGmuP0rZT1QwIxNtWJD8H+f/Nc092v+Qc33Lw+qp2dlHgtki2yTXeKTQ3JKLkidNAgj9+SRPJMX58F5cl6dt0nrlFPObJIfcj4+AdoapEQ=</latexit>

| ii = ↵i|0ii + �i|1iiwith

1

1 1

0

1

0

measurement(in X-basis)

measurement(in X-basis)

Completely random, but perfectly correlated

+1/-1 outcomes

A B

16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


<latexit sha1_base64="6eeALHEpyXyj4xGdxHAvtkAwabM=">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</latexit>

C(XA, XB) = hXA ⌦XBi � hXAihXBi = 1

How correlated is a bi-partite state?Quantum mutual information

generalises Shannon mutual information (quantifying dependence between two random variables)

S

16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


von Neumann entropy

16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)



S

16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


von Neumann entropy

16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


Quantum mutual information

16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


Reduced densityoperators


S

16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


von Neumann entropy

1p2(|0i|0i+ |1i|1i)

16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


Maximally entangled states

16.4 Markus Muller


��+↵AB

=1p2

⇣|00iAB + |11iAB

⌘=

1p2

⇣|++iAB + |��iAB

⌘(2)

with |±i =�|0i ± |1i







S(⇢S) = �P


⇢S|A = TrB(⇢S), ⇢S|B = TrA(⇢S). (4)


⇢S|A =1

2

⇣|0ih0|A + |1ih1|A

⌘, ⇢S|A =

1

2

⇣|0ih0|B + |1ih1|B

⌘, (5)


I(⇢S) = S(⇢S|A) + S(⇢S|B)� S(⇢S). (6)


For any uncorrelated state we have<latexit sha1_base64="iO4j+dJnqdUYtlCl3MMYNSCECRk=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRahXsquFPQiFL3oraK1he5Ssmm2Dc0mS5IVytK/4cWDgnj1x3jz35i2e9DWBwOP92aYmRcmnGnjut9OYWV1bX2juFna2t7Z3SvvHzxqmSpCW0RyqToh1pQzQVuGGU47iaI4Djlth6Prqd9+okozKR7MOKFBjAeCRYxgYyX/tuqroezdn6JLt1euuDV3BrRMvJxUIEezV/7y+5KkMRWGcKx113MTE2RYGUY4nZT8VNMEkxEe0K6lAsdUB9ns5gk6sUofRVLZEgbN1N8TGY61Hseh7YyxGepFbyr+53VTE10EGRNJaqgg80VRypGRaBoA6jNFieFjSzBRzN6KyBArTIyNqWRD8BZfXibts5pXr3neXb3SuMrzKMIRHEMVPDiHBtxAE1pAIIFneIU3J3VenHfnY95acPKZQ/gD5/MHwCaQqQ==</latexit>

I(⇢S) = 0

Quantum Dynamics




2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)




2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)




2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)

General time evaluation of a quantum system

Completely positive, trace-preserving (CPT) maps or Kraus maps

with a set of Kraus operators

Closed system dynamics

System

• Coupling to the environment causes decoherence:

pure coherent superposition state

|0�

|1�

dephasing dynamics

System

Environment

Open-system dynamics: dephasing channel

e.g. (effective) magnetic field fluctuations




2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)

16.6 Markus Muller

If one additionally assumes a stationary autocorrelation function of the noise source

⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)

and furthermore a �-correlation of the noise, one obtains that

⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)

Using these properties, one finds*Z t

0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)

where we have defined the dephasing rate � =⌦[B(0)]

2↵.

For an arbitrary initial (pure) state, ⇢(0) = | (0)ih (0)|, with | (0)i = ↵ |0i + � |1i, the stateat time t will be given by an average over the noise realizations

⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.

(13)This allows us to identify this process as the dephasing channel [12]

ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)

i.e., as a quantum operation with the two Kraus operators K0 =p1�p1 and K1 =

ppZ

and the identification p =12(1 � e

� 12�t). Thus, for long times (t ! 1, p ! 1/2), the initial

coherence (off-diagonal elements of the density matrix (13)) completely vanishes and the qubitends in an incoherent mixture of the computational basis states.

2.2 Detection of correlated dynamics

Let us now generalize our previous discussions in Sec. 1.2 about two qubits and consider ageneral bipartite quantum system S = AB undergoing some dynamics given by a completelypositive and trace preserving (CPT) map ES. Without loss of generality we will assume that thedimension of both subsystems A and B is the same, dim(HA) = dim(HB) = d, and thereforedS := dim(HS) = d

2. The dynamics ES is said to be uncorrelated with respect to the subsystemsA and B if it can be decomposed as ES = EA ⌦ EB, with individual CPT maps EA and EB actingon the subsystems A and B, respectively. Otherwise we call it correlated.Simple examples of correlated dynamics from the field of quantum information are, e.g., two-qubit entangling gates, such as the prototypical two-qubit controlled-NOT (or CNOT) gate [12]

CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)

which flips the state of the target qubit (B), |0i $ |1i, if and only if the control qubit (A) is inthe |1i state. For suitably chosen input product states, e.g. | (0)i = |+i1 ⌦ |0i2, this unitary




2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)

Superposition picks up

unknown, fluctuating

relative phases

16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


for Gaussian, stationary and delta-correlated noise

16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


<latexit sha1_base64="dqxasSVRYJKwNk+UAVIU29wvtgE=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoBeh6MVjC9YW2lA220m7drMJuxuhhP4CLx4UxKv/yJv/xm2bg7Y+GHi8N8PMvCARXBvX/XYKa+sbm1vF7dLO7t7+Qfnw6EHHqWLYYrGIVSegGgWX2DLcCOwkCmkUCGwH49uZ335CpXks780kQT+iQ8lDzqixUvO6X664VXcOskq8nFQgR6Nf/uoNYpZGKA0TVOuu5ybGz6gynAmclnqpxoSyMR1i11JJI9R+Nj90Ss6sMiBhrGxJQ+bq74mMRlpPosB2RtSM9LI3E//zuqkJr/yMyyQ1KNliUZgKYmIy+5oMuEJmxMQSyhS3txI2oooyY7Mp2RC85ZdXSfui6tWqntesVeo3eR5FOIFTOAcPLqEOd9CAFjBAeIZXeHMenRfn3flYtBacfOYY/sD5/AEisoz6</latexit>=

16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


<latexit sha1_base64="dqxasSVRYJKwNk+UAVIU29wvtgE=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoBeh6MVjC9YW2lA220m7drMJuxuhhP4CLx4UxKv/yJv/xm2bg7Y+GHi8N8PMvCARXBvX/XYKa+sbm1vF7dLO7t7+Qfnw6EHHqWLYYrGIVSegGgWX2DLcCOwkCmkUCGwH49uZ335CpXks780kQT+iQ8lDzqixUvO6X664VXcOskq8nFQgR6Nf/uoNYpZGKA0TVOuu5ybGz6gynAmclnqpxoSyMR1i11JJI9R+Nj90Ss6sMiBhrGxJQ+bq74mMRlpPosB2RtSM9LI3E//zuqkJr/yMyyQ1KNliUZgKYmIy+5oMuEJmxMQSyhS3txI2oooyY7Mp2RC85ZdXSfui6tWqntesVeo3eR5FOIFTOAcPLqEOd9CAFjBAeIZXeHMenRfn3flYtBacfOYY/sD5/AEisoz6</latexit>=

16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)





2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)

we identify the Kraus map

From the evaluation for an arbitrary single-qubit state

16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


with





2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)

16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


with

For a small time step

one recovers the quantum master equation in Lindblad form

Coherence

<latexit sha1_base64="mrZcSQGLnOgfgwgcut5oul6jE10=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6LHoxWML1hbaUDbbSbt2swm7E6GU/gIvHhTEq//Im//GbZuDtj4YeLw3w8y8MJXCkOd9O4W19Y3NreJ2aWd3b/+gfHj0YJJMc2zyRCa6HTKDUihskiCJ7VQji0OJrXB0O/NbT6iNSNQ9jVMMYjZQIhKckZUa1CtXvKo3h7tK/JxUIEe9V/7q9hOexaiIS2ZMx/dSCiZMk+ASp6VuZjBlfMQG2LFUsRhNMJkfOnXPrNJ3o0TbUuTO1d8TExYbM45D2xkzGpplbyb+53Uyiq6DiVBpRqj4YlGUSZcSd/a12xcaOcmxJYxrYW91+ZBpxslmU7Ih+Msvr5LWRdW/rPp+47JSu8nzKMIJnMI5+HAFNbiDOjSBA8IzvMKb8+i8OO/Ox6K14OQzx/AHzucPdkWNMQ==</latexit>

t

<latexit sha1_base64="xd/Gg8489cloW7z9wvQbmsxvpDY=">AAACCHicbVBNS8NAEN3Ur1q/oh4FWSxCvdSkFPQiFL14rGBtoSlls920SzebsDtRQunNi3/FiwcF8epP8Oa/cdvmoK0PBh7vzTAzz48F1+A431ZuaXlldS2/XtjY3NresXf37nSUKMoaNBKRavlEM8ElawAHwVqxYiT0BWv6w6uJ37xnSvNI3kIas05I+pIHnBIwUtc+jPEFdk8ruATYU7w/AKJU9IA9LgNIT7p20Sk7U+BF4makiDLUu/aX14toEjIJVBCt264TQ2dEFHAq2LjgJZrFhA5Jn7UNlSRkujOa/jHGx0bp4SBSpiTgqfp7YkRCrdPQN50hgYGe9ybif147geC8M+IyToBJOlsUJAJDhCeh4B5XjIJIDSFUcXMrpgOiCAUTXcGE4M6/vEialbJbLbvuTbVYu8zyyKMDdIRKyEVnqIauUR01EEWP6Bm9ojfryXqx3q2PWWvOymb20R9Ynz/XjZff</latexit>

p = 1/2(t ! 1)

<latexit sha1_base64="HK5POp9Q5GdsQpI2dkIEdjEqbYE=">AAACD3icbVA9SwNBEN3z2/gVtbRZDII24U4CWopaWEYwRsiFMLeZi4u7d+funBCO/AIb/4qNhYLY2tr5b9x8FH49GHi8N8PMvChT0pLvf3pT0zOzc/MLi6Wl5ZXVtfL6xqVNcyOwIVKVmqsILCqZYIMkKbzKDIKOFDajm5Oh37xDY2WaXFA/w7aGXiJjKYCc1CnvZDy0UuMtD2MDoggGRW0Q9kBr4LvhKSoCTnudcsWv+iPwvySYkAqboN4pf4TdVOQaExIKrG0FfkbtAgxJoXBQCnOLGYgb6GHL0QQ02nYxemfAd5zS5XFqXCXER+r3iQK0tX0duU4NdG1/e0PxP6+VU3zYLmSS5YSJGC+Kc8Up5cNseFcaFKT6joAw0t3KxTW4WMglWHIhBL9f/kua+9WgVg2C81rl6HiSxwLbYttslwXsgB2xM1ZnDSbYPXtkz+zFe/CevFfvbdw65U1mNtkPeO9fX3ybeA==</latexit>

p ' 1

4�(�t)

<latexit sha1_base64="HK5POp9Q5GdsQpI2dkIEdjEqbYE=">AAACD3icbVA9SwNBEN3z2/gVtbRZDII24U4CWopaWEYwRsiFMLeZi4u7d+funBCO/AIb/4qNhYLY2tr5b9x8FH49GHi8N8PMvChT0pLvf3pT0zOzc/MLi6Wl5ZXVtfL6xqVNcyOwIVKVmqsILCqZYIMkKbzKDIKOFDajm5Oh37xDY2WaXFA/w7aGXiJjKYCc1CnvZDy0UuMtD2MDoggGRW0Q9kBr4LvhKSoCTnudcsWv+iPwvySYkAqboN4pf4TdVOQaExIKrG0FfkbtAgxJoXBQCnOLGYgb6GHL0QQ02nYxemfAd5zS5XFqXCXER+r3iQK0tX0duU4NdG1/e0PxP6+VU3zYLmSS5YSJGC+Kc8Up5cNseFcaFKT6joAw0t3KxTW4WMglWHIhBL9f/kua+9WgVg2C81rl6HiSxwLbYttslwXsgB2xM1ZnDSbYPXtkz+zFe/CevFfvbdw65U1mNtkPeO9fX3ybeA==</latexit>

p ' 1

4�(�t)


<latexit sha1_base64="vUWwe94Ate7Zvncx8VHfE7aMX1c=">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</latexit>

⇢ ' ⇢(t+�t)� ⇢(t)

�t=

�

4(Z⇢Z† � 1

2{⇢, Z†Z})

<latexit sha1_base64="Ey8T+mlgamq4XWBViWc+x4dwkhc=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0mkUC9C0YvHitYW2lA22027dLMJuxOhhP4ELx4UxKt/yJv/xm2bg7Y+GHi8N8PMvCCRwqDrfjuFtfWNza3idmlnd2//oHx49GjiVDPeYrGMdSeghkuheAsFSt5JNKdRIHk7GN/M/PYT10bE6gEnCfcjOlQiFIyile6TK7dfrrhVdw6ySrycVCBHs1/+6g1ilkZcIZPUmK7nJuhnVKNgkk9LvdTwhLIxHfKupYpG3PjZ/NQpObPKgISxtqWQzNXfExmNjJlEge2MKI7MsjcT//O6KYaXfiZUkiJXbLEoTCXBmMz+JgOhOUM5sYQyLeythI2opgxtOiUbgrf88ippX1S9WtXz7mqVxnWeRxFO4BTOwYM6NOAWmtACBkN4hld4c6Tz4rw7H4vWgpPPHMMfOJ8/YMqNrg==</latexit>

p = 0

<latexit sha1_base64="xd/Gg8489cloW7z9wvQbmsxvpDY=">AAACCHicbVBNS8NAEN3Ur1q/oh4FWSxCvdSkFPQiFL14rGBtoSlls920SzebsDtRQunNi3/FiwcF8epP8Oa/cdvmoK0PBh7vzTAzz48F1+A431ZuaXlldS2/XtjY3NresXf37nSUKMoaNBKRavlEM8ElawAHwVqxYiT0BWv6w6uJ37xnSvNI3kIas05I+pIHnBIwUtc+jPEFdk8ruATYU7w/AKJU9IA9LgNIT7p20Sk7U+BF4makiDLUu/aX14toEjIJVBCt264TQ2dEFHAq2LjgJZrFhA5Jn7UNlSRkujOa/jHGx0bp4SBSpiTgqfp7YkRCrdPQN50hgYGe9ybif147geC8M+IyToBJOlsUJAJDhCeh4B5XjIJIDSFUcXMrpgOiCAUTXcGE4M6/vEialbJbLbvuTbVYu8zyyKMDdIRKyEVnqIauUR01EEWP6Bm9ojfryXqx3q2PWWvOymb20R9Ynz/XjZff</latexit>

p = 1/2(t ! 1)

Correlated quantum dynamicsS

16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


Focus on bi-partite systems

16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)


Example: 2-qubit entangling gate

<latexit sha1_base64="1gCmVYA07S/X/SzS7xpmuNpZTkw=">AAAB/nicbZDLSgMxFIbPeK31Nl52boJFEIQykYIui25cVrC20A4lk2ba0ExmSDJCnRZfxY0LBXHrc7jzbUzbWWjrD4GP/5zDOfmDRHBtPO/bWVpeWV1bL2wUN7e2d3bdvf17HaeKsjqNRayaAdFMcMnqhhvBmoliJAoEawSD60m98cCU5rG8M8OE+RHpSR5ySoy1Ou7hyGsrInuCobMRzrHjlryyNxVaBJxDCXLVOu5XuxvTNGLSUEG0bmEvMX5GlOFUsHGxnWqWEDogPdayKEnEtJ9Nrx+jE+t0URgr+6RBU/f3REYirYdRYDsjYvp6vjYx/6u1UhNe+hmXSWqYpLNFYSqQidEkCtTlilEjhhYIVdzeimifKEKNDaxoQ8DzX16ExnkZV8oY31ZK1as8jwIcwTGcAoYLqMIN1KAOFB7hGV7hzXlyXpx352PWuuTkMwfwR87nD1umlPo=</latexit>

|0i+ |1i

1p2(|0i|0i+ |1i|1i) = 1p

2(|+i|+i+ |�i|�i)

Outputs a (maximally)correlated state

Correlated quantum dynamics

<latexit sha1_base64="77sfXLcjrv9FhqD5QMKLBBA8cjM=">AAACEnicbZDNSgMxFIUz9a/Wv1GXboJFqAhlUgq6LLpxWcHaQmcomTTThmYyY5IRynRewY2v4saFgrh15c63MW1noa0HAh/n3svNPX7MmdKO820VVlbX1jeKm6Wt7Z3dPXv/4E5FiSS0RSIeyY6PFeVM0JZmmtNOLCkOfU7b/uhqWm8/UKlYJG71OKZeiAeCBYxgbayeXXEDiUmKstRV91KntSyrTBxXYjHgFJ5NUI6nPbvsVJ2Z4DKgHMogV7Nnf7n9iCQhFZpwrFQXObH2Uiw1I5xmJTdRNMZkhAe0a1DgkCovnV2UwRPj9GEQSfOEhjP390SKQ6XGoW86Q6yHarE2Nf+rdRMdXHgpE3GiqSDzRUHCoY7gNB7YZ5ISzccGMJHM/BWSITYRaRNiyYSAFk9ehnatiupVhG7q5cZlnkcRHIFjUAEInIMGuAZN0AIEPIJn8ArerCfrxXq3PuatBSufOQR/ZH3+AO4jnXw=</latexit>

1p2(|0i+ |1i)

Correlated quantum dynamics


gate creates (maximally) correlated output states such as the Bell state of Eq. (1), therefore theCNOT gate is clearly a correlated quantum dynamics!Similarly, spatially homogeneous or global (magnetic) field fluctuations, acting with the samestrength on a register of two or more qubits, described by a Hamiltonian

HG(t) =1

2B(t)

X

k

Zk (16)

result in spatially correlated dephasing dynamics. This dynamics ES on the qubit register can-not be described by a product of independent dephasing processes, ES 6= ⌦kEk, with Ek actingon the k-th qubit. It is left as an exercise to work out the generalization of Eq. (14) for thisscenario of correlated dephasing dynamics. Again, working with suitably chosen input states,e.g. | (0)i = ⌦k |+ik, should allow one to distinguish between spatially correlated and uncor-related dephasing.In fact, this idea holds true in general: any correlation C(OA,OB) = hOA ⌦OBi � hOAihOBidetected during the time evolution of an initial product state, ⇢S = ⇢A ⌦ ⇢B, witnesses thecorrelated character of the dynamics. However, for this to work, one needs to be lucky or havea priory knowledge about the dynamics and thereby be able to choose suitable observables andinput states, for which correlated dynamics generates non-vanishing correlations in the finalquantum state generated by the dynamics. Furthermore, note that there exist highly correlateddynamics, which cannot be realized by a combination of local processes, which however do notgenerate any such correlation. A simple example is the swap process between two parties. Suchdynamics can either act on internal degrees of freedom, induced, e.g., by the action of a swapgate acting on two qubits [12], or can correspond to (unwanted) external dynamics, caused,e.g., by correlated hopping of atoms in an optical lattice [19] or the melting of an ion Coulombcrystal and subsequent recooling dynamics with a possibly different rearrangement of particlesin trapped-ion architectures [20].

3 Rigorous quantifier for correlations in quantum dynamics

In light of this discussion, let us therefore now discuss a systematic and rigorous method to cap-ture and quantify spatial correlations in quantum dynamics, not requiring any a-priori knowl-edge or assumptions about the dynamics taking place on the composite quantum system.

3.1 Choi-Jamiołkowski isomorphism

The central tool of our construction is the Choi-Jamiołkowski isomorphism [21, 22, 12]. Thisis a one-to-one correspondence of a given quantum dynamics of a system to an equivalentrepresentation in the form of a quantum state in an enlarged Hilbert space. As we will see, thismapping will allow us to use tools developed for the quantification of correlations in quantumstates, as we discussed above in Sec. 1.2, for our purpose of quantifying correlations in thequantum dynamics taking place in the bi-partite system S = AB. For this mapping, consider a

Spatially correlated, global dephasing on an N-qubit register



HG(t) =1

2B(t)

X

k

Zk (16)






16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)




HG(t) =1

2B(t)

X

k

Zk (16)






Test state

… will build up correlations under the correlated dynamics

How correlated are these?

Real particle exchange,correlated hopping

+ + + + + + + + +

ShuttleSeparate/mergeSwap

e.g melting & re-crystalisation of ion Coulomb crystals

There are highly correlated quantum processes that do not create correlations

<latexit sha1_base64="Xut0dwM0uuScPh9QGwr0gNvBjgo=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK1hbaUDbbTbN0P8LuRiihf8GLBwXx6g/y5r9x0+agrQ8GHu/NMDMvSjkz1ve/vcra+sbmVnW7trO7t39QPzx6NCrThHaI4kr3ImwoZ5J2LLOc9lJNsYg47UaT28LvPlFtmJIPdprSUOCxZDEj2BbSQCdqWG/4TX8OtEqCkjSgRHtY/xqMFMkElZZwbEw/8FMb5lhbRjid1QaZoSkmEzymfUclFtSE+fzWGTpzygjFSruSFs3V3xM5FsZMReQ6BbaJWfYK8T+vn9n4OsyZTDNLJVksijOOrELF42jENCWWTx3BRDN3KyIJ1phYF0/NhRAsv7xKuhfN4LIZBPeXjdZNmUcVTuAUziGAK2jBHbShAwQSeIZXePOE9+K9ex+L1opXzhzDH3ifP7btjoA=</latexit>⇢

<latexit sha1_base64="i+6ECKXgP8Ej4XBX55yzw8dkczQ=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKexKIB6DXjxGMA9IljA7mU3GzGOZmRXCkn/w4kFBvPo93vwbJ8keNLGgoajqprsrSjgz1ve/vcLG5tb2TnG3tLd/cHhUPj5pG5VqQltEcaW7ETaUM0lblllOu4mmWEScdqLJ7dzvPFFtmJIPdprQUOCRZDEj2Dqp3TdsJPCgXPGr/gJonQQ5qUCO5qD81R8qkgoqLeHYmF7gJzbMsLaMcDor9VNDE0wmeER7jkosqAmzxbUzdOGUIYqVdiUtWqi/JzIsjJmKyHUKbMdm1ZuL/3m91MbXYcZkkloqyXJRnHJkFZq/joZMU2L51BFMNHO3IjLGGhPrAiq5EILVl9dJ56oa1KpBcF+rNG7yPIpwBudwCQHUoQF30IQWEHiEZ3iFN095L96797FsLXj5zCn8gff5AzR5j1w=</latexit>�<latexit sha1_base64="Xut0dwM0uuScPh9QGwr0gNvBjgo=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK1hbaUDbbTbN0P8LuRiihf8GLBwXx6g/y5r9x0+agrQ8GHu/NMDMvSjkz1ve/vcra+sbmVnW7trO7t39QPzx6NCrThHaI4kr3ImwoZ5J2LLOc9lJNsYg47UaT28LvPlFtmJIPdprSUOCxZDEj2BbSQCdqWG/4TX8OtEqCkjSgRHtY/xqMFMkElZZwbEw/8FMb5lhbRjid1QaZoSkmEzymfUclFtSE+fzWGTpzygjFSruSFs3V3xM5FsZMReQ6BbaJWfYK8T+vn9n4OsyZTDNLJVksijOOrELF42jENCWWTx3BRDN3KyIJ1phYF0/NhRAsv7xKuhfN4LIZBPeXjdZNmUcVTuAUziGAK2jBHbShAwQSeIZXePOE9+K9ex+L1opXzhzDH3ifP7btjoA=</latexit>⇢

<latexit sha1_base64="i+6ECKXgP8Ej4XBX55yzw8dkczQ=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKexKIB6DXjxGMA9IljA7mU3GzGOZmRXCkn/w4kFBvPo93vwbJ8keNLGgoajqprsrSjgz1ve/vcLG5tb2TnG3tLd/cHhUPj5pG5VqQltEcaW7ETaUM0lblllOu4mmWEScdqLJ7dzvPFFtmJIPdprQUOCRZDEj2Dqp3TdsJPCgXPGr/gJonQQ5qUCO5qD81R8qkgoqLeHYmF7gJzbMsLaMcDor9VNDE0wmeER7jkosqAmzxbUzdOGUIYqVdiUtWqi/JzIsjJmKyHUKbMdm1ZuL/3m91MbXYcZkkloqyXJRnHJkFZq/joZMU2L51BFMNHO3IjLGGhPrAiq5EILVl9dJ56oa1KpBcF+rNG7yPIpwBudwCQHUoQF30IQWEHiEZ3iFN095L96797FsLXj5zCn8gff5AzR5j1w=</latexit>�SWAP gate

=

USWAP(⇢⌦ �)U †SWAP = � ⌦ ⇢

Rigorous quantification of correlationsin quantum dynamics

Basic idea




2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)

<latexit sha1_base64="KLg17zP8hjPhPditBy3JHSXbP28=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6LHoxWMFawtNKJvNpF262Q27G6GE/g0vHhTEqz/Gm//GbZuDtj4YeLw3w8y8KONMG8/7dipr6xubW9Xt2s7u3v5B/fDoUctcUexQyaXqRUQjZwI7hhmOvUwhSSOO3Wh8O/O7T6g0k+LBTDIMUzIULGGUGCsFgWE8xiJQIzkd1Bte05vDXSV+SRpQoj2ofwWxpHmKwlBOtO77XmbCgijDKMdpLcg1ZoSOyRD7lgqSog6L+c1T98wqsZtIZUsYd67+nihIqvUkjWxnSsxIL3sz8T+vn5vkOiyYyHKDgi4WJTl3jXRnAbgxU0gNn1hCqGL2VpeOiCLU2JhqNgR/+eVV0r1o+pdN37+/bLRuyjyqcAKncA4+XEEL7qANHaCQwTO8wpuTOy/Ou/OxaK045cwx/IHz+QMhl5I2</latexit>

⇢

Quantum dynamics Unique quantum state

Apply correlation quantifiers for states

Quantification of correlatedness of

dynamics

Choi-Jamiołkowski Isomorphism (1972)

Man-Duen Choi Andrzej Jamiołkowskidynamics , ⇢CJ

State-evolution correspondence (one-to-one)

<latexit sha1_base64="X1WvaPZvUob9HlNRrBUfxvOhtg8=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoMeiF48tWltoQ9lsJ+3azSbsboQS+gu8eFAQr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjBx2nimGLxSJWnYBqFFxiy3AjsJMopFEgsB2Mb2Z++wmV5rG8N5ME/YgOJQ85o8ZKzbt+ueJW3TnIKvFyUoEcjX75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NApObPKgISxsiUNmau/JzIaaT2JAtsZUTPSy95M/M/rpia88jMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZkQ/CWX14l7YuqV6t6XrNWqV/neRThBE7hHDy4hDrcQgNawADhGV7hzXl0Xpx352PRWnDymWP4A+fzB0QgjRA=</latexit>

SPhysical system<latexit sha1_base64="rQsm5YSI9D2jDZkkk18zoEQcJBw=">AAAB/HicbVBNS8NAFHzxs9avWI9eFotQLyWRgl6EohePFa0ttKFsNtt26W4SdjdiCfkrXjwoiFd/iDf/jZs2B20dWBhm3uPNjh9zprTjfFsrq2vrG5ulrfL2zu7evn1QeVBRIgltk4hHsutjRTkLaVszzWk3lhQLn9OOP7nO/c4jlYpF4b2extQTeBSyISNYG2lgV/oC67EUacBEVrs7RZfBwK46dWcGtEzcglShQGtgf/WDiCSChppwrFTPdWLtpVhqRjjNyv1E0RiTCR7RnqEhFlR56Sx7hk6MEqBhJM0LNZqpvzdSLJSaCt9M5knVopeL/3m9RA8vvJSFcaJpSOaHhglHOkJ5EShgkhLNp4ZgIpnJisgYS0y0qatsSnAXv7xMOmd1t1F33dtGtXlV9FGCIziGGrhwDk24gRa0gcATPMMrvFmZ9WK9Wx/z0RWr2DmEP7A+fwDKaJQU</latexit>

dim(S) = d

<latexit sha1_base64="U59fOB8FFRp/8vmwlB1J2ly/WPA=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbRU0mkoMeiF4/1o7bQhrLZTtqlm03Y3Qgl9B948aAgXv1F3vw3btsctPXBwOO9GWbmBYng2rjut1NYWV1b3yhulra2d3b3yvsHjzpOFcMmi0Ws2gHVKLjEpuFGYDtRSKNAYCsYXU/91hMqzWP5YMYJ+hEdSB5yRo2V7u5Pe+WKW3VnIMvEy0kFcjR65a9uP2ZphNIwQbXueG5i/Iwqw5nASambakwoG9EBdiyVNELtZ7NLJ+TEKn0SxsqWNGSm/p7IaKT1OApsZ0TNUC96U/E/r5Oa8NLPuExSg5LNF4WpICYm07dJnytkRowtoUxxeythQ6ooMzackg3BW3x5mbTOq16t6nm3tUr9Ks+jCEdwDGfgwQXU4QYa0AQGITzDK7w5I+fFeXc+5q0FJ585hD9wPn8ApJuNQQ==</latexit>

S0‘copy’ of system S

<latexit sha1_base64="4H7Em4uEuAdKcyfIrwhJ5w5OSL8=">AAAB/XicbVBPS8MwHE3nvzn/VXf0EhzivIxWBnoRhl48TnRusJWRpukWlqQlSYVS5lfx4kFBvPo9vPltTLcedPNB4PHe78fv5fkxo0o7zrdVWlldW98ob1a2tnd29+z9gwcVJRKTDo5YJHs+UoRRQTqaakZ6sSSI+4x0/cl17ncfiVQ0Evc6jYnH0UjQkGKkjTS0qwOO9FjyLKB8Wr87OYWXwdCuOQ1nBrhM3ILUQIH20P4aBBFOOBEaM6RU33Vi7WVIaooZmVYGiSIxwhM0In1DBeJEedks/BQeGyWAYSTNExrO1N8bGeJKpdw3k3lUtejl4n9eP9HhhZdRESeaCDw/FCYM6gjmTcCASoI1Sw1BWFKTFeIxkghr01fFlOAufnmZdM8abrPhurfNWuuq6KMMDsERqAMXnIMWuAFt0AEYpOAZvII368l6sd6tj/loySp2quAPrM8fMQ6URQ==</latexit>

dim(S0) = d


S


S0




2 Quantum dynamics



ES : ⇢S 7! ES(⇢S) =X

i

Ki ⇢S K†i . (7)





HG(t) =1

2B(t)Z. (8)


⌧exp

±i

Z t

0

B(t0)dt

0��

= exp

"�1

2

*✓Z t

0

B(t0)dt

0◆2

+#. (9)

16.8 Markus Muller

Fig. 1: Schematics of the method. Left: the system S is prepared in a maximally entangled state|�SS0i with the auxiliary system S

0. This state is just a product of maximally entangled statesbetween AA

0 and BB0, see Eq. (17). Middle: the system undergoes some dynamics ES. Right:

if and only if this process is correlated with respect to A and B, the total system SS0 becomes

correlated with respect to the bipartition AA0|BB0. The degree of correlation of the dynamics

can then be measured by the normalized mutual information, see Eq. (20).

second d2-dimensional bipartite system S

0= A

0B

0, essentially a “copy” of system S. Next, let|�SS0i be the maximally entangled state between S and S

0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|k`iAB ⌦ |k`iA0B0 . (17)

Here, |ji denotes the state vector with 1 at the j-th position and zero elsewhere, i.e., the canon-ical basis in the d

2-dimensional Hilbert space of S and its “copy” S0. Similarly, |kiA and |liB

denote the canonical basis of the d-dimensional subsystems A, B, and A0, B

0. The Choi-Jamiołkowki representation of some CPT map ES on S is then given by the d

4-dimensionalquantum state

⇢CJS := ES ⌦ 1S0

�|�SS0ih�SS0 |

�. (18)

This means it is obtained by acting with the quantum operation ES on S, and the identity op-eration 1S0 on S

0, as shown schematically in the middle part of Fig. 1. The entire informationabout the dynamical process ES taking place in S is now contained in this unique state ⇢

CJS in

the enlarged d4-dimensional space.

To become familiar with the Choi-Jamiołkowki representation of a quantum process, it is a use-ful exercise to show that for a system S consisting of a single qubit, which undergoes dephasingdynamics as described by Eq. (14), the Choi-Jamiołkowki state (18) reads

⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

Choi-Jamiołkowski state

Maximally entangled state:<latexit sha1_base64="jFxafbGQshdUUq/huWowYH2GA8U=">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</latexit>

|�SS0i = 1pd

dX

j=1

|jjiSS0

<latexit sha1_base64="jFxafbGQshdUUq/huWowYH2GA8U=">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</latexit>

|�SS0i = 1pd

dX

j=1

|jjiSS0

Choi-Jamiołkowski Isomorphism

Example: Single-qubit dephasing channel


S


S0

1p2(|0i|0i+ |1i|1i)


|�SS0i = 1pd

dX

j=1

|jjiSS0 <latexit sha1_base64="dqxasSVRYJKwNk+UAVIU29wvtgE=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoBeh6MVjC9YW2lA220m7drMJuxuhhP4CLx4UxKv/yJv/xm2bg7Y+GHi8N8PMvCARXBvX/XYKa+sbm1vF7dLO7t7+Qfnw6EHHqWLYYrGIVSegGgWX2DLcCOwkCmkUCGwH49uZ335CpXks780kQT+iQ8lDzqixUvO6X664VXcOskq8nFQgR6Nf/uoNYpZGKA0TVOuu5ybGz6gynAmclnqpxoSyMR1i11JJI9R+Nj90Ss6sMiBhrGxJQ+bq74mMRlpPosB2RtSM9LI3E//zuqkJr/yMyyQ1KNliUZgKYmIy+5oMuEJmxMQSyhS3txI2oooyY7Mp2RC85ZdXSfui6tWqntesVeo3eR5FOIFTOAcPLqEOd9CAFjBAeIZXeHMenRfn3flYtBacfOYY/sD5/AEisoz6</latexit>=

16.6 Markus Muller


⌦B(t+⌧)B(t)

↵=

⌦B(⌧)B(0)

↵, (10)


⌦B(⌧)B(0)

↵=

⌦[B(0)]

2↵�(⌧). (11)


0

B(t0)dt

0�2+

=⌦[B(0)]

2↵t = �t, (12)


2↵.


⇢(t) =

Z| (t)ih (t)|P (B) dB = |↵|2 |0ih0|+ |�|2 |1ih1|+ e

� 12�t

�↵�

⇤ |0ih1|+ ↵⇤� |1ih0|

�.


ES : ⇢S 7! (1�p)⇢S + pZ⇢SZ, (14)


ppZ







CNOT = |0ih0|A ⌦ 1B + |1ih1|A ⌦XB (15)



S


S0


|�SS0i = 1pd

dX

j=1

|jjiSS0

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

<latexit sha1_base64="lz+o3aTqZuCRNbMC5dvnUKIwOBs=">AAAB+3icbVBNS8NAFHzxs9avVI9eFotQLyUpBb0IRS8eK1pbaEPZbLbt0t0k7G6UEvtTvHhQEK/+EW/+GzdtDto6sDDMvMebHT/mTGnH+bZWVtfWNzYLW8Xtnd29fbt0cK+iRBLaIhGPZMfHinIW0pZmmtNOLCkWPqdtf3yV+e0HKhWLwjs9iakn8DBkA0awNlLfLvUE1iMp0oCJaeX29KLWt8tO1ZkBLRM3J2XI0ezbX70gIomgoSYcK9V1nVh7KZaaEU6nxV6iaIzJGA9p19AQC6q8dBZ9ik6MEqBBJM0LNZqpvzdSLJSaCN9MZkHVopeJ/3ndRA/OvZSFcaJpSOaHBglHOkJZDyhgkhLNJ4ZgIpnJisgIS0y0aatoSnAXv7xM2rWqW6+67k293LjM+yjAERxDBVw4gwZcQxNaQOARnuEV3qwn68V6tz7moytWvnMIf2B9/gAir5O4</latexit>

dim(S) = 2

Construction of the correlation measure

Resource-theory approach: Correlations as a resource

Correlated dynamics = resource to perform whatever task which can’t be implemented solely by (composing) uncorrelated dynamics


3.2 Construction of a correlation measure

In order to formulate a faithful measure of spatial correlations for dynamics, we adopt a re-source theoretic approach (see, e.g., [23, 24] where this approach is used in the context ofentanglement theory). The idea is that one may consider correlated dynamics as a resourceto perform whatever task that cannot be implemented solely by (composing) uncorrelated evo-lutions EA ⌦ EB. Then, suppose that the system S undergoes some dynamics given by themap ES. Now, consider the (left and right) composition of ES with some uncorrelated mapsLA ⌦ LB and RA ⌦ RB, which act before and after ES, so that the total dynamics is given byE 0S = (LA ⌦ LB)ES(RA ⌦RB). It is clear that any task that we can do with E 0

S by compositionwith uncorrelated maps can also be achieved with ES by composition with uncorrelated maps.Hence, we assert that the amount of correlation in ES is at least as large as in E 0

S. In other words,the amount of correlations of some dynamics does not increase under composition with uncor-related dynamics. This is the fundamental law of this resource theory of spatial correlations fordynamics, and any faithful measure of correlations should satisfy it. For the sake of comparison,in the resource theory of entanglement, entanglement is the resource, and the fundamental lawis that entanglement cannot increase under application of local operations and classical commu-nication (LOCC) [23]. For example, the entangled state of Eq. (1) can be transformed via thelocal unitary XB on qubit B into another Bell state 1p

2(|01iAB+ |10iAB), with the same amount

of entanglement. However, a product state of two qubits, not having any entanglement, cannotbe transformed into an entangled state by local operations such as single-qubit gate operationsor local measurements on A and B, or classical communication between the two single-qubitsubsystems A and B. In this spirit, we introduce a measure of correlations for dynamics [25]via the (normalized) quantum mutual information of the Choi-Jamiołkowski state ⇢CJ

S , Eq. (18),

I(ES) :=I(⇢

CJS )

4 log d:=

1

4 log d

⇣S�⇢CJS |AA0

�+ S

�⇢CJS |BB0

�� S

�⇢CJS

� ⌘. (20)

Here, S(·) := �Tr[(·) log(·)] is again the von Neumann entropy, now evaluated for the reduceddensity operators ⇢

CJS |AA0 := TrBB0(⇢

CJS ) and ⇢

CJS |BB0 := TrAA0(⇢

CJS ); see Fig. 1. In essence,

here we apply the quantum mutual information and von Neumann entropy we have seen inSec. 1.2 for quantum states, now to the Choi-Jamiołkowski state, which is equivalent to thequantum dynamics taking place on system S.But why is the quantity I(ES) a good and faithful measure of how correlated the dynamics givenby ES is? The reason is that it satisfies the following desired properties:

i) The quantity I(ES) = 0 if and only if ES corresponds to uncorrelated dynamics, ES =

EA ⌦ EB. This follows from the fact that the Choi-Jamiołkowski state of an uncorrelatedmap is a product state with respect to the bipartition AA

0|BB0 (no proof given here).

ii) The quantity I(ES) 2 [0, 1]. It is clear that I(ES) � 0, moreover it reaches its maximumvalue when S(⇢

CJS ) is minimal and S

�⇢CJS |AA0

�+S

�⇢CJS |BB0

�is maximal. Both conditions

are met when ⇢CJS is a maximally entangled state with respect to the bipartition AA

0|BB0,leading to I(⇢

CJS ) = 2 log d

2.








S , Eq. (18),

I(ES) :=I(⇢

CJS )

4 log d:=

1

4 log d

⇣S�⇢CJS |AA0

�+ S

�⇢CJS |BB0

�� S

�⇢CJS

� ⌘. (20)



CJS ) and ⇢









�⇢CJS |AA0

�+S

�⇢CJS |BB0




CJS ) = 2 log d

2.

Fundamental law of the resource theoryThe amount of correlations of some dynamics does not increase under composition with uncorrelated dynamics

<latexit sha1_base64="f0xR//isDbyKLJGfSQ8xps9Zh1Q=">AAACAHicbVBPS8MwHE3nvzn/VQUvXoJD8DRaGehxKILHic4NtlLSLN3CkrQkqTBqD34VLx4UxKsfw5vfxrTrQTcfBB7v/X78Xl4QM6q043xblaXlldW16nptY3Nre8fe3btXUSIx6eCIRbIXIEUYFaSjqWakF0uCeMBIN5hc5n73gUhFI3GnpzHxOBoJGlKMtJF8+2DAkR5jxNKrzC+45Olt5tt1p+EUgIvELUkdlGj79tdgGOGEE6ExQ0r1XSfWXoqkppiRrDZIFIkRnqAR6RsqECfKS4v8GTw2yhCGkTRPaFiovzdSxJWa8sBM5gnVvJeL/3n9RIfnXkpFnGgi8OxQmDCoI5iXAYdUEqzZ1BCEJTVZIR4jibA2ldVMCe78lxdJ97ThNhuue9Osty7KPqrgEByBE+CCM9AC16ANOgCDR/AMXsGb9WS9WO/Wx2y0YpU7++APrM8fVTGWwg==</latexit>

ES<latexit sha1_base64="f0xR//isDbyKLJGfSQ8xps9Zh1Q=">AAACAHicbVBPS8MwHE3nvzn/VQUvXoJD8DRaGehxKILHic4NtlLSLN3CkrQkqTBqD34VLx4UxKsfw5vfxrTrQTcfBB7v/X78Xl4QM6q043xblaXlldW16nptY3Nre8fe3btXUSIx6eCIRbIXIEUYFaSjqWakF0uCeMBIN5hc5n73gUhFI3GnpzHxOBoJGlKMtJF8+2DAkR5jxNKrzC+45Olt5tt1p+EUgIvELUkdlGj79tdgGOGEE6ExQ0r1XSfWXoqkppiRrDZIFIkRnqAR6RsqECfKS4v8GTw2yhCGkTRPaFiovzdSxJWa8sBM5gnVvJeL/3n9RIfnXkpFnGgi8OxQmDCoI5iXAYdUEqzZ1BCEJTVZIR4jibA2ldVMCe78lxdJ97ThNhuue9Osty7KPqrgEByBE+CCM9AC16ANOgCDR/AMXsGb9WS9WO/Wx2y0YpU7++APrM8fVTGWwg==</latexit>

ES<latexit sha1_base64="o9NkN0oygFafR5GSITEJ9uQuKu0=">AAACAHicbVBPS8MwHE3nvzn/VQUvXoJD8DRaGehx6sWDhwnODbZS0izdwpK0JKkwag9+FS8eFMSrH8Ob38a060E3HwQe7/1+/F5eEDOqtON8W5Wl5ZXVtep6bWNza3vH3t27V1EiMengiEWyFyBFGBWko6lmpBdLgnjASDeYXOV+94FIRSNxp6cx8TgaCRpSjLSRfPtgwJEeY8TSm8wvuOTpRebbdafhFICLxC1JHZRo+/bXYBjhhBOhMUNK9V0n1l6KpKaYkaw2SBSJEZ6gEekbKhAnykuL/Bk8NsoQhpE0T2hYqL83UsSVmvLATOYJ1byXi/95/USH515KRZxoIvDsUJgwqCOYlwGHVBKs2dQQhCU1WSEeI4mwNpXVTAnu/JcXSfe04TYbrnvbrLcuyz6q4BAcgRPggjPQAtegDToAg0fwDF7Bm/VkvVjv1sdstGKVO/vgD6zPH0S8lrc=</latexit>

LA

<latexit sha1_base64="u4gVXt8IyuCgjd84RkvtMQj2tkE=">AAACAHicbVBPS8MwHE3nvzn/VQUvXoJD8DRaGehxzIsHDxOcG2ylpFm6hSVpSVJh1B78Kl48KIhXP4Y3v41p14NuPgg83vv9+L28IGZUacf5tiorq2vrG9XN2tb2zu6evX9wr6JEYtLFEYtkP0CKMCpIV1PNSD+WBPGAkV4wvcr93gORikbiTs9i4nE0FjSkGGkj+fbRkCM9wYilN5lfcMnTdubbdafhFIDLxC1JHZTo+PbXcBThhBOhMUNKDVwn1l6KpKaYkaw2TBSJEZ6iMRkYKhAnykuL/Bk8NcoIhpE0T2hYqL83UsSVmvHATOYJ1aKXi/95g0SHl15KRZxoIvD8UJgwqCOYlwFHVBKs2cwQhCU1WSGeIImwNpXVTAnu4peXSe+84TYbrnvbrLfaZR9VcAxOwBlwwQVogWvQAV2AwSN4Bq/gzXqyXqx362M+WrHKnUPwB9bnD0ZClrg=</latexit>

LB

<latexit sha1_base64="T4+4uERKU7NOwcfVoidSupwW2Os=">AAACAHicbVBPS8MwHE3nvzn/VQUvXoJD8DRaGehx6sXjFOcGWylplm5hSVqSVBi1B7+KFw8K4tWP4c1vY9r1oJsPAo/3fj9+Ly+IGVXacb6tytLyyupadb22sbm1vWPv7t2rKJGYdHDEItkLkCKMCtLRVDPSiyVBPGCkG0yucr/7QKSikbjT05h4HI0EDSlG2ki+fTDgSI8xYult5hdc8vQi8+2603AKwEXilqQOSrR9+2swjHDCidCYIaX6rhNrL0VSU8xIVhskisQIT9CI9A0ViBPlpUX+DB4bZQjDSJonNCzU3xsp4kpNeWAm84Rq3svF/7x+osNzL6UiTjQReHYoTBjUEczLgEMqCdZsagjCkpqsEI+RRFibymqmBHf+y4uke9pwmw3XvWnWW5dlH1VwCI7ACXDBGWiBa9AGHYDBI3gGr+DNerJerHfrYzZascqdffAH1ucPTiKWvQ==</latexit>RA

<latexit sha1_base64="/wIkLWs9wl9L/IGPtW/1cuwvwLk=">AAACAHicbVBPS8MwHE3nvzn/VQUvXoJD8DRaGehxzIvHKc4NtlLSLN3CkrQkqTBqD34VLx4UxKsfw5vfxrTrQTcfBB7v/X78Xl4QM6q043xblZXVtfWN6mZta3tnd8/eP7hXUSIx6eKIRbIfIEUYFaSrqWakH0uCeMBIL5he5X7vgUhFI3GnZzHxOBoLGlKMtJF8+2jIkZ5gxNLbzC+45Gk78+2603AKwGXilqQOSnR8+2s4inDCidCYIaUGrhNrL0VSU8xIVhsmisQIT9GYDAwViBPlpUX+DJ4aZQTDSJonNCzU3xsp4krNeGAm84Rq0cvF/7xBosNLL6UiTjQReH4oTBjUEczLgCMqCdZsZgjCkpqsEE+QRFibymqmBHfxy8ukd95wmw3XvWnWW+2yjyo4BifgDLjgArTANeiALsDgETyDV/BmPVkv1rv1MR+tWOXOIfgD6/MHT6iWvg==</latexit>RB

‘Correlatedness’

<latexit sha1_base64="Sd6WvNk4kGhKEtMmuHC56Rkip2Y=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoMeiF48VrC20oWy2k3bpbhJ3N0IJ/QtePCiIV3+QN/+NmzYHbX0w8Hhvhpl5QSK4Nq777ZTW1jc2t8rblZ3dvf2D6uHRg45TxbDNYhGrbkA1Ch5h23AjsJsopDIQ2AkmN7nfeUKleRzdm2mCvqSjiIecUZNL/RE+Dqo1t+7OQVaJV5AaFGgNql/9YcxSiZFhgmrd89zE+BlVhjOBs0o/1ZhQNqEj7FkaUYnaz+a3zsiZVYYkjJWtyJC5+nsio1LrqQxsp6RmrJe9XPzP66UmvPIzHiWpwYgtFoWpICYm+eNkyBUyI6aWUKa4vZWwMVWUGRtPxYbgLb+8SjoXda9R97y7Rq15XeRRhhM4hXPw4BKacAstaAODMTzDK7w50nlx3p2PRWvJKWaO4Q+czx+kmI50</latexit>�

Construction of the correlation measure

Resource-theory approach: Correlations as a resource








S , Eq. (18),

I(ES) :=I(⇢

CJS )

4 log d:=

1

4 log d

⇣S�⇢CJS |AA0

�+ S

�⇢CJS |BB0

�� S

�⇢CJS

� ⌘. (20)



CJS ) and ⇢









�⇢CJS |AA0

�+S

�⇢CJS |BB0




CJS ) = 2 log d

2.

Fundamental law of the resource theory

Correlated dynamics = resource to perform whatever task which can’t be implemented solely by (composing) uncorrelated dynamics








S , Eq. (18),

I(ES) :=I(⇢

CJS )

4 log d:=

1

4 log d

⇣S�⇢CJS |AA0

�+ S

�⇢CJS |BB0

�� S

�⇢CJS

� ⌘. (20)



CJS ) and ⇢









�⇢CJS |AA0

�+S

�⇢CJS |BB0




CJS ) = 2 log d

2.

The amount of correlations of some dynamics does not increase under composition with uncorrelated dynamics

I(E) � I(E 0) par!al orderGood Quantifier:

*L. Li, K. Bu and Z.-W. Liu, Quantifying the resource content of quantum channels: An operational approach, arXiv:1812.02572*Y. Liu and X. Yuan, Operational Resource Theory of Quantum Channels, arXiv:1904.02680*Z.-W. Liu and A. Winter, Resource theories of quantum channels and the universal role of resource erasure, arXiv:1904.04201.

Resource theory of entanglement

ALICE BOBXA YB

Entanglement as a resource …

does not increaseunder…

… Local Operations …

… and Classical Communication

(LOCC paradigm)

1p2(|0i|0i+ |1i|1i)

<latexit sha1_base64="TAvfGHkdlXRxnBbfh24/cIEjaEo=">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</latexit>

! 1p2(|1i|0i+ |0i|1i)

<latexit sha1_base64="TAvfGHkdlXRxnBbfh24/cIEjaEo=">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</latexit>

! 1p2(|1i|0i+ |0i|1i)<latexit sha1_base64="d/yWPoAANxo+JK8dK1UNY4v/Z1M=">AAAB9HicbVBNSwMxFHxbv2r9qnr0EiyCp7IrBT1WvXis4NpCdynZNNuGJtklyQpl6d/w4kFBvPpjvPlvzLZ70NaBwDDzHm8yUcqZNq777VTW1jc2t6rbtZ3dvf2D+uHRo04yRahPEp6oXoQ15UxS3zDDaS9VFIuI0240uS387hNVmiXywUxTGgo8kixmBBsrBf4gENiMlcivZ4N6w226c6BV4pWkASU6g/pXMExIJqg0hGOt+56bmjDHyjDC6awWZJqmmEzwiPYtlVhQHebzzDN0ZpUhihNlnzRorv7eyLHQeioiO1kk1MteIf7n9TMTX4U5k2lmqCSLQ3HGkUlQUQAaMkWJ4VNLMFHMZkVkjBUmxtZUsyV4y19eJd2Lptdqet59q9G+Kfuowgmcwjl4cAltuIMO+EAghWd4hTcnc16cd+djMVpxyp1j+APn8wfJhpH9</latexit>

UA

Correlation measure for dynamics

S

Bi-partite system:

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

Maximally entangled state:

<latexit sha1_base64="9Y816/kLUVSmc2iLOKEuyA6c0EE=">AAACD3icbZDNSgMxFIUz9a/Wv1GXboKlUDdlRgq6EWrduKxgbaEdSiaTaUOTzJBkhDL0Cdz4Km5cKIhbt+58GzPtLGzrgcDhu/eSe48fM6q04/xYhbX1jc2t4nZpZ3dv/8A+PHpQUSIxaeOIRbLrI0UYFaStqWakG0uCuM9Ixx/fZPXOI5GKRuJeT2LicTQUNKQYaYMGdqXPkR5JngaUT6vXZ/AKLpBmRoKBXXZqzkxw1bi5KYNcrYH93Q8inHAiNGZIqZ7rxNpLkdQUMzIt9RNFYoTHaEh6xgrEifLS2TlTWDEkgGEkzRMazujfiRRxpSbcN53Zqmq5lsH/ar1Eh5deSkWcaCLw/KMwYVBHMMsGBlQSrNnEGIQlNbtCPEISYW0SLJkQ3OWTV03nvObWa657Vy83mnkeRXACTkEVuOACNMAtaIE2wOAJvIA38G49W6/Wh/U5by1Y+cwxWJD19Qu0GpsH</latexit>

dim(A) = dim(B) = d

<latexit sha1_base64="58HK0EXcVsRnVHAYmpPh567Z05w=">AAAB/HicbVBNS8NAEN34WetXrEcvi0X0VBIp6LHVi8cK1haaEDbbabt0swm7G7HE/BUvHhTEqz/Em//GbZuDtj4YeLw3w8y8MOFMacf5tlZW19Y3Nktb5e2d3b19+6Byr+JUUmjTmMeyGxIFnAloa6Y5dBMJJAo5dMLx9dTvPIBULBZ3epKAH5GhYANGiTZSYFeevNaIBVmzeZp7koghh8CuOjVnBrxM3IJUUYFWYH95/ZimEQhNOVGq5zqJ9jMiNaMc8rKXKkgIHZMh9AwVJALlZ7Pbc3xilD4exNKU0Him/p7ISKTUJApNZ0T0SC16U/E/r5fqwaWfMZGkGgSdLxqkHOsYT4PAfSaBaj4xhFDJzK2YjogkVJu4yiYEd/HlZdI5r7n1muve1quNqyKPEjpCx+gMuegCNdANaqE2ougRPaNX9Gbl1ov1bn3MW1esYuYQ/YH1+QNDjpRm</latexit>

|�AA0i <latexit sha1_base64="j3WPBKKREFVr4vI3A0wPnu8QCM8=">AAAB/HicbVBNS8NAEN3Ur1q/Yj16WSyip5JIQY+lXjxWsLbQhrDZTtqlm03Y3Ygl5q948aAgXv0h3vw3btsctPXBwOO9GWbmBQlnSjvOt1VaW9/Y3CpvV3Z29/YP7MPqvYpTSaFDYx7LXkAUcCago5nm0EskkCjg0A0m1zO/+wBSsVjc6WkCXkRGgoWMEm0k364+Ddpj5met1lk+kESMOPh2zak7c+BV4hakhgq0fftrMIxpGoHQlBOl+q6TaC8jUjPKIa8MUgUJoRMygr6hgkSgvGx+e45PjTLEYSxNCY3n6u+JjERKTaPAdEZEj9WyNxP/8/qpDq+8jIkk1SDoYlGYcqxjPAsCD5kEqvnUEEIlM7diOiaSUG3iqpgQ3OWXV0n3ou426q5726g1W0UeZXSMTtA5ctElaqIb1EYdRNEjekav6M3KrRfr3fpYtJasYuYI/YH1+QNGq5Ro</latexit>

|�BB0i

<latexit sha1_base64="HcusibRoWfUJN03vpMWebKkddIM=">AAAB/3icbVBNS8NAFHzxs9avqHjysliEeilJKehFKHrxWNHaQhvLZrNtl+4mYXcjlFDwr3jxoCBe/Rve/Ddu2hy0dWBhmHmPNzt+zJnSjvNtLS2vrK6tFzaKm1vbO7v23v69ihJJaJNEPJJtHyvKWUibmmlO27GkWPictvzRVea3HqlULArv9DimnsCDkPUZwdpIPfuwK7AeSpEGTEzKt6foAgUP1Z5dcirOFGiRuDkpQY5Gz/7qBhFJBA014VipjuvE2kux1IxwOil2E0VjTEZ4QDuGhlhQ5aXT+BN0YpQA9SNpXqjRVP29kWKh1Fj4ZjILq+a9TPzP6yS6f+6lLIwTTUMyO9RPONIRyrpAAZOUaD42BBPJTFZEhlhiok1jRVOCO//lRdKqVtxaxXVvaqX6Zd5HAY7gGMrgwhnU4Roa0AQCKTzDK7xZT9aL9W59zEaXrHznAP7A+vwBWTyU4g==</latexit>

dim(S) = d2

Choi-Jamiołkowski Isomorphism

Choi-Jamiołkowski state

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

Correlation measure for dynamics

S

Bi-partite system:<latexit sha1_base64="9Y816/kLUVSmc2iLOKEuyA6c0EE=">AAACD3icbZDNSgMxFIUz9a/Wv1GXboKlUDdlRgq6EWrduKxgbaEdSiaTaUOTzJBkhDL0Cdz4Km5cKIhbt+58GzPtLGzrgcDhu/eSe48fM6q04/xYhbX1jc2t4nZpZ3dv/8A+PHpQUSIxaeOIRbLrI0UYFaStqWakG0uCuM9Ixx/fZPXOI5GKRuJeT2LicTQUNKQYaYMGdqXPkR5JngaUT6vXZ/AKLpBmRoKBXXZqzkxw1bi5KYNcrYH93Q8inHAiNGZIqZ7rxNpLkdQUMzIt9RNFYoTHaEh6xgrEifLS2TlTWDEkgGEkzRMazujfiRRxpSbcN53Zqmq5lsH/ar1Eh5deSkWcaCLw/KMwYVBHMMsGBlQSrNnEGIQlNbtCPEISYW0SLJkQ3OWTV03nvObWa657Vy83mnkeRXACTkEVuOACNMAtaIE2wOAJvIA38G49W6/Wh/U5by1Y+cwxWJD19Qu0GpsH</latexit>

dim(A) = dim(B) = d



|�BB0i

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

I

Choi-Jamiołkowski IsomorphismBi-partite system:








S , Eq. (18),

I(ES) :=I(⇢

CJS )

4 log d:=

1

4 log d

⇣S�⇢CJS |AA0

�+ S

�⇢CJS |BB0

�� S

�⇢CJS

� ⌘. (20)



CJS ) and ⇢









�⇢CJS |AA0

�+S

�⇢CJS |BB0




CJS ) = 2 log d

2.

Normalised quantum mutual information of the Choi-Jamiołkowski state:

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)








S , Eq. (18),

I(ES) :=I(⇢

CJS )

4 log d:=

1

4 log d

⇣S�⇢CJS |AA0

�+ S

�⇢CJS |BB0

�� S

�⇢CJS

� ⌘. (20)



CJS ) and ⇢









�⇢CJS |AA0

�+S

�⇢CJS |BB0




CJS ) = 2 log d

2.








S , Eq. (18),

I(ES) :=I(⇢

CJS )

4 log d:=

1

4 log d

⇣S�⇢CJS |AA0

�+ S

�⇢CJS |BB0

�� S

�⇢CJS

� ⌘. (20)



CJS ) and ⇢









�⇢CJS |AA0

�+S

�⇢CJS |BB0




CJS ) = 2 log d

2.


dim(S) = d2


Properties of the quantifier I(E) := 1

4 log dI(⇢CJ)

Fundamental Law:

I(E) � I[(LA ⌦ LB)E(RA ⌦RB)]

= for local unitaries↪ UA ⌦ UB

Faithfullness:

I(E) = 0 , E = EA ⌦ EB

Correlated Dynamics If and only if I(⇢CJ) > 0

I(E) 2 [0, 1] Normalisation:








S , Eq. (18),

I(ES) :=I(⇢

CJS )

4 log d:=

1

4 log d

⇣S�⇢CJS |AA0

�+ S

�⇢CJS |BB0

�� S

�⇢CJS

� ⌘. (20)



CJS ) and ⇢









�⇢CJS |AA0

�+S

�⇢CJS |BB0




CJS ) = 2 log d

2. maximal for a maximally entangled state w/ respect to partition








S , Eq. (18),

I(ES) :=I(⇢

CJS )

4 log d:=

1

4 log d

⇣S�⇢CJS |AA0

�+ S

�⇢CJS |BB0

�� S

�⇢CJS

� ⌘. (20)



CJS ) and ⇢









�⇢CJS |AA0

�+S

�⇢CJS |BB0




CJS ) = 2 log d

2.

Two copies of the quantum system needed?No… mathematical construction only.

S



|�BB0i

I

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)


dim(S) = d2

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

Reconstruction of Via quantum process tomography

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

Equivalent to a quantum state tomography on

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

Example: N-qubit-system S<latexit sha1_base64="2xu+y6pc69efiulAp/AR8bvzxfk=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSJ4Kkkp6EUoevEkFYwttLFsNpt26WY37G6EEvobvHhQEK/+H2/+G7dtDtr6YODx3gwz88KUM21c99tZWV1b39gsbZW3d3b39isHhw9aZopQn0guVSfEmnImqG+Y4bSTKoqTkNN2OLqe+u0nqjST4t6MUxokeCBYzAg2VvKjy/rjbb9SdWvuDGiZeAWpQoFWv/LViyTJEioM4VjrruemJsixMoxwOin3Mk1TTEZ4QLuWCpxQHeSzYyfo1CoRiqWyJQyaqb8ncpxoPU5C25lgM9SL3lT8z+tmJr4IcibSzFBB5ovijCMj0fRzFDFFieFjSzBRzN6KyBArTIzNp2xD8BZfXibtes1r1DzvrlFtXhV5lOAYTuAMPDiHJtxAC3wgwOAZXuHNEc6L8+58zFtXnGLmCP7A+fwBosOOZA==</latexit>

d = 2N<latexit sha1_base64="zazAdysP/LB4JExDaJc1biRKAkk=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRahHiy7UtBj0YsnqWBtYbuWbJptQ7PJkmSFsvRnePGgIF79M978N6btHrT1QcLjvRlm5oUJZ9q47rdTWFldW98obpa2tnd298r7Bw9aporQFpFcqk6INeVM0JZhhtNOoiiOQ07b4eh66refqNJMinszTmgQ44FgESPYWMmvP96iqv3OvNNeueLW3BnQMvFyUoEczV75q9uXJI2pMIRjrX3PTUyQYWUY4XRS6qaaJpiM8ID6lgocUx1ks5Un6MQqfRRJZZ8waKb+7shwrPU4Dm1ljM1QL3pT8T/PT010GWRMJKmhgswHRSlHRqLp/ajPFCWGjy3BRDG7KyJDrDAxNqWSDcFbPHmZtM9rXr3meXf1SuMqz6MIR3AMVfDgAhpwA01oAQEJz/AKb45xXpx352NeWnDynkP4A+fzByhqj7A=</latexit>

4N (4N � 1) real parameters

N=1: 12, N=2: 240, N=3: 4032, …

Maximally Correlated Dynamics

Resource theory approachMaximally correlated dynamics cannot be obtained by

composing some dynamical map with uncorrelated dynamics

All non-factorable unitaries satisfy

this criterion!U 6= UA ⌦ UB

<latexit sha1_base64="ifmIrqxwfoCAcbEuvsTtK+nWtJ8=">AAAB93icbZC7TsMwFIadcivlFi4bi0WFxFQlCAnGUhbGIpFSqYkixz1prTpOsB2kEvVZYELAxnvwArwNbskALf/0+fy/dS5RxpnSjvNlVZaWV1bXquu1jc2t7R17d6+j0lxS8GjKU9mNiALOBHiaaQ7dTAJJIg530ehq6t89gFQsFbd6nEGQkIFgMaNEm1JoH3i+gHvshZd+qlkCymArtOtOw5kJL4JbQh2Vaof2p99PaZ6A0JQTpXquk+mgIFIzymFS83MFGaEjMoCeQUFMo6CYTT/Bx3EqsR4Cnr1/ZwuSKDVOIpNJiB6qeW9a/M/r5Tq+CAomslyDoCZivDjnWKd4egTcZxKo5mMDhEpmpsR0SCSh2pyqZtZ355ddhM5pw3Ua7s1ZvdkqD1FFh+gInSAXnaMmukZt5CGKHtEzekPv1th6sl6s159oxSr/7KM/sj6+ASfVkhU=</latexit><latexit sha1_base64="ifmIrqxwfoCAcbEuvsTtK+nWtJ8=">AAAB93icbZC7TsMwFIadcivlFi4bi0WFxFQlCAnGUhbGIpFSqYkixz1prTpOsB2kEvVZYELAxnvwArwNbskALf/0+fy/dS5RxpnSjvNlVZaWV1bXquu1jc2t7R17d6+j0lxS8GjKU9mNiALOBHiaaQ7dTAJJIg530ehq6t89gFQsFbd6nEGQkIFgMaNEm1JoH3i+gHvshZd+qlkCymArtOtOw5kJL4JbQh2Vaof2p99PaZ6A0JQTpXquk+mgIFIzymFS83MFGaEjMoCeQUFMo6CYTT/Bx3EqsR4Cnr1/ZwuSKDVOIpNJiB6qeW9a/M/r5Tq+CAomslyDoCZivDjnWKd4egTcZxKo5mMDhEpmpsR0SCSh2pyqZtZ355ddhM5pw3Ua7s1ZvdkqD1FFh+gInSAXnaMmukZt5CGKHtEzekPv1th6sl6s159oxSr/7KM/sj6+ASfVkhU=</latexit><latexit sha1_base64="ifmIrqxwfoCAcbEuvsTtK+nWtJ8=">AAAB93icbZC7TsMwFIadcivlFi4bi0WFxFQlCAnGUhbGIpFSqYkixz1prTpOsB2kEvVZYELAxnvwArwNbskALf/0+fy/dS5RxpnSjvNlVZaWV1bXquu1jc2t7R17d6+j0lxS8GjKU9mNiALOBHiaaQ7dTAJJIg530ehq6t89gFQsFbd6nEGQkIFgMaNEm1JoH3i+gHvshZd+qlkCymArtOtOw5kJL4JbQh2Vaof2p99PaZ6A0JQTpXquk+mgIFIzymFS83MFGaEjMoCeQUFMo6CYTT/Bx3EqsR4Cnr1/ZwuSKDVOIpNJiB6qeW9a/M/r5Tq+CAomslyDoCZivDjnWKd4egTcZxKo5mMDhEpmpsR0SCSh2pyqZtZ355ddhM5pw3Ua7s1ZvdkqD1FFh+gInSAXnaMmukZt5CGKHtEzekPv1th6sl6s159oxSr/7KM/sj6+ASfVkhU=</latexit><latexit sha1_base64="ifmIrqxwfoCAcbEuvsTtK+nWtJ8=">AAAB93icbZC7TsMwFIadcivlFi4bi0WFxFQlCAnGUhbGIpFSqYkixz1prTpOsB2kEvVZYELAxnvwArwNbskALf/0+fy/dS5RxpnSjvNlVZaWV1bXquu1jc2t7R17d6+j0lxS8GjKU9mNiALOBHiaaQ7dTAJJIg530ehq6t89gFQsFbd6nEGQkIFgMaNEm1JoH3i+gHvshZd+qlkCymArtOtOw5kJL4JbQh2Vaof2p99PaZ6A0JQTpXquk+mgIFIzymFS83MFGaEjMoCeQUFMo6CYTT/Bx3EqsR4Cnr1/ZwuSKDVOIpNJiB6qeW9a/M/r5Tq+CAomslyDoCZivDjnWKd4egTcZxKo5mMDhEpmpsR0SCSh2pyqZtZ355ddhM5pw3Ua7s1ZvdkqD1FFh+gInSAXnaMmukZt5CGKHtEzekPv1th6sl6s159oxSr/7KM/sj6+ASfVkhU=</latexit>

Not very discrimina!ve

Consider ins"ad:

I(Emax) = max<latexit sha1_base64="ZtWRRwl5Dvmh8aAtdZ7zoYx1gyE=">AAACBnicbVDNSsNAGNz4W+tf1aOXxSLUS0lE0ItQFEFvFewPNKF82W7apbtJ2N1IS8hdX0ZPot58AF/At3Fbc9DWOc3OzMI348ecKW3bX9bC4tLyymphrbi+sbm1XdrZbaookYQ2SMQj2fZBUc5C2tBMc9qOJQXhc9ryh5cTv3VPpWJReKfHMfUE9EMWMALaSN1S2fVBpjdZxRWgBwR4epV1U1cKLGCUHZ0beWRSdtWeAs8TJydllKPeLX26vYgkgoaacFCq49ix9lKQmhFOs6KbKBoDGUKfdgwNQVDlpdMyGT4MIon1gOLp+3c2BaHUWPgmMzlWzXoT8T+vk+jgzEtZGCeahsREjBckHOsITzbBPSYp0XxsCBDJzJWYDEAC0Wa5oqnvzJadJ83jqmNXnduTcu0iH6KA9tEBqiAHnaIaukZ11EAEPaJn9IberQfryXqxXn+iC1b+Zw/9gfXxDXuJmQs=</latexit><latexit sha1_base64="ZtWRRwl5Dvmh8aAtdZ7zoYx1gyE=">AAACBnicbVDNSsNAGNz4W+tf1aOXxSLUS0lE0ItQFEFvFewPNKF82W7apbtJ2N1IS8hdX0ZPot58AF/At3Fbc9DWOc3OzMI348ecKW3bX9bC4tLyymphrbi+sbm1XdrZbaookYQ2SMQj2fZBUc5C2tBMc9qOJQXhc9ryh5cTv3VPpWJReKfHMfUE9EMWMALaSN1S2fVBpjdZxRWgBwR4epV1U1cKLGCUHZ0beWRSdtWeAs8TJydllKPeLX26vYgkgoaacFCq49ix9lKQmhFOs6KbKBoDGUKfdgwNQVDlpdMyGT4MIon1gOLp+3c2BaHUWPgmMzlWzXoT8T+vk+jgzEtZGCeahsREjBckHOsITzbBPSYp0XxsCBDJzJWYDEAC0Wa5oqnvzJadJ83jqmNXnduTcu0iH6KA9tEBqiAHnaIaukZ11EAEPaJn9IberQfryXqxXn+iC1b+Zw/9gfXxDXuJmQs=</latexit><latexit sha1_base64="ZtWRRwl5Dvmh8aAtdZ7zoYx1gyE=">AAACBnicbVDNSsNAGNz4W+tf1aOXxSLUS0lE0ItQFEFvFewPNKF82W7apbtJ2N1IS8hdX0ZPot58AF/At3Fbc9DWOc3OzMI348ecKW3bX9bC4tLyymphrbi+sbm1XdrZbaookYQ2SMQj2fZBUc5C2tBMc9qOJQXhc9ryh5cTv3VPpWJReKfHMfUE9EMWMALaSN1S2fVBpjdZxRWgBwR4epV1U1cKLGCUHZ0beWRSdtWeAs8TJydllKPeLX26vYgkgoaacFCq49ix9lKQmhFOs6KbKBoDGUKfdgwNQVDlpdMyGT4MIon1gOLp+3c2BaHUWPgmMzlWzXoT8T+vk+jgzEtZGCeahsREjBckHOsITzbBPSYp0XxsCBDJzJWYDEAC0Wa5oqnvzJadJ83jqmNXnduTcu0iH6KA9tEBqiAHnaIaukZ11EAEPaJn9IberQfryXqxXn+iC1b+Zw/9gfXxDXuJmQs=</latexit><latexit sha1_base64="ZtWRRwl5Dvmh8aAtdZ7zoYx1gyE=">AAACBnicbVDNSsNAGNz4W+tf1aOXxSLUS0lE0ItQFEFvFewPNKF82W7apbtJ2N1IS8hdX0ZPot58AF/At3Fbc9DWOc3OzMI348ecKW3bX9bC4tLyymphrbi+sbm1XdrZbaookYQ2SMQj2fZBUc5C2tBMc9qOJQXhc9ryh5cTv3VPpWJReKfHMfUE9EMWMALaSN1S2fVBpjdZxRWgBwR4epV1U1cKLGCUHZ0beWRSdtWeAs8TJydllKPeLX26vYgkgoaacFCq49ix9lKQmhFOs6KbKBoDGUKfdgwNQVDlpdMyGT4MIon1gOLp+3c2BaHUWPgmMzlWzXoT8T+vk+jgzEtZGCeahsREjBckHOsITzbBPSYp0XxsCBDJzJWYDEAC0Wa5oqnvzJadJ83jqmNXnduTcu0iH6KA9tEBqiAHnaIaukZ11EAEPaJn9IberQfryXqxXn+iC1b+Zw/9gfXxDXuJmQs=</latexit>

Characterization of

Maximally Correlated Dynamics

I(Emax) = 1

I(E) = 1 ) E(⇢) = U⇢U†, UU† = 1

Example: despi" #e fact #at it does not crea" correla!ons!I(USWAP) = 1

16.10 Markus Muller

iii) The fundamental law of the resource theory for correlations in quantum dynamics is sat-isfied, namely that

I(ES) � I�(LA ⌦ LB)ES(RA ⌦RB)

�, (21)

stating that the amount of correlations of the dynamics ES decreases or at most staysthe same, if the dynamics is composed with uncorrelated dynamics. Stated differently,if a process is a composition of a correlated and an uncorrelated part, the amount ofcorrelations in the composition has to be equal or smaller than the amount of correlationthat is inherent to the correlated part. Here, equality in the above inequality is reached forcomposition with uncorrelated unitary dynamics, LA(·) = UA(·)U †

A, LB(·) = UB(·)U †B,

RA(·) = VA(·)V †A, and RB(·) = VB(·)V †

B.

Leaving aside the normalization factor 1/(4 log d), the quantifier (20) can be intuitively under-stood as the amount of information that is needed to distinguish the actual dynamics ES fromthe individual dynamics of its parts ES1 ⌦ ES2 [12]. Namely, the information that is lost whenES1 ⌦ ES2 is taken as an approximation of ES. The normalized quantity I 2 [0, 1] quantifies thisinformation relative to the maximum value it can take on all possible processes.For clarity, we remark that the use of an ancilla system S

0 is merely underlying the mathe-matical construction of the isomorphism. It is not required in an experimental determinationof I . Rather than reconstructing the Choi-Jamiołkowski state ⇢CJ

S from quantum state tomog-raphy [12] on the enlarged system SS

0, one can equivalently determine ⇢CJS by reconstructing

the dynamics ES by means of quantum process tomography on the physical system S alone.For a system S of N qubits, due to the Choi-Jamiołkowski isomorphism the number of realparameters to determine, 4N(4N�1), is in both cases the same and grows exponentially withthe number of qubits.

3.3 Maximally correlated quantum dynamics

Before computing I for some cases of physical interest it is worth studying which dynamicsachieve the maximum value Imax = 1. From the resource theory point of view, these dynamicscan be considered as maximally correlated since they cannot be constructed from other mapsby composition with uncorrelated maps [because of Eq. (21)]. One can show the followingproperty of maximally correlated dynamics:

Theorem 1. If for a map ES the property that I(ES) = 1 holds, it must be unitary ES(·) =

US(·)U †S, USU

†S = 1.

Proof. As aforementioned, the maximum value, I(ES) = 1, is reached if and only if ⇢CJS is a

maximally entangled state with respect to the bipartition AA0|BB0, | (AA0)|(BB0)i. Then

ES ⌦ 1S0�|�SS0ih�SS0 |

�= | (AA0)|(BB0)ih (AA0)|(BB0)| (22)

is a pure state. Therefore ES must be unitary as the Choi-Jamiołkowski state is pure if and onlyif it represents a unitary map.

S SWAP

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

=

maximallyentangled

16.8 Markus Muller








0= A

0B


0,

|�SS0i :=1

d

d2X

j=1

|jjiSS0 =1

d

dX

k,`=1

|kìAB ⌦ |kìA0B0 . (17)






⇢CJS := ES ⌦ 1S0


�. (18)



CJS in



⇢CJS =

1

2

⇣|00ih00|SS0 + |11ih11|SS0

⌘+

1

2(1�2p)

⇣|00ih11|SS0 + |11ih00|SS0

⌘. (19)

(Non-)Maximally Correlated Dynamics

Remark: highly correlating dynamics such as the CNOT gate or maps for dissipative preparation of Bell-states are not maximally correlated according to !I

<latexit sha1_base64="UwZrSnVnikazlouaibRSaDucCD8=">AAAB6HicbZDLSgNBEEVr4ivGV9Slm8YguAozIugy6EZ3EcwDkiHUdCpJm54H3T1CGPIPuhJ15/f4A/6NnTgLTbyr03VvQ90KEim0cd0vp7Cyura+UdwsbW3v7O6V9w+aOk4VpwaPZazaAWqSIqKGEUZSO1GEYSCpFYyvZ37rkZQWcXRvJgn5IQ4jMRAcjR21ugGq7HbaK1fcqjsXWwYvhwrkqvfKn91+zNOQIsMlat3x3MT4GSojuKRpqZtqSpCPcUgdixGGpP1svu6UnQxixcyI2Pz9O5thqPUkDGwmRDPSi95s+J/XSc3g0s9ElKSGIm4j1hukkpmYzVqzvlDEjZxYQK6E3ZLxESrkxt6mZOt7i2WXoXlW9dyqd3deqV3lhyjCERzDKXhwATW4gTo0gMMYnuEN3p0H58l5cV5/ogUn/3MIf+R8fANDTY0n</latexit><latexit sha1_base64="UwZrSnVnikazlouaibRSaDucCD8=">AAAB6HicbZDLSgNBEEVr4ivGV9Slm8YguAozIugy6EZ3EcwDkiHUdCpJm54H3T1CGPIPuhJ15/f4A/6NnTgLTbyr03VvQ90KEim0cd0vp7Cyura+UdwsbW3v7O6V9w+aOk4VpwaPZazaAWqSIqKGEUZSO1GEYSCpFYyvZ37rkZQWcXRvJgn5IQ4jMRAcjR21ugGq7HbaK1fcqjsXWwYvhwrkqvfKn91+zNOQIsMlat3x3MT4GSojuKRpqZtqSpCPcUgdixGGpP1svu6UnQxixcyI2Pz9O5thqPUkDGwmRDPSi95s+J/XSc3g0s9ElKSGIm4j1hukkpmYzVqzvlDEjZxYQK6E3ZLxESrkxt6mZOt7i2WXoXlW9dyqd3deqV3lhyjCERzDKXhwATW4gTo0gMMYnuEN3p0H58l5cV5/ogUn/3MIf+R8fANDTY0n</latexit><latexit sha1_base64="UwZrSnVnikazlouaibRSaDucCD8=">AAAB6HicbZDLSgNBEEVr4ivGV9Slm8YguAozIugy6EZ3EcwDkiHUdCpJm54H3T1CGPIPuhJ15/f4A/6NnTgLTbyr03VvQ90KEim0cd0vp7Cyura+UdwsbW3v7O6V9w+aOk4VpwaPZazaAWqSIqKGEUZSO1GEYSCpFYyvZ37rkZQWcXRvJgn5IQ4jMRAcjR21ugGq7HbaK1fcqjsXWwYvhwrkqvfKn91+zNOQIsMlat3x3MT4GSojuKRpqZtqSpCPcUgdixGGpP1svu6UnQxixcyI2Pz9O5thqPUkDGwmRDPSi95s+J/XSc3g0s9ElKSGIm4j1hukkpmYzVqzvlDEjZxYQK6E3ZLxESrkxt6mZOt7i2WXoXlW9dyqd3deqV3lhyjCERzDKXhwATW4gTo0gMMYnuEN3p0H58l5cV5/ogUn/3MIf+R8fANDTY0n</latexit><latexit sha1_base64="UwZrSnVnikazlouaibRSaDucCD8=">AAAB6HicbZDLSgNBEEVr4ivGV9Slm8YguAozIugy6EZ3EcwDkiHUdCpJm54H3T1CGPIPuhJ15/f4A/6NnTgLTbyr03VvQ90KEim0cd0vp7Cyura+UdwsbW3v7O6V9w+aOk4VpwaPZazaAWqSIqKGEUZSO1GEYSCpFYyvZ37rkZQWcXRvJgn5IQ4jMRAcjR21ugGq7HbaK1fcqjsXWwYvhwrkqvfKn91+zNOQIsMlat3x3MT4GSojuKRpqZtqSpCPcUgdixGGpP1svu6UnQxixcyI2Pz9O5thqPUkDGwmRDPSi95s+J/XSc3g0s9ElKSGIm4j1hukkpmYzVqzvlDEjZxYQK6E3ZLxESrkxt6mZOt7i2WXoXlW9dyqd3deqV3lhyjCERzDKXhwATW4gTo0gMMYnuEN3p0H58l5cV5/ogUn/3MIf+R8fANDTY0n</latexit>

I(E) = 1 ) E(⇢) = U⇢U†, UU† = 1

<latexit sha1_base64="JoC8oiOPjVKGnTsdilfDXqH0fNo=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkUI9FLx5bsLbQhrLZTtq1m03Y3Qgl9Bd48aAgXv1H3vw3btsctPXBwOO9GWbmBYng2rjut1PY2Nza3inulvb2Dw6PyscnDzpOFcM2i0WsugHVKLjEtuFGYDdRSKNAYCeY3M79zhMqzWN5b6YJ+hEdSR5yRo2VWu6gXHGr7gJknXg5qUCO5qD81R/GLI1QGiao1j3PTYyfUWU4Ezgr9VONCWUTOsKepZJGqP1sceiMXFhlSMJY2ZKGLNTfExmNtJ5Gge2MqBnrVW8u/uf1UhNe+xmXSWpQsuWiMBXExGT+NRlyhcyIqSWUKW5vJWxMFWXGZlOyIXirL6+TzlXVq1U9r1WrNG7yPIpwBudwCR7UoQF30IQ2MEB4hld4cx6dF+fd+Vi2Fpx85hT+wPn8AQ7xjO0=</latexit>

0

<latexit sha1_base64="VMykZDNl5NJDTtTSu6ShVDeyVyQ=">AAAB6XicbVA9SwNBEJ2LXzF+RS1tFoNgFW5F0DJoY5mAMYHkCHubuWTN3t6xuyeEkF9gY6Egtv4jO/+Nm+QKTXww8Hhvhpl5YSqFsb7/7RXW1jc2t4rbpZ3dvf2D8uHRg0kyzbHJE5nodsgMSqGwaYWV2E41sjiU2ApHtzO/9YTaiETd23GKQcwGSkSCM+ukBu2VK37Vn4OsEpqTCuSo98pf3X7CsxiV5ZIZ06F+aoMJ01ZwidNSNzOYMj5iA+w4qliMJpjMD52SM6f0SZRoV8qSufp7YsJiY8Zx6DpjZodm2ZuJ/3mdzEbXwUSoNLOo+GJRlEliEzL7mvSFRm7l2BHGtXC3Ej5kmnHrsim5EOjyy6ukdVGll1VKG5eV2k2eRxFO4BTOgcIV1OAO6tAEDgjP8Apv3qP34r17H4vWgpfPHMMfeJ8/EHaM7g==</latexit>

1

<latexit sha1_base64="skm7aiG86la/kS6U4QpcCK5/6ck=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRbBU92Ugh6LXjxWtLbQLiWbZtvQJLskWaEs/QlePCiIV/+QN/+NabsHbX0w8Hhvhpl5YSK4sb7/7RXW1jc2t4rbpZ3dvf2D8uHRo4lTTVmLxiLWnZAYJrhiLcutYJ1EMyJDwdrh+Gbmt5+YNjxWD3aSsECSoeIRp8Q66R5f1Prlil/150CrBOekAjma/fJXbxDTVDJlqSDGdLGf2CAj2nIq2LTUSw1LCB2TIes6qohkJsjmp07RmVMGKIq1K2XRXP09kRFpzESGrlMSOzLL3kz8z+umNroKMq6S1DJFF4uiVCAbo9nfaMA1o1ZMHCFUc3croiOiCbUunZILAS+/vEratSquVzG+q1ca13keRTiBUzgHDJfQgFtoQgsoDOEZXuHNE96L9+59LFoLXj5zDH/gff4A7jiNYw==</latexit>

1/2








S , Eq. (18),

I(ES) :=I(⇢

CJS )

4 log d:=

1

4 log d

⇣S�⇢CJS |AA0

�+ S

�⇢CJS |BB0

�� S

�⇢CJS

� ⌘. (20)



CJS ) and ⇢









�⇢CJS |AA0

�+S

�⇢CJS |BB0




CJS ) = 2 log d

2.

SWAP

What about the CNOT-gate?

I[(UA ⌦ UB)USWAP(VA ⌦ VB)] = 1

9U 0, such that I(U 0) = 1, but U 0 6= (UA ⌦ UB)USWAP(VA ⌦ VB)9U 0, such that I(U 0) = 1, but U 0 6= (UA ⌦ UB)USWAP(VA ⌦ VB)

U 0 = | "#ih#" |+ i(| ##ih"# |+ | #"ih## |+ | ""ih"" |)

Physical Application: SuperradianceTwo two-level atoms radiating in the EM vacuum

HS = !02 �

z1 + !0

2 �z2

d⇢

dt= �i[HS , ⇢] +

X

j,k=1,2

ajk

��k ⇢�

+j � 1

2{�+j �

�k , ⇢}

�

Details in the lecture notes

|0�

|1�1 2

ajk = �0 [j0(xjk) + P2(cos ✓jk)j2(xjk)] '(�0�jk, if r � 1/!

�0, if r ⌧ 1/!

16.12 Markus Muller

corresponds to the modes of the radiation field and is given by

HE =

X

k

X

�=1,2

!ka†�(k)a�(k), (24)

where k and � stand for the wave vector and the two polarization degrees of freedom, respec-tively. We have taken natural units ~ = c = 1. The dispersion relation in free space is !k = |k|,and the field operators a†�(k) and a�(k) describe the creation and annihilation of photons withwave vector k and polarization vector e�. These fulfill k · e� = 0 and e� · e�0 = ��,�0 .The atom-field interaction is described in dipole approximation by the Hamiltonian

HSE = �X

j=1,2

⇥��j d ·E(rj) + �

+j d

⇤ ·E(rj)⇤. (25)

Here, d is the dipole matrix element of the atomic transition, rj denotes the position of the j-thatom, and the raising and lowering operators �+

j and ��j are defined as �+

j = (��j )

†= |eij jhg|

for its exited |eij and ground |gij states. Furthermore, the electric field operator is given (inGaussian units)

E(r) = i

X

k,�

r2⇡!k

V e�(k)⇣a�(k)e

ik·r � a†�(k)e

�ik·r⌘, (26)

where V denotes the quantization volume. Under a series of standard assumptions known asthe Markovian weak-coupling limit [3] the dynamics of the atoms is governed by a Lindbladmaster equation of the form

d⇢S

dt= L(⇢S) = �i

!2

�Z1 + Z2, ⇢S

�+

X

i,j=1,2

ajk

⇣��k ⇢S�

+j � 1

2{�+j �

�k , ⇢S}

⌘. (27)

After taking the continuum limit ( 1VP

k ! 1(2⇡)3

Rd3k) and performing the integrals, the coef-

ficients ajk are given by (see e.g. Sec. 3.7.5 of Ref. [3])

ajk = �0

�j0(xjk) + P2(cos ✓jk)j2(xjk)

�. (28)

Here, �0 = 43!

3|d|2, and j0(x) and j2(x) are spherical Bessel functions [26]

j0(x) =sin x

x, j2(x) =

✓3

x3� 1

x

◆sin x� 3

x2cos x, (29)

andP2(cos ✓) =

1

2

�3 cos

2✓ � 1

�(30)

is a Legendre polynomial, with

xjk = !|rj�rk| and cos2(✓jk) =

|d · (rj�rk)|2|d|2|rj�rk|2

. (31)

Notice that if the distance between atoms r = |r1 � r2|, is much larger than the wavelengthassociated with the atomic transition r � 1/!, we have ajk ' �0�ij and only the diagonal

ajk = �0 [j0(xjk) + P2(cos ✓jk)j2(xjk)] '(�0�jk, if r � 1/!

�0, if r ⌧ 1/!

<latexit sha1_base64="w7SVVuHGLsHavPvpV66I0CvwSzU=">AAAB9nicbVBNS8NAFHypX7V+VT16WSyCp5JIoR6LXjxWsLbQxLLZbNKlm03Y3Sgl9H948aAgXv0t3vw3btoctHVgYZh5w3s7fsqZ0rb9bVXW1jc2t6rbtZ3dvf2D+uHRvUoySWiPJDyRAx8rypmgPc00p4NUUhz7nPb9yXXh9x+pVCwRd3qaUi/GkWAhI1gb6UEiN4qQy00gwLVRvWE37TnQKnFK0oAS3VH9yw0SksVUaMKxUkPHTrWXY6kZ4XRWczNFU0wmOKJDQwWOqfLy+dUzdGaUAIWJNE9oNFd/J3IcKzWNfTMZYz1Wy14h/ucNMx1eejkTaaapIItFYcaRTlBRAQqYpETzqSGYSGZuRWSMJSbaFFWU4Cx/eZX0L5pOq+k4t61G56rsowoncArn4EAbOnADXegBAQnP8Apv1pP1Yr1bH4vRilVmjuEPrM8f0TyR4g==</latexit>

r � �<latexit sha1_base64="HbCOl9mR/CUhL49AYkiZJItcdTo=">AAAB9nicbVBNS8NAFHypX7V+VT16WSyCp5JIQY9FLx4rWCu0sWw2L+3SzSbsbpQS+j+8eFAQr/4Wb/4bt20O2jrwYJh5w76dIBVcG9f9dkorq2vrG+XNytb2zu5edf/gTieZYthmiUjUfUA1Ci6xbbgReJ8qpHEgsBOMrqZ+5xGV5om8NeMU/ZgOJI84o8ZKD4r0hLBjAyGt9Ks1t+7OQJaJV5AaFGj1q1+9MGFZjNIwQbXuem5q/Jwqw5nASaWXaUwpG9EBdi2VNEbt57OrJ+TEKiGJEmVHGjJTfydyGms9jgO7GVMz1IveVPzP62YmuvBzLtPMoGTzh6JMEJOQaQUk5AqZEWNLKFPc3krYkCrKjC1qWoK3+OVl0jmre4265900as3Loo8yHMExnIIH59CEa2hBGxgoeIZXeHOenBfn3fmYr5acInMIf+B8/gDgzZHs</latexit>

r ⌧ �

First Application: Superradiance

Two two-level atoms radiating in the EM vacuumHS = !0

2 �z1 + !0

2 �z2

d⇢

dt= �i[HS , ⇢] +

X

j,k=1,2

ajk

��k ⇢�

+j � 1

2{�+j �

�k , ⇢}

�

� � � � � ��

��

��

��

��

��

��

��

� ��

��

��

��

��

|0�

|1�

|0�

|1�

Independent decay

<latexit sha1_base64="SgkIrBspywV9zlyfRiZQW18i41w=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnBeIHkCHubTbJkb/fYnRPCkR9hY6Egtv4dO/+Nm+QKTXww8Hhvhpl5cSqFRd//9kpr6xubW+Xtys7u3v5B9fDo0erMMN5iWmrTjqnlUijeQoGSt1PDaRJLHsbj25kfPnFjhVYPOEl5lNChEgPBKDop7GoUCbe9as2v+3OQVRIUpAYFmr3qV7evWZZwhUxSazuBn2KUU4OCST6tdDPLU8rGdMg7jirqlkT5/NwpOXNKnwy0caWQzNXfEzlNrJ0ksetMKI7ssjcT//M6GQ6uo1yoNEOu2GLRIJMENZn9TvrCcIZy4ghlRrhbCRtRQxm6hCouhGD55VUSXtSDy3oQ3F/WGjdFHmU4gVM4hwCuoAF30IQWMBjDM7zCm5d6L96797FoLXnFzDH8gff5Ax6Gj+Y=</latexit>⌦

<latexit sha1_base64="w7SVVuHGLsHavPvpV66I0CvwSzU=">AAAB9nicbVBNS8NAFHypX7V+VT16WSyCp5JIoR6LXjxWsLbQxLLZbNKlm03Y3Sgl9H948aAgXv0t3vw3btoctHVgYZh5w3s7fsqZ0rb9bVXW1jc2t6rbtZ3dvf2D+uHRvUoySWiPJDyRAx8rypmgPc00p4NUUhz7nPb9yXXh9x+pVCwRd3qaUi/GkWAhI1gb6UEiN4qQy00gwLVRvWE37TnQKnFK0oAS3VH9yw0SksVUaMKxUkPHTrWXY6kZ4XRWczNFU0wmOKJDQwWOqfLy+dUzdGaUAIWJNE9oNFd/J3IcKzWNfTMZYz1Wy14h/ucNMx1eejkTaaapIItFYcaRTlBRAQqYpETzqSGYSGZuRWSMJSbaFFWU4Cx/eZX0L5pOq+k4t61G56rsowoncArn4EAbOnADXegBAQnP8Apv1pP1Yr1bH4vRilVmjuEPrM8f0TyR4g==</latexit>

r � �

First Application: SuperradianceTwo two-level atoms radiating in the EM vacuum

HS = !02 �

z1 + !0

2 �z2

d⇢

dt= �i[HS , ⇢] +

X

j,k=1,2

ajk

��k ⇢�

+j � 1

2{�+j �

�k , ⇢}

�

� � � � � ��

��

��

��

��

��

��

��

� ��

��

��

��

��<latexit sha1_base64="HbCOl9mR/CUhL49AYkiZJItcdTo=">AAAB9nicbVBNS8NAFHypX7V+VT16WSyCp5JIQY9FLx4rWCu0sWw2L+3SzSbsbpQS+j+8eFAQr/4Wb/4bt20O2jrwYJh5w76dIBVcG9f9dkorq2vrG+XNytb2zu5edf/gTieZYthmiUjUfUA1Ci6xbbgReJ8qpHEgsBOMrqZ+5xGV5om8NeMU/ZgOJI84o8ZKD4r0hLBjAyGt9Ks1t+7OQJaJV5AaFGj1q1+9MGFZjNIwQbXuem5q/Jwqw5nASaWXaUwpG9EBdi2VNEbt57OrJ+TEKiGJEmVHGjJTfydyGms9jgO7GVMz1IveVPzP62YmuvBzLtPMoGTzh6JMEJOQaQUk5AqZEWNLKFPc3krYkCrKjC1qWoK3+OVl0jmre4265900as3Loo8yHMExnIIH59CEa2hBGxgoeIZXeHOenBfn3fmYr5acInMIf+B8/gDgzZHs</latexit>

r ⌧ �

collective jump operator<latexit sha1_base64="xGRi7YoqGBSHP7k+7vSX3P0ArfE=">AAACBHicbZDNSgMxFIUz9a/Wv1F3ugkWQZCWSSnoRii6EVcVrC200yGTZtrQTGZIMkIZCm58FTcuFMStD+HOtzFtZ6GtBwIf597LzT1+zJnSjvNt5ZaWV1bX8uuFjc2t7R17d+9eRYkktEEiHsmWjxXlTNCGZprTViwpDn1Om/7walJvPlCpWCTu9Cimboj7ggWMYG0szz64gRewo1g/xB7qluBpxpVuybOLTtmZCi4CyqAIMtU9+6vTi0gSUqEJx0q1kRNrN8VSM8LpuNBJFI0xGeI+bRsUOKTKTac3jOGxcXowiKR5QsOp+3sixaFSo9A3nSHWAzVfm5j/1dqJDs7dlIk40VSQ2aIg4VBHcBII7DFJieYjA5hIZv4KyQBLTLSJrWBCQPMnL0KzUkbVMkK31WLtMssjDw7BETgBCJyBGrgGddAABDyCZ/AK3qwn68V6tz5mrTkrm9kHf2R9/gCdIZYL</latexit>

J = ��1 + ��

2

|0�

|1�

<latexit sha1_base64="GDF9OgdTFx7T53vj6AoBW3tMPHs=">AAAB8nicbVA9SwNBEJ2LXzF+RS1tFoNgFW5FMGXAxjKCMcHkCHubuWTJ3t6xuyeEmH9hY6Egtv4aO/+Nm+QKTXww8Hhvhpl5YSqFsb7/7RXW1jc2t4rbpZ3dvf2D8uHRvUkyzbHJE5nodsgMSqGwaYWV2E41sjiU2ApH1zO/9YjaiETd2XGKQcwGSkSCM+ukhydKu5qpgcReueJX/TnIKqE5qUCORq/81e0nPItRWS6ZMR3qpzaYMG0FlzgtdTODKeMjNsCOo4rFaILJ/OIpOXNKn0SJdqUsmau/JyYsNmYch64zZnZolr2Z+J/XyWxUCyZCpZlFxReLokwSm5DZ+6QvNHIrx44wroW7lfAh04xbF1LJhUCXX14lrYsqvaxSentZqdfyPIpwAqdwDhSuoA430IAmcFDwDK/w5hnvxXv3PhatBS+fOYY/8D5/AL5SkMA=</latexit>

|11i<latexit sha1_base64="AQRfF2Q2XtgO0Vfy1M1yLwpd10g=">AAACAXicbVBNSwMxEM3Wr1q/Vj148BIsgiCUjRTsseDFYwVrC+1SsulsG5rNLklWKNte/CtePCiIV/+FN/+NabsHbX0w8PLeDJl5QSK4Np737RTW1jc2t4rbpZ3dvf0D9/DoQcepYtBksYhVO6AaBJfQNNwIaCcKaBQIaAWjm5nfegSleSzvzTgBP6IDyUPOqLFSzz2ZeKSrqBwIwJd4Qrz80XPLXsWbA68SkpMyytHouV/dfszSCKRhgmrdIV5i/Iwqw5mAaambakgoG9EBdCyVNALtZ/MDpvjcKn0cxsqWNHiu/p7IaKT1OApsZ0TNUC97M/E/r5OasOZnXCapAckWH4WpwCbGszRwnytgRowtoUxxuytmQ6ooMzazkg2BLJ+8SlpXFVKtEHJXLddreR5FdIrO0AUi6BrV0S1qoCZiaIqe0St6c56cF+fd+Vi0Fpx85hj9gfP5A5uRlY8=</latexit>

|01i+ |10i<latexit sha1_base64="QqOVZj3Vb3UH1Gcol8kmRHxMiz0=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mk0B4LXjxWsLbYhrLZTtqlm03Y3Qgl9l948aAgXv013vw3btsctPXBwOO9GWbmBYng2rjut1PY2Nza3inulvb2Dw6Pyscn9zpOFcM2i0WsugHVKLjEtuFGYDdRSKNAYCeYXM/9ziMqzWN5Z6YJ+hEdSR5yRo2VHp5ct6+oHAkclCtu1V2ArBMvJxXI0RqUv/rDmKURSsME1brnuYnxM6oMZwJnpX6qMaFsQkfYs1TSCLWfLS6ekQurDEkYK1vSkIX6eyKjkdbTKLCdETVjverNxf+8XmrChp9xmaQGJVsuClNBTEzm75MhV8iMmFpCmeL2VsLGVFFmbEglG4K3+vI66VxVvVrV825rlWYjz6MIZ3AOl+BBHZpwAy1oAwMJz/AKb452Xpx352PZWnDymVP4A+fzB7s5kL4=</latexit>

|00i<latexit sha1_base64="hpGARNx2iph8KdBQvb3RKZh+mVY=">AAACBXicbZDNSgMxFIUz/tb6N+qym2AR6sKSSMFuhIIbcVXB2kI7lEyaaUMzmSHJCGXahRtfxY0LBXHrO7jzbUzbWWjrgcDHufdyc48fC64NQt/Oyura+sZmbiu/vbO7t+8eHN7rKFGUNWgkItXyiWaCS9Yw3AjWihUjoS9Y0x9eTevNB6Y0j+SdGcXMC0lf8oBTYqzVdQs3pTHCHUVkX7AzOMYo49NL1HWLqIxmgsuAMyiCTPWu+9XpRTQJmTRUEK3bGMXGS4kynAo2yXcSzWJCh6TP2hYlCZn20tkRE3hinR4MImWfNHDm/p5ISaj1KPRtZ0jMQC/WpuZ/tXZigqqXchknhkk6XxQkApoIThOBPa4YNWJkgVDF7V8hHRBFqLG55W0IePHkZWiel3GljPFtpVirZnnkQAEcgxLA4ALUwDWogwag4BE8g1fw5jw5L8678zFvXXGymSPwR87nD7H/lqE=</latexit>

J(|01i � |10i) = 0

<latexit sha1_base64="Goa9Nk47plNMmwREaticeMgMb+Q=">AAACAHicbZDNSgMxFIUz9a/Wv1HBjZtgEdxYJlKwy4IblxWsLbRDyaR32tBMZkgyQpl24au4caEgbn0Md76NaTsLbT0Q+Dj3Xu7NCRLBtfG8b6ewtr6xuVXcLu3s7u0fuIdHDzpOFYMmi0Ws2gHVILiEpuFGQDtRQKNAQCsY3czqrUdQmsfy3owT8CM6kDzkjBpr9dyTiUe6isqBgEs8IV7OPbfsVby58CqQHMooV6PnfnX7MUsjkIYJqnWHeInxM6oMZwKmpW6qIaFsRAfQsShpBNrP5vdP8bl1+jiMlX3S4Ln7eyKjkdbjKLCdETVDvVybmf/VOqkJa37GZZIakGyxKEwFNjGehYH7XAEzYmyBMsXtrZgNqaLM2MhKNgSy/OVVaF1VSLVCyF21XK/leRTRKTpDF4iga1RHt6iBmoihCXpGr+jNeXJenHfnY9FacPKZY/RHzucPQ7GVZw==</latexit>

|01i � |10i

dark state

Ion-trap quantum computer

Courtesy: Innsbruck ion-trap group

Second application: Noise characterisationin a quantum computer

Innsbruck linear ion-trap lab

Vacuum chamber

Ions confined in a string by a Paul trap

endcaps DC

AC RF fields

Solution: Effective confinement by combining static and oscillating electric fields.

1b. Quantum Computing with Trapped Ions

• qubits: long-lived internal electronic or spin states

• single qubit gate: laser

• two-qubit gate: phonons as quantum data bus

• measurement: quantum jumps

Friday, April 3, 2009

ions form Coulomb crystals (e.g. linear chains)

phonons as information bus: entangling quantum gates

small oscillations around equilibrium positions: collective vibrations (phonons)

laser cooling

Charged particles can’t be trapped by static electric fields only (Earnshaw’s theorem).

Mechanical analog

Ion Coulomb crystals

Universität Innsbruck

Loading your desired number of ions

S1/2

D5/240Ca+

Qubit transi+on

Physical qubits are encoded in (meta-)stable electronic states

70 µm

R. Blatt group(Innsbruck)

|1i

|0i

Simplified electronic level scheme

The qubit register

70 µm


Single-qubit quantum gates

can be realised by tightly focused, near-resonant laser beams applied to individual ions

Simplified electronic level scheme

+++ ......

U(�) = exp

✓�i

�

2X

◆Example:

�

⌦

|gi

|ei

|1i

|0i

S1/2

D5/240Ca+

|1i

|0i

Qubit transi+on

Laser

Single-qubit gate operations

|0�

|1�

laser pulses

+++

Key idea:

Use collective vibrational modes (phonons) as a quantum information bus

Lasers pulses that couple electronic (qubit) states of different ions to the collective vibrational modes can mediate entangling interactions between two or more qubits

Two- or multi-ion entangling gates + arbitrary single-qubit rotations = universal set of quantum gates

Entangling gate operations

... 14-qubit entanglement, T. Monz et al. PRL 106,130506 (2011)

... fidelity > 99.3 % or higher for 2 qubits, e.g. Benhelm et al. Nat. Phys. 4, 463 (2008)

... entanglement in a 20-qubit register, Friis et al. Phys. Rev. X 8, 021012 (2018)

P1/2

S1/2

D5/2

40Ca+

Detec+on ≈ 5 ms

|0i

|1i70 µm |0� |1�


Readout / measurement of qubits: Quantum states can be discriminated via laser-induced fluorescence light, recorded by a CCD camera

Working principle

Electron in is excited to the P-state, decays quickly, under an emission of a photon, excited again… “cycling transition”

Electron in does not couple to the laser light. |0i

|1ibright ion = qubit in |1i

dark ion = qubit in |0i

Measurement of the qubit register

P1/2

S1/2

D5/2

40Ca+

Detec+on ≈ 5 ms

|0i

|1i

5µm

|00i

|11i

|01i

|10i

Collect fluorescence with CCD camera (picture) Repeat 100-200 times

Example, repeat 100 times:

50% 50%

...

| i

?or|1i 1p

2(|0i+ |1i)

|1i 1p2(|0i � |1i)

Measurement of populations in and|0i |1i

How do we know which state we have?Rotate qubits before the measurement

reconstruction ofquantum states and dynamics(“state and process tomography”)

Example: 2 ions

Quantum state and process reconstruction

-1/21/2

-1/21/2

-3/2-5/2

3/25/2

|0i

|1i

Characterisation of noise correlations in a real quantum computer

Usual preferred choice: ‘clock qubit’ (First-order insensitive to magnetic field fluctuations)

16.16 Markus Muller

Fig. 4: Electronic level scheme of 40Ca+. The green and blue squares and circles indicate dif-ferent qubit encodings, denoted A and B, respectively. Squares are marking the qubit state |1iwhereas the state |0i is highlighted with circles. The corresponding frequency shifts of the tran-sitions caused by the magnetic field are �2.80MHz/G and +3.36MHz/G for the qubits markedwith green and blue symbols respectively. For configuration 1 described in the enumeration inthe main text for both qubits the encoding marked in green is used. The asymmetry in scenario2 is introduced by encoding one of the qubits in the states illustrated in blue. For the thirdconfiguration both qubits again use the encoding marked in green and the spontaneous decayfrom |0i to |1i is enhanced. Figure from [28].

Let us now discuss how the temporal development of the spatial correlation estimator I can beused to determine the degree of spatial correlations in a two-qubit register. For this, we performfull quantum process tomography on qubit registers with varying degree of correlations. Theelectronic hyperfine level structure of the 40Ca+ (see Fig. 4) is rich enough to allow the experi-mentalists to choose and investigate the noise characteristics for qubits encoded in various pairsof computational basis states. Here, the idea is that the degree of noise correlations betweenindividual qubits can be tuned by encoding them in Zeeman states with differing magnetic fieldsusceptibility. As a consequence, different sensitivities to noise from magnetic field fluctuationsis expected. Concretely, there exist multiple possibilities to encode a qubit in the Zeeman levelsof the 4S1/2 and 3D5/2 states as shown in Fig. 4. The susceptibility of the qubits to the magneticfield ranges from �2.80MHz/G to +3.36MHz/G, which allows the experimentalists to tune notonly the coherence time of the individual qubits but also the correlations between qubits, whenmagnetic field fluctuations are the dominant source of noise.Understanding the dephasing dynamics, and in particular noise correlations, in registers con-taining qubits in different encodings is essential in the context of error mitigation and quantumerror correction: this understanding will be needed to determine the viability of an approach tobuild, e.g., functional logical qubits formed of entangled ensembles of physical qubits, whichcan be used to fight errors by means of quantum error correction techniques.

Two Ions: two possible encodings

16.16 Markus Muller



16.16 Markus Muller



Different susceptibility to magnetic field changes

16.18 Markus Muller

0 1 2 3 4 5t/⌧

0

2

4

6

8

10

12

14

16

18

I(%

)

asym.sym.uncorrel.

Fig. 5: Dynamics of the spatial correlation quantifier I for different qubit encodings.Three cases are depicted: Both qubits encoded in |4S1/2, mS=�1/2i $ |3D5/2, mS=�5/2i(blue triangles), one qubit encoded in |4S1/2, mS=�1/2i $ |3D5/2, mS=�5/2i and|3D1/2, mS=�1/2i $ |3D5/2, mS=�5/2i (green circles) and both qubits subject to uncor-related dynamics via spontaneous decay (red diamonds). The horizontal axis is normalized tothe coherence time for the first two cases and to the decay time for the third case. Results froma Monte Carlo based numerical simulations with 500 samples are depicted with shaded areasin the corresponding color. Figure from [28].

The small quantum register consisting of only two qubits allows one to perform full processtomography [12] to fully reconstruct the dynamics ES in the two-qubit system. From this, thecorrelation measure I (see Eq. (20)) can be directly determined. In the present platform, theamplitude of the magnetic field fluctuations is non-stationary as it depends on the entire lab-oratory environment, e.g., due to fluctuating currents flowing through wires, which thereforecannot be controlled accurately. However, the apparatus allows one to engineer a stationarymagnetic field noise as the dominating noise source (a situation where laser and magnetic fieldnoise have to be taken into account is described in [28]). Thus we could control and tune thesingle qubit coherence time. The stationary magnetic field noise is engineered by our experi-mental colleagues by applying a white-noise current to the coils that generate the magnetic fieldat the ions’ positions. The noise amplitude is set such that the coherence time of the qubit en-coded in

��4S1/2, mS=�1/2↵

and��3D5/2, mS=�5/2

↵is reduced from 59(3)ms to 1.98(7)ms.

The increase of magnetic field noise by a factor of ⇡ 30 ensures that laser phase-noise is neg-ligible on these timescales. From the measured data, a process matrix fully describing ES wasreconstructed using an iterative maximum likelihood method to ensure trace preservation andpositivity of the process matrix.


4.3 Experimental determination of spatial dynamical correlations

In the following we will consider dephasing dynamics that is caused by a magnetic field actingon a string of two ions. The various qubit-states have different susceptibilities to magnetic fieldfluctuations, given by the Lande g factors gi of the involved Zeeman substates. The phase thatqubit i accumulates during the time evolution is therefore given by

�i(t) =

Z t

0

d⌧B(⌧)µbgi

with the time-dependent magnitude of the magnetic field B(⌧) and the Bohr magneton µb. Themagnetic field fluctuations can be modeled by multiple random implementations of B(t). Thetime evolution for a single implementation can then be expressed as

U(�1) = exp��i�1(�

z1+g�

z2)�

(33)

with the ratio of the Lande factors g=g2/g1. In order to estimate the dynamics under a dephasingdecay, one needs to average the evolution over many noise realizations with random phases.In the experiment we investigated the following qubit configurations that implement dephasingand spontaneous decay dynamics:

1. Configuration 1: For the realization of maximally correlated dephasing dynamics, bothqubits are encoded in the

��4S1/2, mS=�1/2↵

and��3D5/2, mS=�5/2

↵states. This en-

coding is referred to as encoding A hereinafter, and corresponds to the green markers inFig. 4. Both qubits have a susceptibility to the magnetic field of �2.80MHz/G, leadingto identical susceptibility coefficients (g = 1) (see Eq. (33)).

2. Configuration 2: To introduce an asymmetric dephasing dynamics, one qubit is encodedin A and the second is encoded in the states

��3D1/2, mS=�1/2↵

and��3D5/2, mS=�5/2

↵

respectively. This encoding is referred to as encoding B hereinafter, and correspondsto the blue markers in Fig. 4. Their different susceptibilities to magnetic field noise of�2.80MHz/G and +3.36MHz/G introduce unequal dephasing and therefore are expectedto affect correlations between the qubits, corresponding to the susceptibility coefficients(g = �0.83).

3. Configuration 3: Uncorrelated dynamics can be engineered in this experimental sys-tem by introducing spontaneous decay. In this scenario, both qubits are encoded in En-coding A. A laser pulse resonant with the 3D5/2 $ 4P3/2 transition at 854 nm short-ens the effective lifetime of the exited state by inducing a spontaneous decay to the4S1/2,mS = �1/2 level via the short-lived 3P3/2,mS = �3/2 level. Since spontaneousemission of visible photons by the ions at a distance of several micrometers correspondsto an uncorrelated noise process, as we have seen in Sec. 3, this controllable pump processimplements an uncorrelated noise process that can be modeled as spontaneous decay. Theeffective lifetime depends on the laser power and is in our case set to be Tspont = 7(1)µs.




�i(t) =

Z t

0

d⌧B(⌧)µbgi


U(�1) = exp��i�1(�

z1+g�

z2)�

(33)



��4S1/2, mS=�1/2↵

and��3D5/2, mS=�5/2

↵states. This en-



��3D1/2, mS=�1/2↵

and��3D5/2, mS=�5/2

↵



Qubit i accumulates phase




�i(t) =

Z t

0

d⌧B(⌧)µbgi


U(�1) = exp��i�1(�

z1+g�

z2)�

(33)



��4S1/2, mS=�1/2↵

and��3D5/2, mS=�5/2

↵states. This en-



��3D1/2, mS=�1/2↵

and��3D5/2, mS=�5/2

↵



Time evolution of a single implementation

both qubits in same encoding (Maximally correlated noise)

Laser-induced spontaneous emission (uncorrelated)

Qubits in different encodings(Partially correlated)

| ""i

| "#i

| #"i

| ##i | ##i

ei�1 | "#i

ei�2 | #"i

ei(�1+�2)| ""i

Harnessing spatially correlated noiseDecoherence-free subspaces

Random B(t)

uncorrelated

| ##i

ei�| "#i

ei�| #"i

e2i�| ""i



HG(t) =1

2B(t)

X

k

Zk (16)






Fully correlated

| ##i

e2i�| ""i

a| "#i+ b| #"i ei�(a| "#i+ b| #"i)

�

invariant

Decoherence Free Subspace

ei�| "#i

ei�| #"i

Decoherence-free subspacesHarnessing spatially correlated noise



HG(t) =1

2B(t)

X

k

Zk (16)






Random B(t)

� �

<latexit sha1_base64="gZPMBfA7dbR/L11sBmU0vqi6pw4=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCFw8eKlhbSEPZbCft0s0m7G6EEvszvHhQEK/+GW/+G7dtDtr6YODx3gwz88JUcG1c99spra1vbG6Vtys7u3v7B9XDowedZIphmyUiUd2QahRcYttwI7CbKqRxKLATjq9nfucRleaJvDeTFIOYDiWPOKPGSv6T21NUDgX2b/vVmlt35yCrxCtIDQq0+tWv3iBhWYzSMEG19j03NUFOleFM4LTSyzSmlI3pEH1LJY1RB/n85Ck5s8qARImyJQ2Zq78nchprPYlD2xlTM9LL3kz8z/MzEzWCnMs0MyjZYlGUCWISMvufDLhCZsTEEsoUt7cSNqKKMmNTqtgQvOWXV0nnou5d1j3v7rLWbBR5lOEETuEcPLiCJtxAC9rAIIFneIU3xzgvzrvzsWgtOcXMMfyB8/kDoIuRQw==</latexit>

|0iL<latexit sha1_base64="SGaXzxQ2C6ZzZcE8eda+xRll3oc=">AAAB83icbVBNSwMxEM3Wr1q/qh69BIvgqWxEsMeCFw8eKlhb2C4lm862odlkSbJCWfszvHhQEK/+GW/+G9N2D9r6YODx3gwz86JUcGN9/9srra1vbG6Vtys7u3v7B9XDowejMs2gzZRQuhtRA4JLaFtuBXRTDTSJBHSi8fXM7zyCNlzJeztJIUzoUPKYM2qdFDyRnqZyKKB/26/W/Lo/B14lpCA1VKDVr371BoplCUjLBDUmIH5qw5xqy5mAaaWXGUgpG9MhBI5KmoAJ8/nJU3zmlAGOlXYlLZ6rvydymhgzSSLXmVA7MsveTPzPCzIbN8KcyzSzINliUZwJbBWe/Y8HXAOzYuIIZZq7WzEbUU2ZdSlVXAhk+eVV0rmok8s6IXeXtWajyKOMTtApOkcEXaEmukEt1EYMKfSMXtGbZ70X7937WLSWvGLmGP2B9/kDohmRRA==</latexit>

|1iL

take 2 physical qubits

for 1 protected ‘logical’ qubit

Entanglement-based magnetometry in a trapped-ion quantum computer

conceived using single-qubit and two-qubit CNOT gates[25,26], it would be desirable to devise trapped-ion circuitsthat exploit MS gates directly and to study how errorspropagate on those circuits to demonstrate fault tolerance.We address these points by presenting a detailed descrip-tion of a generic MS-based toolbox for QEC. We apply thistoolbox to the seven-qubit topological color code withtrapped ions, either using a non-fault-tolerant stabilizerreadout with seven data qubits and one additional ancillaryqubit (i.e., 7! 1-qubit scheme) based on multiqubit orsequential two-qubit MS gates, or by using fault-tolerantstabilizer readout via MS-based schemes that realize theequivalent of the CNOT DiVincenzo-Shor (DVS) protocolfor 7! 5 qubits [25] or the DiVincenzo-Aliferis (DVA)protocol for 7! 4 qubits [26]. Although we have focusedon this particular code, we remark that this trapped-ionQEC toolbox for stabilizer readout can be generalized toany other stabilizer QEC code of interest and scaled tolarger-size codes in a modular fashion.The MS-based stabilizer readout is used, in combination

with some of the elementary operations of Tables II and IV,as a building block for the development of trapped-ionQEC protocols in Sec. VI. As already outlined above, weexplore different scenarios according to varying experi-mental capabilities:

(1) Shuttling-based protocol.— Here, we considertrapped-ion crystals with either a single or twospecies of ions species, i.e., data and ancillary qubitsbeing encoded in the same or different atomicspecies. We develop sequences of crystal-reconfig-uration operations and stabilizermappings to performa full QEC cycle on a single logical qubit.We explorehow the ability of crystal recooling by sympatheticcooling via the ancillary ion at intermediate stagesaffects the performance of the protocol.

(2) Hiding-based protocol.— Here, we consider theprotocols realized in a static ion crystal. Qubitsare selectively addressed by shelving inactive ionsvia spectroscopic decoupling and recoupling pulses,and combined with stabilizer mappings to perform afull QEC cycle on a single logical qubit. We considerencoding of data and ancillary qubits in two differentspecies and the possibility to apply recooling afterthe readout.

These QEC protocols are complemented with the errormodel introduced in Sec. IV, which improves upon custom-ary circuit-error models that consider a unique quantumchannel for all elementary operations in a QEC cycle. Thisallows us to perform a detailed study that goes beyondstandard, albeit not very realistic, assumptions: (i) Weconsider that the different gates (including the identity), thestate preparation, and the measurements do not take thesame amount of time. (ii) We use distinct error channelsaffecting the different stages of the QEC protocols. Forinstance, idle qubits are subjected to dephasing in atrapped-ion setup, whereas single- and multiqubit gatesare subjected to depolarizing noise. More importantly,(iii) the different channels are not all characterized by aunique error probability. Certainly, single- and multiqubitgates do not have the same error in any known experimentalplatform. We use a microscopic modeling of the ioncrystals to derive the particular expressions or values ofthe corresponding error rates for each operation. Therefore,our treatment not only goes beyond models that do notconsider, or simplify, the occurrence of errors on thesyndrome readout, but it also goes beyond the standardso-called circuit-level noise model, which typically makesthese over-simplifications.These sections set the stage for a large-scale numerical

analysis that investigates the performance of such protocolsin Sec. VII. The criterion introduced for beneficial or usefulQEC is used to quantify the three essential requirementsthat will need to be met in forthcoming experiments fortrapped-ion QEC: (i) sufficiently small natural physicalerror rates from fundamental error sources, (ii) detectionand dynamical correction of errors at a fast enough rate, and(iii) sufficiently accurate realizations of unavoidably imper-fect error-correction routines so that there is still an overallgain of applying (imperfect) QEC procedures.Finally, in Sec. VIII, we present our conclusions.

FIG. 1. The Sandia HOA2 trap as a QEC platform: In ourenvisioned scheme, 40Ca! ions (blue and red dots) are cotrappedwith 88Sr! ions (green dots) in a quantum zone divided into threestorage regions S1, S2, S3 and two manipulations zones M1, M2.Some of the 40Ca! ions can be used as data qubits to encodequantum information according to a QEC code (blue dots), whileothers (red dots) can be used as ancilla qubits for syndromeextraction. The 88Sr! ions (green dots) are used as sympatheticcoolants to reduce the number of phonons prior to the entanglinggates. Possible crystal-reconfiguration operations are shown inthe panel in the lower-right corner: (a) splitting of an ion crystal,(b) shuttling of an ion and subsequent merging with another ionto form a crystal, and (c) rotation (swapping) of a mixed speciescrystal. Schematics of the trap are adapted from a micrographin Ref. [21].

ASSESSING THE PROGRESS OF TRAPPED-ION … PHYS. REV. X 7, 041061 (2017)

041061-3

+ + + + + + + + +

ShuttleSeparate/mergeSwap

Harnessing spatially correlated noise

Entanglement-based magnetometry 16.20 Markus Muller

measure magnetic-field gradients while rejecting common-mode fluctuations [26–28].Quantum entanglement can be harnessed to extend

sensing capabilities [29,30]. Entangled Greenberger–Horne–Zeilinger or NOON states can, in principle, yielda sensitivity beyond the standard quantum limit [31–33].However, an increased sensitivity also implies an increasednoise-induced decoherence [34]. Hence, the beneficialeffect of entanglement is generally compromised unlessmeasurement schemes are designed to reject noise in favorof the desired signal. With trapped ions, entangledsensor states of the type !j!"i" ei!j"!i#=

!!!2

phave been

employed to measure local magnetic-field gradients [35,36]as well as the magnetic dipole interaction between theconstituents’ valence electrons [37].In this article, we present a magnetic gradiometer, where

entangled ions are moved to different locations x1 and x2along the trap axis of a segmented linear Paul trap. The dcmagnetic-field difference !B!x1; x2# between the ionlocations can be inferred from the phase accumulation rateof these sensor states via the linear Zeeman effect

!"!x1; x2#dc ! _!dc $g#B!

!B!x1; x2#: !1#

Since the net magnetic moment of the two constituent ionsvanishes, common-mode noise is rejected such that thecoherence time can exceed 20 s [35,36,38,39]. Combinedwith the fine-positioning capabilities offered by trappedions, this enables magnetic-field sensing in a parameterregimewhich could previously not be accessed:We sense dcfield differences at around 300 fT precision and 12 pT=

!!!!!!Hz

p

sensitivity, and the spatial resolution is limited by the size ofthe ion’s ground-state wave function of about 13 nm.In Sec. II, we describe the procedure for measuring the

relative phase ! of sensor states, apply it to determine phaseaccumulation rates !"!x1; x2# in Sec. III, and discuss thelimitations in Sec. IV. An efficient measurement schemeutilizing Bayesian frequency estimation is presented inSec. V. In Sec. VI, we extend our sensing scheme to inferboth dc and ac magnetic-field differences from the mea-sured phase accumulation rates. Finally, in Sec. VII, wecompare our results to state-of-the-art magnetic-field meas-urement techniques and discuss applications of our sensor.

II. EXPERIMENTAL PROCEDURE

We trap two 40Ca" ions in a segmented linear Paul trap[40], featuring 32 control electrode pairs along the trapaxis x. The distance between the center of neighboringelectrodes is 200 #m. A dc trapping voltage of !6 V leadsto an oscillation frequency of the ions of about 1.5 MHzalong the trap axis, corresponding to a 1$ width of theground-state wave function of about 13 nm.A quantizing magnetic field at an angle of 45° to the trap

axis is created by Sm2Co17 permanent magnets, splitting

the ground-state Zeeman sublevels j"i! jS1=2; mj $ ! 12i

and j!i! jS1=2; mj $ " 12i by about 2% ! 10.4 MHz. The

trap setup is shielded from ambient magnetic-field fluctua-tions by a #-metal magnetic shielding enclosure, yielding acoherence time of about 300 ms [41] in a Ramsey-typeexperiment.Laser cooling, coherent spin manipulations, and read-out

[42] take place in the laser interaction zone (LIZ) of the trap(Fig. 1). An experimental cycle starts with Doppler lasercooling a two-ion crystal on the S1=2 # P1=2 cyclingtransition near 397 nm. All collective transverse modesof vibration of the ion crystal are cooled close to themotional ground state via resolved sideband cooling on thestimulated Raman transition between j!i and j"i. Stateinitialization to j!!i is achieved via frequency-selectivepumping utilizing the narrow S1=2 # D5=2 quadrupoletransition near 729 nm.A pair of copropagating laser beams, detuned by 2% !

300 GHz from the cycling transition, serves to drive spinrotations without coupling to motional degrees of freedom.After state initialization, a %=2 pulse on both ions createsthe superposition state j!!i" ij!"i" ij"!i ! j""i. Then,an entangling geometric phase gate [43] is carried out.A spin-dependent optical dipole force transiently excitescollective vibrations only for parallel spin configurations,such that the j!!i and j""i states acquire a phase of %=2.

LIZ

Prepare sensor state

L R

Wait

Rotate basis

Shelve

Read-out

729

nm

D5/2

S1/2

P1/2 Ram

an

397nm

S1/2T

ime

397

nm

S1/2

P1/2866 nm

D3/2

Trap axis

FIG. 1. Experimental procedure for measurements of inhomo-geneous magnetic fields. After creation of the sensor state at theLIZ, the two constituent ions are separated and shuttled to thedesired trap segments L and R. In order to measure theaccumulated phase during the interrogation time T, the ionsare individually shuttled to the LIZ to perform basis rotations thatallow for state read-out via electron shelving and fluorescencedetection in either the X1X2 or X1Y2 basis. For basis rotations,electron shelving, and fluorescence detection, the relevant energylevels are shown.

T. RUSTER et al. PHYS. REV. X 7, 031050 (2017)

031050-2

Fig. 6: Experimental procedure implemented in [29] for measurements of inhomogenous mag-netic fields in a segmented ion trap. After the creation of the sensor Bell state by means ofsingle- and two-qubit gates in the laser interaction zone (LIZ), the two constituent ions areseparated and shuttled to the desired trap segments L and R. In order to measure the accumu-lated phase during the interrogation time T , the ions are individually shuttled back to the LIZto perform basis rotations that allow for state read-out via electron shelving and fluorescencedetection in either the X1X2 or Y1Y2 measurement basis. For basis rotations, electron shelving,and fluorescence detection, the relevant energy levels are shown in the small inset figures at theright. Figure reproduced from [29].

The key idea of how this works can be understood by considering a fluctuating magnetic fieldwhich acts with exactly the same magnitude, and thus perfectly spatially correlated on allqubits, as discussed above and described by Hamiltonian of Eq. (16). A single-qubit super-position state | ik = ↵ |0ik + � |1ik will under such noise dephase over time and end up in aclassical mixture |⇢ik = |↵|2 |0ih0|k + |�|2 |1ih1|k. If we, however, consider instead a Bell stateof two qubits,

| i12 = ↵ |01i12 + � |10i12 , (34)

we find that under the time evolution generated by the collective dephasing Hamiltonian Eq. (16)such a superposition remains an eigenstate of the time evolution operator at all times. Or in otherwords, no relative phase in the superposition state (34) is accumulated. Therefore, under thiscorrelated dephasing noise, the basis states |01i12 and |10i12 span a two-dimensional so-calleddecoherence-free subspace (DFS): this is a subspace of the two-qubit Hilbert space, within


16.26 Markus Muller

[23] M.B. Plenio and S. Virmani, Quant. Inf. Comp. 7, 1 (2007)

[24] F.G.S.L. Brandao and M.B. Plenio, Nat. Phys. 4, 873 (2008)

[25] A. Rivas and M. Muller, New J. Phys. 17, 062001 (2015)

[26] M. Abramowitz and I.A. Stegun: Handbook of Mathematical Functions(Dover Publications, New York, 1972)

[27] P. Schindler, et al., New J. Phys. 15, 123012 (2013)

[28] L. Postler, et al., Quantum 2, 90 (2018)

[29] T. Ruster, et al., Phys. Rev. X 7, 031050 (2017)

Bell state as sensor<latexit sha1_base64="TAvfGHkdlXRxnBbfh24/cIEjaEo=">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</latexit>

! 1p2(|1i|0i+ |0i|1i)

<latexit sha1_base64="JiLy0QXHkXmsp7yycJnyf7jcOrc=">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</latexit>

! 1p2(|1i|0i+ ei�|1i|0i)

magn. field inhomogeneities


measurements, i.e., j!!T; x1; x2" ! !meas!T; x1; x2"j < ". Inorder to check if the phase has been incremented ordecremented properly, we verify that the residuals of allpoints are well below ". Figure 2 shows an examplemeasurement at maximum ion distance d # 6.2 mm andthe residuals #! for each point. In this measurement, phasesof over 40 000 rad have been accumulated during inter-rogation times of up to Tmax # 1.5 s, but the residuals j#!jof all measurement points are well below ".The maximum interrogation time Tmax is ultimately

limited by the coherence time Tcoh of the sensor state.The coherence time is therefore analyzed in the followingsection.

IV. COHERENCE TIMES

We characterize the coherence time Tcoh of the sensorstate for two settings: The ions are kept (i) in a commonpotential well at a distance of about 4.2 $m and (ii) inseparate harmonic wells at the maximum possible distanceof 6.2 mm. The coherence time is inferred from measure-ments of the contrast C for varying interrogation times T.For each interrogation time, we repeat the experimental

procedure 400 times for each of the two measurementoperators.For case (i), a coherence time Tcoh > 12.5 s is observed

[Fig. 3(a)]. In this regime, residual heating of the radialmodes of motion compromises the fidelity of electronshelving and therefore the spin read-out. In separatemeasurements, we characterized the spin read-out effi-ciency for the input states j!!i and j""i, and confirmedthat the observed contrast loss is entirely caused byread-out.For the maximum possible ion distance, a Gaussian

contrast decay is observed, with a coherence time in the1–2-s range. For Gaussian contrast decay, the best sensi-tivity for our phase measurement scheme is achieved at aninterrogation time corresponding to a contrast of 0.85(see Ref. [44]).The contrast decay at maximum ion distance is presum-

ably caused by a slow drift of the magnetic-field minimumposition along the trap axis. In order to verify ourpresumption, we measured this drift consecutively fortwo different ion separation distances of d # 6.2 mmand d # 3.2 mm over the course of 6 hours [Fig. 3(b)].For the former case, a typical drift rate of 1 Hz=h isobserved. We verified that this drift rate corresponds to acontrast decay within 2s. For an ion distance ofd # 3.2 mm, the drift rate is suppressed by a factor ofabout 1.6 as compared to the maximum ion distance. Thespatial dependence of the observed drift rates is consistentwith movement of the ion trap relative to the magnetic fieldin the 200-nm range, equivalent to thermal expansion of ourvacuum chamber due to a temperature change ofroughly 0.1 °C.

V. BAYESIAN FREQUENCY ESTIMATION

In order to speed up the incremental measurementscheme for determining !%!x1; x2" described in Sec. III,we implement an adaptive scheme for frequency estimationbased on a Bayesian experiment design algorithm [51,52].In general, such algorithms control the choice of a

measurement parameter—in our particular case, the inter-rogation time—which, in each measurement run, guaran-tees the optimum gain of information on the parameter to bedetermined. These algorithms are beneficial in situationswhere only a few parameters are to be determined, anaccurate model relating the design parameters to themeasurement outcome holds, and the measurement is“expensive” in terms of resources such as time.In Bayesian statistics, for a given phase measurement to

be carried out, the combined result of all previous mea-surements is expressed with the prior probability distribu-tion function (PDF) p!!%;!0". Initially, we assume auniformly distributed prior PDF, limited to a reasonableparameter range !% " f!%min;!%maxg and !0 " f!"; "g.After a phase measurement with the outcome fn;mg, thecombined result is described by the posterior PDF,

Read-out limit

(a)

(b)

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

Sen

sor

stat

e co

ntra

st C

Interrogation time T (s)

-1.5

-1

-0.5

0

0 50 100 150 200 250 300 350

Fre

quen

cy s

hift

(Hz)

Elapsed time (min)

d = 6.2 mmd = 3.2 mm

d = 6.2 mmd = 4.2 µm

FIG. 3. (a) Sensor state contrast C versus interrogation time T atthe maximum ion distance of d # 6.2 mm (red dots) and at an iondistance of d # 4.2 $m (blue squares). For illustration, the blackcurve and gray region indicate a third-order polynomial fit to aseparate read-out fidelity measurement and its confidence bands.(b) Simultaneous drift of the measured frequency difference forion distances d # 6.2 mm (blue circles) and d # 3.2 mm (purpletriangles) over a duration of about 6 hours with an interrogationtime of T # 150 ms. For d # 3.2 mm, the measured drift issuppressed by a factor of about 1.6 as compared to the maximumion distance.


031050-4

Fig. 7: (a) Sensor state contrast C as a function of the interrogation time during which the twoions of the sensor Bell state (34) are exposed to the magnetic fields at their respective positions,spatially separated by a distance of d = 6.2 mm (red dots) and d = 4.2 µm (blue squares). Forillustration, the black curve and gray region indicate a third-order polynomial fit to a separateread-out fidelity measurement and its confidence bands (see [29]). (b) Simultaneous drift of themeasured frequency difference for ion distances d = 6.2 mm (blue circles) and d = 3.2 mm(purple triangles) over a duration of about 6 hours with an interrogation time of T = 150 ms.Figure reproduced from [29].

which the noise acts trivially and quantum information can be stored and protected for longertimes than in single physical qubits. Alternatively, one can view the state of Eq. (34) as aminimal “logical qubit” formed of two physical qubits, with effective logical basis states |0iL =

|01i12 and |1iL = |10i12, and which offers protection against spatially correlated dephasingnoise. Using such two-qubit DFS spaces in two ions, quantum information and entanglementcan be preserved for timescales of minutes, as impressively demonstrated already in 2005 [9],



HG(t) =1

2B(t)

X

k

Zk (16)







measurements, i.e., j!!T; x1; x2" ! !meas!T; x1; x2"j < ". Inorder to check if the phase has been incremented ordecremented properly, we verify that the residuals of allpoints are well below ". Figure 2 shows an examplemeasurement at maximum ion distance d # 6.2 mm andthe residuals #! for each point. In this measurement, phasesof over 40 000 rad have been accumulated during inter-rogation times of up to Tmax # 1.5 s, but the residuals j#!jof all measurement points are well below ".The maximum interrogation time Tmax is ultimately

limited by the coherence time Tcoh of the sensor state.The coherence time is therefore analyzed in the followingsection.

IV. COHERENCE TIMES

We characterize the coherence time Tcoh of the sensorstate for two settings: The ions are kept (i) in a commonpotential well at a distance of about 4.2 $m and (ii) inseparate harmonic wells at the maximum possible distanceof 6.2 mm. The coherence time is inferred from measure-ments of the contrast C for varying interrogation times T.For each interrogation time, we repeat the experimental

procedure 400 times for each of the two measurementoperators.For case (i), a coherence time Tcoh > 12.5 s is observed

[Fig. 3(a)]. In this regime, residual heating of the radialmodes of motion compromises the fidelity of electronshelving and therefore the spin read-out. In separatemeasurements, we characterized the spin read-out effi-ciency for the input states j!!i and j""i, and confirmedthat the observed contrast loss is entirely caused byread-out.For the maximum possible ion distance, a Gaussian

contrast decay is observed, with a coherence time in the1–2-s range. For Gaussian contrast decay, the best sensi-tivity for our phase measurement scheme is achieved at aninterrogation time corresponding to a contrast of 0.85(see Ref. [44]).The contrast decay at maximum ion distance is presum-

ably caused by a slow drift of the magnetic-field minimumposition along the trap axis. In order to verify ourpresumption, we measured this drift consecutively fortwo different ion separation distances of d # 6.2 mmand d # 3.2 mm over the course of 6 hours [Fig. 3(b)].For the former case, a typical drift rate of 1 Hz=h isobserved. We verified that this drift rate corresponds to acontrast decay within 2s. For an ion distance ofd # 3.2 mm, the drift rate is suppressed by a factor ofabout 1.6 as compared to the maximum ion distance. Thespatial dependence of the observed drift rates is consistentwith movement of the ion trap relative to the magnetic fieldin the 200-nm range, equivalent to thermal expansion of ourvacuum chamber due to a temperature change ofroughly 0.1 °C.

V. BAYESIAN FREQUENCY ESTIMATION

In order to speed up the incremental measurementscheme for determining !%!x1; x2" described in Sec. III,we implement an adaptive scheme for frequency estimationbased on a Bayesian experiment design algorithm [51,52].In general, such algorithms control the choice of a

measurement parameter—in our particular case, the inter-rogation time—which, in each measurement run, guaran-tees the optimum gain of information on the parameter to bedetermined. These algorithms are beneficial in situationswhere only a few parameters are to be determined, anaccurate model relating the design parameters to themeasurement outcome holds, and the measurement is“expensive” in terms of resources such as time.In Bayesian statistics, for a given phase measurement to

be carried out, the combined result of all previous mea-surements is expressed with the prior probability distribu-tion function (PDF) p!!%;!0". Initially, we assume auniformly distributed prior PDF, limited to a reasonableparameter range !% " f!%min;!%maxg and !0 " f!"; "g.After a phase measurement with the outcome fn;mg, thecombined result is described by the posterior PDF,

Read-out limit

(a)

(b)

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

Sen

sor

stat

e co

ntra

st C

Interrogation time T (s)

-1.5

-1

-0.5

0

0 50 100 150 200 250 300 350

Fre

quen

cy s

hift

(Hz)

Elapsed time (min)

d = 6.2 mmd = 3.2 mm

d = 6.2 mmd = 4.2 µm

FIG. 3. (a) Sensor state contrast C versus interrogation time T atthe maximum ion distance of d # 6.2 mm (red dots) and at an iondistance of d # 4.2 $m (blue squares). For illustration, the blackcurve and gray region indicate a third-order polynomial fit to aseparate read-out fidelity measurement and its confidence bands.(b) Simultaneous drift of the measured frequency difference forion distances d # 6.2 mm (blue circles) and d # 3.2 mm (purpletriangles) over a duration of about 6 hours with an interrogationtime of T # 150 ms. For d # 3.2 mm, the measured drift issuppressed by a factor of about 1.6 as compared to the maximumion distance.


031050-4

Fig. 7: (a) Sensor state contrast C as a function of the interrogation time during which the twoions of the sensor Bell state (34) are exposed to the magnetic fields at their respective positions,spatially separated by a distance of d = 6.2 mm (red dots) and d = 4.2 µm (blue squares). Forillustration, the black curve and gray region indicate a third-order polynomial fit to a separateread-out fidelity measurement and its confidence bands (see [29]). (b) Simultaneous drift of themeasured frequency difference for ion distances d = 6.2 mm (blue circles) and d = 3.2 mm(purple triangles) over a duration of about 6 hours with an interrogation time of T = 150 ms.Figure reproduced from [29].

which the noise acts trivially and quantum information can be stored and protected for longertimes than in single physical qubits. Alternatively, one can view the state of Eq. (34) as aminimal “logical qubit” formed of two physical qubits, with effective logical basis states |0iL =

|01i12 and |1iL = |10i12, and which offers protection against spatially correlated dephasingnoise. Using such two-qubit DFS spaces in two ions, quantum information and entanglementcan be preserved for timescales of minutes, as impressively demonstrated already in 2005 [9],

Entanglement-based magnetometry


magn. field inhomogeneities

Several parties

Correlated Dynamics in Multipartite Setting

I(ES) : =1

2M log dS⇥⇢CJS

��(⇢CJS |S1S0

1)⌦ · · ·⌦ (⇢CJ

S |SMS0M)⇤

=1

2M log d

("MX

i=1

S(⇢CJS |SiS0

i)

#� S(⇢CJ

S )

)


Fig. 8: Schematic illustration of the multipartite correlation measure. (A) Choi-Jamiołkowskirepresentation of the dynamics. The system is prepared in a product of maximally entangledstates of 2M parties {Sj|S0

j} and the dynamics affects only the subsystems Sj . If and only ifthe dynamics are correlated, the bipartitions {SiS

0i|SjS

0j} will be entangled, yielding a nonzero

correlation measure I . (B) Schematic depiction of the procedure to estimate a lower bound of I .There, the system is initially prepared in a separable state ⇢S1 ⌦ ⇢S2 · · ·⌦ ⇢SM and correlationsin the dynamics show up as correlations C (see Eq. (38)) in the measurement of suitably chosenobservables Oj . Figure reproduced from [28].

dynamics can then be assessed by

I(ES) :=1

2M log dS

⇣⇢CJS

��⇢CJS |S1S01 ⌦ . . .⌦ ⇢

CJS |SMS0M

⌘

:=1

2M log d

("MX

i=1

S�⇢CJS |SiS0i

�#� S

�⇢CJS

�), (36)

where ⇢CJS |SiS0i = Tr{8Sj 6=iS0j 6=i}(⇢

CJS ).

A lower bound for the multipartite setting can be applied as shown in Fig. 8(B), by measuringcorrelations. Mathematically the same steps as in the bipartite case [see Eq. (35)] can be applied,resulting in

I(E) � 1

4M ln d

C2⇢0(O1, . . . ,OM)

kO1k2 . . . kOMk2 . (37)

Here, ⇢(t) is the joint state after the evolution of an initial product state, O1, . . . ,OM are localobservables for the parties S1, . . . , SM , respectively, and the correlation function is

C⇢(t)(O1, . . . ,OM) = hO1 . . .OMi⇢(t) � hO1i⇢(t) . . . hOMi⇢(t). (38)

This multipartite bound makes investigating correlation dynamics accessible in systems that aretoo large for full quantum process tomography, as here the number of measurements increasesonly linearly compared to the exponential scaling for full quantum process tomography.

with

I(ES) : =1

2M log dS⇥⇢CJS

��(⇢CJS |S1S0

1)⌦ · · ·⌦ (⇢CJ

S |SMS0M)⇤

=1

2M log d

("MX

i=1

S(⇢CJS |SiS0

i)

#� S(⇢CJ

S )

)

Lower Bound EstimationThe exact experimental determination of I becomes

impractical as the number of systems increases

Lower estimators

C2E(⇢)(X1, . . . , XM ) = hX1 . . . XM iE(⇢) � hX1iE(⇢) . . . hXM iE(⇢)

I(E) � 1

4M ln d

C2E(⇢)(X1, . . . , XM )

kX1k2 . . . kXMk2 , ⇢ =MO

i=1

⇢i

Operator norm

Segmented ion traps

Couple linear traps to build larger 2D trap arrays as scalable qubit registers

Linear Paul trap

Vision of scalable quantum computers 16.24 Markus Muller

Fig. 9: Illustration of one scalable route from macroscopic linear Paul traps (upper left) to-wards large-scale ion-trap quantum processors. Ions can be stored in segmented traps (upperright), where ion crystals can be controlled locally, and ions can be split, moved around andmerged with ion-crystals in different trapping regions. This allows one to control increasinglylarger qubit registers with high flexibility. Such linear traps can be coupled via junctions, alongwhich ions can be moved from one trap into neighboring zones, where they can be stored (S) ormanipulated (M). This will allow one to assemble traps into larger two-dimensional trap arrays,which can be used to host and control large registers of qubits for quantum error correction andeventually large-scale fault-tolerant quantum computation.

In summary, based on the mapping of quantum dynamics to quantum states in an enlargedHilbert space via the Choi-Jamiołkowski isomorphism, in this lecture we have discussed a rig-orous and systematic method to quantify the amount of spatial correlations in general quantumdynamics. Furthermore, we have applied the theoretical concepts developed to paradigmaticphysical models and demonstrated their usefulness for the characterization of noise in experi-mental quantum processors. We expect that noise characterization techniques such as the onesdiscussed in this lecture will be of fundamental importance for the study of dynamics in a largevariety of quantum systems. From a practical and more applied standpoint, such tools are likelyto be essential to make further progress in developing and characterizing increasingly larger andscalable qubits registers, as shown for trapped ions in Fig. 9, to make the dream of large-scalequantum computers and simulators become a reality.

Acknowledgments. I would like to thank my collaborators, Angel Rivas from the ComplutenseUniversity in Madrid, for the joint theory work of Refs. [25,28], as well as my colleagues fromRainer Blatt’s experimental ion-trapping group at the University of Innsbruck for the joint andjoyful collaboration that gave rise to the results of Ref. [28].

W. Hensinger group University of Sussex

Vision of scalable quantum computers

Understanding and mitigating correlated noise,and quantum error correction will be crucial

Daniel NiggPhilipp Schindler

Thomas Monz

Rainer BlattLukas Postler Alex Erhard Roman Stricker

New J. Phys. 17 (2015) 062001 doi:10.1088/1367-2630/17/6/062001

FAST TRACK COMMUNICATION

Quantifying spatial correlations of general quantum dynamics

Ángel Rivas andMarkusMüllerDepartamento de Física Teórica I, UniversidadComplutense, E-28040Madrid, Spain

E-mail: [email protected]

Keywords: open quantum systems, spatially correlated dynamics, correlation quanti!ers

AbstractUnderstanding the role of correlations in quantum systems is both a fundamental challenge as well asof high practical relevance for the control ofmulti-particle quantum systems.Whereas a lot of researchhas been devoted to study the various types of correlations that can be present in the states of quantumsystems, in this workwe introduce a general and rigorousmethod to quantify the amount ofcorrelations in the dynamics of quantum systems. Using a resource-theoretical approach, weintroduce a suitable quanti!er and characterize the properties of correlated dynamics. Furthermore,we benchmark ourmethod by applying it to the paradigmatic case of two atomsweakly coupled to theelectromagnetic radiation !eld, and illustrate its potential use to detect and assess spatial noisecorrelations in quantumcomputing architectures.

1. Introduction

Quantum systems can display awide variety of dynamical behaviors, in particular depending on how the systemis affected by its coupling to the surrounding environment. One interesting feature which has attractedmuchattention is the presence ofmemory effects (non-Markovianity) in the time evolution. These typically arise forstrong enough coupling between the system and its environment, or when the environment is structured, suchthat the assumptions of thewell-knownweak-coupling limit [1–3] are no longer valid.Whereasmemory effects(or time correlations) can be present in any quantum system exposed to noise, another extremely relevantfeature, whichwewill focus on in this work, are correlations in the dynamics of different parts ofmulti-partitequantum systems. Since different parties of a partition are commonly, though not always, identi!edwithdifferent places in space, without loss of generality wewill in the following refer to these correlations betweendifferent subsystems of a larger system as spatial correlations.

Spatial correlations in the dynamics give rise to awide plethora of interesting phenomena ranging fromsuper-radiance [4] and super-decoherence [5] to sub-radiance [6] and decoherence-free subspaces [7–11].Moreover, clarifying the role of spatial correlations in the performance of a large variety of quantumprocesses,such as e.g. quantum error correction [12–17], photosynthesis and excitation transfer [18–28], dissipative phasetransitions [29–33] and quantummetrology [34] has been and still is an active area of research.

Along the last few years, numerousworks have aimed at quantifying up towhich extent quantumdynamicsdeviates from theMarkovian behavior, see e.g. [35–43].However,much less attention has been paid to developquanti!ers of spatial correlations in the dynamics, although someworks e.g. [44, 45] have addressed this issuefor some speci!cmodels. Thismay be partially due to thewell-known fact that undermany, though not allpractical circumstances, dynamical correlations can be detected by studying the time evolution of correlationfunctions of properly chosen observables A! and B! , acting respectively on the two partiesA andB of interest.For instance, in the context of quantum computing, sophisticatedmethods towitness the correlated character ofquantumdynamics, have been developed and implemented in the laboratory [45]. Indeed, any correlationC ( , )A B A B A B= ! " # $ ! #! #! ! ! ! ! ! detected during the time evolution of an initial product state,

A B! ! != " , witnesses the correlated character of the dynamics. However, note that there exist highlycorrelated dynamics, which cannot be realized by a combination of local processes, which do not generate any

OPEN ACCESS

RECEIVED

19March 2015

REVISED

14April 2015

ACCEPTED FOR PUBLICATION

29April 2015

PUBLISHED

18 June 2015

Content from this workmay be used under theterms of theCreativeCommonsAttribution 3.0licence.

Any further distribution ofthis workmustmaintainattribution to theauthor(s) and the title ofthework, journal citationandDOI.

© 2015 IOPPublishing Ltd andDeutsche PhysikalischeGesellschaft

Thanks!

Angel Rivas

1 p 2 (|0i|0i + |1i|1i) Two-qubit entangled state A B 1 measurement (in Z-basis) 0 1 completely...

Documents

Transcript of 1 p 2 (|0i|0i + |1i|1i) Two-qubit entangled state A B 1 measurement (in Z-basis) 0 1 completely...