Three problems from quantum opticstomtyc/habilitation.pdf · basic unit of quantum information is a...

Institute of Theoretical Physics and Astrophysics

Faculty of Science, Masaryk University

Brno, Czech Republic

Three problems from quantum optics

(habilitation thesis)

Tomas Tyc

Brno 2005

Contents

1 Introduction 5

1.1 Quantum state sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Homodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Fermion coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Some important terms and concepts of quantum optics 9

2.1 Field operators and quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Linear mode transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Coherent states of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 POVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Quantum state sharing 14

3.1 Continuous-variable quantum state sharing in the Schrodinger picture . . . . . . . . . 15

3.1.1 Encoding the quantum secret . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Extraction of the quantum secret . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.3 Example: the (2,3) threshold scheme . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.4 Optimizing the secret extraction . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.5 Finite squeezing in dealer’s encoding procedure . . . . . . . . . . . . . . . . . . 19

3.2 Heisenberg picture of continuous-variable quantum state sharing . . . . . . . . . . . . 20

3.2.1 Encoding the secret . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.2 Extraction of the secret state by players 1 and 2 . . . . . . . . . . . . . . . . . 21

3.2.3 Extraction of the secret state by players 1 and 3 . . . . . . . . . . . . . . . . . 22

3.3 Experimental realization of the (2, 3) threshold scheme . . . . . . . . . . . . . . . . . . 23

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Homodyne detection 25

4.1 Homodyne detection as a phase-sensitive method . . . . . . . . . . . . . . . . . . . . . 25

4.2 Why homodyne detection measures the field quadrature . . . . . . . . . . . . . . . . . 26

4.3 POVM calculation using the SU(2) Wigner functions . . . . . . . . . . . . . . . . . . . 27

4.4 POVM calculation using the Glauber-SudarshanP -representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4.1 Properties of the series expressing the probability P jm . . . . . . . . . . . . . . 30

4.5 Strong local oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Fermion coherent states 34

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 The options for introducing coherent states of light . . . . . . . . . . . . . . . . . . . . 34

5.3 Fermion analogy of the boson coherent state . . . . . . . . . . . . . . . . . . . . . . . . 36

5.4 Properties of fermion correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3

5.4.1 Correlators of chaotic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4

Chapter 1

Introduction

This thesis is concerned with three problems from the field of quantum optics. In their choice Iwas not motivated by attempting to explain quantum optics systematically but I rather talk aboutproblems I was working on in the past five years. The thesis is based on a set of articles that havebeen published in international physical journals and are attached at the end of the thesis.

The first topic, quantum secret sharing, deals with protection of quantum information that isphysically realized by a quantum-optical system. The second topic, theory of homodyne detection, isconcerned with the description of one of the most important detection methods in quantum optics.The third topic, fermion coherent states, deals with generalization to fermion fields of coherent states,one of the key concepts of quantum optics. Each of these topics is described briefly below and indetail in a separated chapter.

I have tried to write this thesis clearly so that a physicist not specialized in quantum optics canunderstand it, and also that it can be of some use to a non-physicist. To some extent the thesisre-tells the papers it is based on; at the same time, I have tried to include all the important resultsand explain the steps that led to them so that a reader does not have to look into the papers toooften. This is also the reason why I have included Chapter 2 that explains some important termsand concepts of quantum optics that are used in this thesis.

As I said, all the topics this thesis deals with are directly connected to quantum optics. This areaof physics is, as is clear from its name, the quantum theory of light. In many situations light behavesas a wave governed by the laws of classical physics but sooner or later one comes across a situationwhere the classical description is completely unsatisfactory and the quantum nature of light presentsitself in the full extent. It is enough just to sit down by a fireplace and think what is the colorof the light emitted by the glowing coals. Classical physics would give us a completely wrong answerin the form of the “ultraviolet catastrophe” [1] while quantum optics allows to find the spectralcomposition of the emitted light in a full agreement with the observation. And of course, quantumoptics offers much more. The consequences of the quantum nature of light are vast and many ofthem are very practical. We just remind of the laser, which is a source of light commonly usedfor precise measurements, communication, reading information media, for medical therapy etc., andwhich can work thanks to the quantum properties of light. Also, quantum optics enables realizationof various cryptographic protocols, the security of which is guaranteed by the very laws of nature andnot e.g. by just computational difficulty. Last but not least, it is in quantum optics where the lawsof quantum physics often appear in a crystalline pure form and so it enables us to deeper understandthe rules that the world around us is governed by.

1.1 Quantum state sharing

At the present time, the importance of quantum optics for practical implementation of quantuminformation protocols is growing as quantum states of light belong to the best carriers of quantum

5


information [2]. Moreover, the experimental effort for realizing certain quantum-information proto-cols such as quantum teleportation has been most successful in quantum optics [3, 4, 5]. Quantuminformation theory is a fast-developing interdisciplinary field that offers options that would other-wise be impossible or very difficult [6]. For example, quantum cryptography provides nowadaysa practically usable method for an unconditionally secure information transfer without the risk ofeavesdropping [7, 8]. At the same time, processing of quantum information in quantum computersenables solving problems that would take an incomparably longer time on a classical computer (e.g.billions of years compared to a few minutes) [9, 10], and simulating quantum systems that is highlyineffective on a classical computer [11]. There are several algorithms that have been proposed forquantum computers that are designed for solving very specific problems such as large number factor-ization or search in a database [12]. However, these algorithms have a relatively limited use and sonew algorithms that would exploit the full potential of quantum computers are still to be discovered.Similarly, quantum computers themselves are waiting for their practical realization.

Quantum information differs significantly from its counterpart, the classical information. Thebasic unit of quantum information is a quantum bit (qubit). A qubit can have, just as a classicalbit, the values 0 and 1, but it can also be in a so-called superposition of these two values. Thesuperposition is a way of simultaneous existence of the two options that a human has no directexperience of, which makes it hard to imagine. A qubit is realized practically by a two-level quantumsystem, e.g. the spin state of an electron, photon polarization or even by the options “there isa photon in mode k” and “there is no photon in mode k”. In principle, one can perform similar logicaloperations with qubits as with classical bits. However, it is not possible to copy (or clone) them,which is a fundamental difference compared to classical information that can be copied arbitrarily.The impossibility of copying quantum information is an important consequence of linearity of thelaws of quantum physics and it became known as the no-cloning theorem [13]. It is also connectedto the fact that it is not possible to read quantum information completely even if one possesses thesystem carrying it; there is always some information that escapes, no matter what measurements oneperforms on the system [14].

With the expansion of quantum information theory, there is a growing interest for its storage,transfer and protection against misuse. More specifically, quantum teleportation enables to trans-fer quantum information between stations where it cannot be be sent physically (i.e., that are notconnected by a so-called quantum channel) [3, 5]. On the other hand, for protecting quantum infor-mation one can use the protocol of quantum state sharing that enables the access to the informationonly based on collaboration between several participants; without such collaboration, the access isdenied completely.

In the last few years I have been working on the theory of quantum state sharing. With co-workersI have achieved several results, the most important of which was proposing optical quantum statesharing scheme and its experimental realization at the Australian National University in Canberra.

Theory of quantum state sharing and the experimental realization is discussed in Chapter 3. Itis based on the following articles:(1) Tomas Tyc and Barry C. Sanders, ”How to share a continuous-variable quantum secret by opticalinterferometry”, Physical Review A 65, 042310 (2002)(2) Tomas Tyc, David Rowe and Barry Sanders, ”Efficient sharing of a continuous-variable quantumsecret”, Journal of Physics A 36, 7625 (2003)(3) Andrew M. Lance, Thomas Symul, Warwick P. Bowen, Tomas Tyc, Barry C. Sanders and PingKoy Lam, ”Continuous variable (2,3) threshold quantum secret sharing schemes”, New Journalof Physics 5, 4 (2003)(4) Andrew M. Lance, Thomas Symul, Warwick P. Bowen, Barry C. Sanders, Tomas Tyc, TimothyC. Ralph and Ping K. Lam, ”Continuous Variable Quantum State Sharing via Quantum Disentan-glement”, Physical Review A 71, 33814 (2005).

6

Chapter 1. Introduction

1.2 Homodyne detection

Modern quantum optics is, to a large extent, an experimental discipline for which the precise mea-surement of a quantum state of light is of a key importance. One of the essential detection methods isthe homodyne detection based on interference of the measured light beam with a beam of well-knownproperties (the so-called local oscillator). Homodyne detection is a phase-sensitive method and itenables a direct measurement of quadratures, basic quantities used for describing the quantized elec-tromagnetic field. Many important quantum-optical experiments are literally based on homodynedetection that has become a standard experimental tool. Homodyne detection is the ultimate detec-tion method in experiments with squeezed light, in quantum teleportation and cryptography withso-called continuous variables and in many other situations.

Therefore it is surprising that until recently the full quantum description of homodyne detec-tion was missing, especially the knowledge of POVM (its meaning will be explained in Chapter 2,Sec. 2.4). The theory of homodyne detection was based on indirect calculations employing charac-teristic functions and quadrature moments of the electromagnetic field [15, 16, 17, 18, 19, 20], butno direct derivation of the probability distribution of homodyne detector output was known. WhenI discussed this problem with colleagues at a conference in Vienna in 2000, we decided to work onfinding the POVM for homodyne detection and directly calculating the corresponding probabilitydistribution. In the following two years we have managed to find the POVM including correctionterms that enables to describe the detection even in the non-ideal conditions of a weak local oscillator.This result is important both for theoretical understanding of homodyne detection and for practicalapplication in which a weak reference field has to be used.

Theory of homodyne detection that I have developed in collaboration with Barry Sanders isexplained in Chapter 4 and it was published in the paper(5) Tomas Tyc and Barry C. Sanders, ”Operational formulation of homodyne detection”, Journalof Physics A 37, 7341 (2004).

1.3 Fermion coherent states

For describing the quantized electromagnetic field, quantum optics uses an extended mathematicalformalism that is sometimes very elegant. One of the most important representations of quantumstates and operators is provided by coherent states of light that possess many useful physical andmathematical properties [21]. These states exhibit high coherence, they are close to classical statesof light, do not change their character when subject to a linear mode transformation at a beamsplitter etc. Thanks to the so-called overcompleteness of the set of coherent states they can beemployed even for describing situations in which coherent states themselves do not take part, andsimplify calculations significantly.

Coherent states became best known in their connection with the electromagnetic field. Photons,the quanta of this field, belong to bosons, the group of particles that tend to gather in the samequantum state. This property is a consequence of their quantum indistinguishability – it is notpossible, even in principle, to distinguish two particles of the same sort. For the other type of particles,the fermions, quantum indistinguishability has just the opposite effect: it is not possible to find morethan one fermion in the same state. This is expressed by the Pauli exclusion principle, one of the mostfundamental statements in quantum physics. Thanks to the common properties of all bosons, one caneasily extend coherent states to arbitrary boson fields. This raises a natural question: is it possibleto generalize the concept of coherent states also to fermion fields, for example electrons or neutrons?Such a generalization is indeed possible and was performed several decades ago [22, 23, 24] based onthe so-called Grassmann numbers. However, the analogy with boson coherent states is just partial andnot direct. The largest problem is that the algebra of non-commuting Grassmann numbers providescoherent states without any physical interpretation. Therefore I was thinking about introducing

7


fermion coherent states in a direct analogy to the boson case without employing the Grassmannnumbers. When discussing this with my colleagues at Macquarie University in Sydney, we attackedthis problem and have achieved several results. We have shown that the desired generalization isnot possible and that the Grassmann variables are probably the only possibility how to introducefermion coherent states meaningfully. A side effect of our effort was deriving several theorems thatare valid for fermion correlation functions and that have no analogy for boson fields.

The problem of fermion coherent states is discussed in Chapter 5 that is based on the paper(6) Tomas Tyc and Barry C. Sanders, “Investigating complex fermion coherent states”, at the presenttime in the review process in New Journal of Physics.

8

Chapter 2

Some important terms and conceptsof quantum optics

In this chapter we remind of some important terms that will be used in this thesis. We do not intendto provide a complete introduction but rather mention some basic quantities and relations betweenthem so that a reader who is not trained in the area of quantum optics could read the followingchapters without having to look often into a quantum optics textbook.

2.1 Field operators and quadratures

Basic quantities used for describing the quantized electromagnetic field are creation and annihilationoperators a†, a that are generally called field operators. The creation and annihilation operator raisesand lowers, respectively, the number of quanta (photons) in the field. If |n〉 denotes the state with nphotons (nth Fock state), then it holds

a|0〉 = 0,

a|n〉 =√n |n− 1〉 (n = 1, 2, 3, . . . ) (2.1)

a†|n〉 =√n+ 1 |n+ 1〉 (n = 0, 1, 2, . . . )

The creation and annihilation operators satisfy the commutation relations

[ai, a†j ] = δij 1, [ai, aj ] = 0, [a†i , a

†j ] = 0, (2.2)

where the indexes i, j distinguish individual modes, that is, the ways of the possible field oscillations.It is well-known that the electromagnetic field is equivalent to a system of harmonic oscillators

and a single mode corresponds to a single harmonic oscillator. We will not show here this equivalenceas it is explained in most textbooks of quantum optics (see e.g. [21]). For a given mode of the fieldone can define dimensionless position and momentum operators of the corresponding oscillator as

x =a+ a†√

2, p =

a− a†i√2. (2.3)

The operators x and p satisfy the canonical commutation relation [x, p] = i 1 that follows from therelations (2.2). The quantities x and p are called quadratures of the field and they are fundamental fordescribing quantum-optical phenomena with so-called continuous variables. The word “continuous”is related to the fact that the spectrum of the quadratures is continuous in contrast to the discretespectrum of the photon-number operator n = a†a. Instead of x and p, scaled quadratures X± are

1We take the Planck constant ~ to be equal to unity; otherwise the commutator would be equal to i~

9


often used that are defined as X+ =√2 x and X− =

√2 p. Along with x and p one can also define

a general quadrature

xϕ = x cosϕ+ p sinϕ =ae−iϕ + a†eiϕ√

2(2.4)

as a linear combination of x and p that can be interpreted as a rotated position in the phase spaceof the given mode.

Continuous-variable quantum information protocols correspond to analog systems in the classicalinformation theory, while discrete-variable protocols correspond to digital systems. As the digitaltechnology has some clear advantages over the analog technology, discrete variables are often preferredalso in quantum information theory. However, continuous variables have other advantages such aspossibility of manipulating with quantum information by linear optical elements.

For the mathematical description of mode transformation one can employ the continuous basisof the Hilbert space, namely the basis of the eigenstates |x〉 of the quadrature operator x such that

x|x〉 = x|x〉. (2.5)

Some operations with the states |x〉may be problematic from the rigorous mathematical point of viewas these states cannot be properly normalized, which is the case of some calculations in Chapter 3.However, it turns out that by a relatively simple way, namely introducing the so-called Gelfandtriplet [25, 26] one can make the explained theory mathematically rigorous.

2.2 Linear mode transformation

It is well-known that physical quantities that characterize a quantum system can change in time.The time evolution of a quantum system can be described in two different ways, namely using theSchrodinger and Heisenberg pictures. The Schrodinger description views the operators correspondingto physical quantities as fixed and the evolution is given by changing the quantum state in the Hilbertspace. On the other hand, the Heisenberg description considers the quantum state to be fixed andthe evolution is ascribed to the operators of the physical quantities. Both approaches are equivalent,and none of them should be regarded as “more correct”.

The time evolution of a given mode of the electromagnetic field proceeds spontaneously due to thenonzero energy of the mode and due to its interaction with various optical elements. The spontaneousevolution is of little interest as it is given by a uniform phase change; we will concentrate on theevolution caused by the optical elements. A beam splitter, phase shifter and squeezer are typicalsuch elements. They have one or two input modes and the same number of output modes that canbe considered as the transformed input modes. In the Schrodinger picture the elements transformthe quantum state while in the Heisenberg picture they transform the field operators describing themodes.

An important class of mode transformations is formed by linear canonical (symplectic) trans-formations for which the output quadratures can be expressed as linear combinations of the inputquadratures. We consider here a special case only, namely the so-called point transformations, forwhich the positions and momenta transform separately and do not mix. Such as general transforma-tion of m modes can be expressed in the Heisenberg picture as

x′i =m∑

j=1

Tij xj , p′i =m∑

j=1

Sij pj (i = 1, . . . ,m) , (2.6)

where the matrices T and S satisfy S = (T−1)T. The corresponding transformation in the Schrodingerpicture changes the eigenstate of the quadratures x1, . . . , xm according to

|x1〉1|x2〉2 · · · |xm〉m →√

| detT |∣

∣

∣

∣

∣

m∑

k=1

T1kxk

⟩

1

∣

∣

∣

∣

∣

m∑

k=1

T2kxk

⟩

2

· · ·∣

∣

∣

∣

∣

m∑

k=1

Tmkxk

⟩

m

. (2.7)

10

Chapter 2. Some important terms and concepts of quantum optics

The indexes at the kets | . . . 〉 label the modes of the field and the factor√

| detT | ensures the correctnormalization of the states or, in other words, the unitarity of the transformation (2.7).

As can be seen by comparing Equations (2.6) and (2.7), the eigenvalues in the Schrodinger picturetransform in the same way as the quadratures in the Heisenberg picture. This must be so because theeigenvalues are measurable quantities and both pictures have to provide an equivalent description.

There is a special class of so-called passive transformations among (2.6) and (2.7) for which thematrices T and S are orthogonal. These transformations can be realized experimentally by passiveoptical elements only, i.e., linear mode couplers (usually beam splitters) and phase shifters. On theother hand, realizing a non-orthogonal transformation requires employing active elements such asoptical parametric oscillators and it is much more challenging experimentally.

The simplest example of passive mode transformation is a phase shift that does not, however,belong to the the point transformations as it mixes position and momentum, and therefore we willnot consider it here. Another example is mixing two modes on a beam splitter (e.g. a half-silveredglass) that can be expressed as

(

x′1x′2

)

=

(

cos θ/2 − sin θ/2sin θ/2 cos θ/2

)(

x1

x2

)

. (2.8)

For a symmetric beam splitter with both transmissivity and reflectivity equal to 50%, it holds θ = π/2and therefore

x′1 =x1 − x2√

2, x′2 =

x1 + x2√2

. (2.9)

The simplest example of the active transformation is a single-mode squeezing operation:

x′ =x

s, p′ = sp , (2.10)

where s is the squeezing factor. For |s| > 1, the operation (2.10) squeezes the quadrature x andfor |s| < 1 it squeezes p. In practice the squeezing operation is realized e.g. by a degenerate down-conversion in an optical parametric oscillator (OPA) pumped by a beam of double frequency. Withsome probability amplitude a pump photon in the nonlinear crystal realizing OPA can change intoa pair of photons of the mode being transformed, and the opposite process is also possible. If onetransforms the vacuum, that is, the state with the wavefunction

ψvac(x) = 〈x|vac〉 =14√πe−x

2/2 (2.11)

by the single-mode squeezer, then the output state in the Schrodinger picture will be

ψs(x) = 〈x|s〉 =√

|s|4√π

e−s2x2/2. (2.12)

Clearly, this state differs from the vacuum (2.11) for s 6= ±1 and hence its expansion in the Fockbasis 〈n|s〉 must have nonzero coefficients for some n > 0. Therefore the state |s〉 contains photonsthat were added by the transformation (2.10), which is where the name “active” comes from. Is isnot hard to show that 〈n|s〉 6= 0 for n even and 〈n|s〉 = 0 for n odd. This is related to the realizationof the squeezing transformation mentioned above – photons emerge in pairs and if there was nophoton in the field initially, there can only be an even number of them after the squeezing operation.

One can show [27] that an arbitrary matrix T from Eq. (2.6) can be decomposed as T = O2DO1,where the matrices O1 and O2 are orthogonal and the matrix D = diag (d1, . . . , dm) is diagonal.Therefore the transformation (2.6) or (2.7) can be realized in three steps (see Figure 2.1): the firstand last steps are passive transformations corresponding to the matrices O1 and O2, respectively. Themiddle step consists of m single-mode squeezing operations corresponding to the diagonal elementsof the matrix D and scaling the quadrature xi and pi by the factor di and 1/di, respectively. Thus thenumber of active elements needed for realizing an arbitrary symplectic transformation of m modesdoes not exceed the number of modes.

11


23

m

1

PI PI

1’

3’

m’

2’SS

S

S

Figure 2.1: Decomposition of a general symplectic transformation of m modes: first the modes are combinedin a passive interferometer (PI), then each mode undergoes a squeezing transformation (S) individually andfinally the modes are combined in another passive interferometer.

2.3 Coherent states of light

Coherent states play an important role in quantum optics for their numerous useful physical andmathematical properties. First of all, they are states that are closest to classical states of light andthat exhibit large coherence. Coherent states have the minimum product of uncertainties of thequadratures x and p; in both x-, and p-representations they are Gaussian wavepackets. Anotheruseful property of coherent states is their elegant transformation on a linear mode coupler. Co-herent states also have interesting mathematical properties that enable constructing representationsvery useful for describing quantized electromagnetic field. One of them is the Glauber-SudarshanP -representation [28, 29, 30] that will be discussed in a moment. Thanks to their physical and math-ematical properties, coherent states are useful for various optical measurements, as local oscillatorsfor homodyne detection, for pumping down-converters and squeezers, as testing states for quantumteleportation, quantum state sharing etc.

With respect to what we just said about the importance of coherent states, it may be surprisingthat they have not yet been realized at optical frequencies as was emphasized by K. Mølmer [31]and B. C. Sanders and T. Rudolph [32]. Hence, coherent states are a “convenient fiction” ratherthan a physical reality. For example, laser light is not in a coherent state as one often hears in thecommunity of quantum opticians, but rather in a mixture of coherent states with equal amplitudeand with the phase distributed uniformly over the interval 〈0, 2π) [21]. However, when describing anexperiment with a laser source using coherent states instead of their mixtures, one does not makea serious mistake; the result expected by the theory is the same in both cases because the measuredbeam and the reference beam (local oscillator) are derived from the same source and hence have afixed relative phase. This way, most experiments that one should, strictly speaking, describe usingmixtures of coherent states can equivalently be described using pure coherent states.

Coherent states can be defined by several equivalent ways that will be discussed in Chapter 5;here we mention just the most common definition. Coherent state is the eigenstate of the annihilationoperator a, i.e., the state satisfying a|α〉 = α|α〉 for some complex number α. This definition yieldsimmediately the expansion of the coherent state in the Fock basis:

|α〉 = e−|α|2/2

∞∑

n=0

αn√n!|n〉 . (2.13)

The photon number distribution in the coherent state |α〉 is Poissonian with both the mean andvariance equal to |α|2.

It is an important property of coherent states that they provide the following decomposition of theunit operator:

1 =1

π

∫

|β〉〈β| d2β , (2.14)

where the integration runs over the whole complex plane. At the same time, no two coherent statesare orthogonal as |〈α|β〉|2 = e−|α−β|

2. These two properties have an interesting consequence – any

12

Chapter 2. Some important terms and concepts of quantum optics

state |ψ〉 from the Hilbert space can be expressed as a superposition of coherent states by an infinitenumber of ways. To show this, assume that |ψ〉 itself is a coherent state. Then

|α〉 =∫

δ2(β − α) |β〉 d2β (2.15)

certainly holds, where δ2(β) = δ(Reβ) δ(Imβ) is the two-dimensional Dirac delta-function. At thesame time, using the unit operator expansion (2.14) one arrives at

|α〉 = 1

π

∫

|β〉〈β|α〉 d2β =1

π

∫

eβ∗α−(|β|2+|α|2)/2 |β〉 d2β. (2.16)

Equations (2.15) and (2.16) give two different and valid expansions of the state |α〉 in terms of coherentstates.

Similarly, a general density matrix ρ of the mode can be expressed in many different ways asfollows,

ρ =

∫ ∫

ρ(β, γ)|β〉〈γ| d2β d2γ. (2.17)

There is so much freedom in the choice of the function ρ(β, γ) that it enables something seeminglyimpossible: one can choose it such that ρ(β, γ) = 0 for β 6= γ, that is, the density matrix can beexpressed in terms of coherent states in a diagonal way:

ρ =

∫

P (β)|β〉〈β| d2β. (2.18)

P (β) is the so-called Glauber-Sudarshan P -function; it has some unusual properties and for manystates it is a distribution rather than an ordinary function. This is quite natural with respect tothe very strong requirement of diagonality of ρ(β, γ). For a coherent state ρ = |α〉〈α| one hasP (β) = δ2(β−α), for a thermal state with the mean photon number N it is P (β) = 1

πN e−|β|2/N and

for a Fock state |n〉 the P -function is proportional to the nth derivative of the Dirac delta-functionδ2(β). The P -function forms the basis for mathematical description of homodyne detection as willbe discussed in Chapter 4.

2.4 POVM

POVM (positive operator-valued measure) is a set of positive-semidefinite Hermitian operators Πi

that characterizes completely a given quantum-mechanical measurement. If ρ is the density matrixof the system, the probability of the ith measurement output is given by pi = Tr (ρΠi). The operatorsΠi satisfy the unit operator decomposition

∑

i Πi = 1.The POVM is a generalization of a projective quantum-mechanical measurement. A general

measurement can be performed by adding an ancilla system in a known state to the measuredsystem and making a projective measurement on the composite system [33].

13

Chapter 3

Quantum state sharing

Quantum state sharing is an important quantum-information protocol. Its goal is to protect a quan-tum information (called quantum secret) that is distributed among a group of parties (called players)against its misuse by unauthorized groups of players and to enable the access to this informationto other, authorized groups. Initially, the dealer who owns the quantum information in the formof a quantum state of a given system encodes this state into an entangled state of n quantum sys-tems and distributes these systems to the individual players. The encoding is performed in such away that for any authorized group of players there exists a unitary operation that the players canapply to their systems (called shares) and in this way obtain one system in the same state as wasthe original secret. This is called secret reconstruction or extraction. On the other hand, the densitymatrix of the systems of the players from any unauthorized group is independent of the secret andhence the unauthorized groups cannot get any information about the secret, no matter what opera-tions they perform with their shares. At first sight, it might seem odd that such a protocol can existat all.

It is important to note that the quantum secret may be in a mixed state and it can even beentangled with another quantum system. In this case, the entanglement is recovered after the secretextraction. This way it is possible to share e.g. just a component of a quantum state of a largersystem.

The method of the secret encoding is a public information and it is closely related to the so-calledaccess structure, which is the set of all authorized groups of players that should be able to extractthe secret. The access structure cannot be arbitrary but it must satisfy certain conditions. Anobvious rule is that when adding a player to an authorized group, it remains authorized. Anothercondition says that there cannot exist two separated (disjoint) authorized groups of players. If thiswere possible, one could create two copies of the extracted secret from a single original secret state,and this way effectively clone a quantum state. However, cloning quantum states is impossible, ashas been shown by W. K. Wootters and W. H. Zurek [13] (the no-cloning theorem). The conditiondoes not apply to classical secret sharing, which the classical analogy of quantum state sharing; thesecret in this case is a classical information that can be copied or cloned arbitrarily.

Among quantum state sharing schemes there is an important class of the so-called self-dual accessstructures with the following property: for every division of all players into two groups, exactly onegroup is authorized. It turns out that any access structure that is not self-dual can be derived fromsome self-dual one by discarding one or more shares [34]. Therefore exploring only self-dual structuresis sufficient for describing quantum state sharing. Another important class of access structures arethe so-called threshold schemes for which it is only the number of players in the group that determineswhether the group is authorized or not. For the (k, n) threshold scheme there are n players in totaland any k of them are authorized to extract the secret. It can be seen easily that self-dual structuresare those for which n = 2k − 1 holds; in the following we will consider these structures only.

Quantum state sharing can be implemented is quantum systems described by both discrete and

14

Chapter 3. Quantum state sharing

(a) (b) (c)

Figure 3.1: Three examples of access structures; only the minimal authorized sets are shown. The accessstructure in (a) is not allowed in quantum state sharing as two disjoint groups of players can access the secret;however, it is allowed in classical secret sharing; the access structure in (b) is allowed also in the quantumcase, and (c) shows the access structure of the (2, 3) threshold scheme in which any two players can access thesecret.

continuous variables. In discrete variables where the secret is realized as qubits (or more generallyqudits), the encoding can effectively be designed by employing properties of matrices over finitenumber fields, and the theory of quantum state sharing is well developed [35, 36, 34]. The the-ory of quantum state sharing with continuous variables was developed by me, B. C. Sanders andD. J. Rowe at Macquarie University in Sydney [37, 38]. Later we have, together with co-workersat Australian National University in Canberra, proposed [39] and realized successfully [40, 41] anexperiment that demonstrated quantum state sharing for the first time. The proposed scheme wasdesigned for optical implementation and the quantum systems carrying the secret and the shareswere realized as modes of the electromagnetic field. The fundamental quantities used for describingthe quantum system are the field quadratures that have been discussed in Sec. 2.1.

Originally, we have formulated the theory of continuous-variable quantum state sharing in theSchrodinger picture [37, 38] in analogy to the discrete-variable case. However, later the Heisenbergpicture was preferred [39, 40, 41] (to compare both pictures, see Sec. 2.2). In the following section wewill describe the Schrodinger approach and in Sec. 3.2 the Heisenberg approach to continuous-variablequantum state sharing.

3.1 Continuous-variable quantum state sharing in the Schrodingerpicture

In the following we explain continuous-variable quantum state sharing in the Schrodinger picture onthe example of the (k, 2k− 1) threshold scheme that has total 2k− 1 players and any k of them canextract the secret. Generalization of the protocol to an arbitrary access structure is straightforward.

3.1.1 Encoding the quantum secret

The first step in the protocol is the encoding of the quantum secret into an entangled state of 2k− 1modes of the field and distributing these modes to the players. The initial state of the dealer isformed by 2k − 1 modes of the electromagnetic field: the first of them is the quantum secret

|ψ〉 =∫

Rψ(x) |x〉 dx (3.1)

and the remaining 2k−2 are squeezed vacuum states. Half of them, that is k−1, are squeezed in thequadrature p, so they are the states from Eq. (2.12) with s < 1, and the other half are squeezed inthe quadrature x so they are the states |s〉 with s > 1. In order for the secret extraction to be perfect,the squeezing must be infinite, which corresponds to the limits s → 0 and s → ∞, respectively. Inthis case one can express both states as

∫

R|x〉 dx and |0〉 . (3.2)

15


In the following we will assume this ideal case of infinite squeezing. The more realistic situationof finite squeezing will be discussed in sections 3.1.5 and 3.2 that talks about the (2, 3) thresholdscheme and its experimental realization.

In the ideal situation of infinite squeezing, the initial state of the dealer is

|Φ0〉 =∫

Rk

ψ(x1) |x1〉1|x2〉2 · · · |xk〉k|0〉k+1 · · · |0〉2k−1 dkx , (3.3)

and the indexes of the kets mark modes of the field. The dealer then applies a particular symplectictransformation (see Eq. (2.7)) to the state |Φ0〉 to create the following entangled state:

|Φ〉 =∫

Rk

ψ(x1) |L1(x)〉1|L2(x)〉2 · · · |L2k−1(x)〉2k−1dkx . (3.4)

Here x denotes the set of variables x1, x2, . . . , xk and Li(x) with i = 1, . . . , 2k−1 are linear combina-tions of the variables x1, x2, . . . , xk that satisfy a certain condition that ensures that any k players canextract the secret. The condition is that any k elements from the 2k-element set {x1, L1, . . . , L2k−1}must be linearly independent. By a proper choice of L1(x), . . . , L2k−1(x) one can ensure that thetransformation |Φ0〉 → |Φ〉 is orthogonal. This means that the dealer does not need active operationsfor encoding the quantum secret but only for creating the initial squeezed states.

3.1.2 Extraction of the quantum secret

Next we show how a group of k players can extract the quantum secret. Without loss of generalitywe can assume that the first k players collaborate, in the opposite case we can relabel the players.

When thinking about the linear combinations L1(x), . . . , L2k−1(x) of the variables x1, . . . , xk, itis useful to view these objects as vectors in a k-dimensional vector space V with the basis vectorsx1, . . . , xk. This makes our considerations much clearer. It then follows from our assumptionsabout x1, L1(x), . . . , L2k−1(x) from last section that the vectors L1, L2, . . . , Lk as well as the vectorsx1, Lk+1, . . . , L2k−1 are linearly independent. At the same time, in both groups there are k vectors,which is the same number as the dimension of the vector space V . Therefore there must exista non-singular matrix T such that

T

L1

L2...Lk

=

x1

Lk+1...

L2k−1

(3.5)

holds. The existence of the matrix T implies the existence of a unitary operator U(T ) acting on themodes 1, 2, . . . , k as follows:

U(T )|L1〉1|L2〉2 · · · |Lk〉k =√

| detT | |x1〉1|Lk+1〉2 · · · |L2k−1〉k . (3.6)

Now, if players 1, 2, . . . , k apply the operation U to their shares, the total state of all shares willbe

U |Φ〉 = J√

| detT |∫

Rk

ψ(x1) |x1〉1|Lk+1〉2 · · · |L2k−1〉k|Lk+1〉k+1 · · · |L2k−1〉2k−1dx1dLk+1 · · ·dL2k−1

= J√

| detT | |ψ〉1 |Θ〉2,k+1|Θ〉3,k+2 · · · |Θ〉k,2k−1 . (3.7)

In the integral in Eq. (3.7) we changed the integration variables x1, . . . , xk to x1, Lk+1, . . . , L2k−1, wehave denoted the Jacobian of this transformation by J and have defined a two-mode state

|Θ〉ij ≡∫

R|x〉i|x〉j dx . (3.8)

16


Eq. (3.7) shows that the first player’s share is left in the state |ψ〉, so the secret is extracted in itsoriginal form in mode 1. The shares of players 2, 3, . . . , k form strongly entangled pairs |Θ〉ij withthe shares of players k + 1, . . . , 2k − 1 who did not participate in the extraction process.

The state |Θ〉ij is the EPR (Einstein-Podolsky-Rosen) state that A. Einstein and co-workers usedin 1935 in their famous paper [42] attacking completeness of quantum mechanics. As can be seenfrom Eq. (3.8), in the EPR state the quadratures xi, xj are perfectly correlated; at the same time,the quadratures pi, pj are perfectly anticorrelated. The EPR state |Θ〉ij plays an important role inmany continuous-variable quantum-information protocols, e.g. in quantum teleportation [3, 4, 5].

It still remains to show that unauthorized groups cannot obtain any information about the quan-tum secret. For this it is enough to know that the access structure of the (k, 2k−1) threshold schemeis self-dual, so the complement of any unauthorized group is a group that can extract the secretperfectly. This itself denies any information leakage to the unauthorized group as such a leakagewould prevent the authorized group from perfect extraction of the secret. This is because any infor-mation about a quantum state that escapes to the environment changes the state of the system. Thisis a general property of quantum states and it forms the basis for important quantum-informationprotocols, in particular of quantum key distribution [7]. To show that a group of k−1 players cannotget any information about the secret, one can also calculate the trace of the total state of all shares|Φ〉〈Φ| over the shares of the remaining k shares. It is not hard to show that the resulting densitymatrix is independent of the secret |ψ〉, so it cannot provide any information about it.

3.1.3 Example: the (2,3) threshold scheme

In this section we illustrate the quantum state sharing protocol on the example of the (2, 3) thresholdscheme in which there are three players in total and any two of them can obtain the quantum secretby collaboration. This scheme is important in that it has been realized experimentally, which will bediscussed in Sec. 3.2 in detail.

The initial state of the dealer |Φ0〉 consists of the quantum secret |ψ〉 and two states squeezedinfinitely in the quadratures p and x, respectively:

|Φ0〉 =∫

R2

ψ(x1) |x1〉1|x2〉2|0〉3 dx1 dx2 . (3.9)

The dealer chooses the following linear combinations L1, L2, L3 according to Eq. (3.4):

L1 =x1√2+x2

2, L2 =

x1√2− x2

2, L3 =

x2√2

(3.10)

and employing a passive transformation (2.7) with the orthogonal matrix

T =

1√2

12

12

1√2−1

2 −12

0 1√2− 1√

2

(3.11)

he encodes |Φ0〉 into the three-share entangled state

|Φ〉 =∫

R2

ψ(x1)

∣

∣

∣

∣

x1√2+x2

2

⟩

1

∣

∣

∣

∣

x1√2− x2

2

⟩

2

∣

∣

∣

∣

x2√2

⟩

3

dx1 dx2 . (3.12)

The dealer then distributes the shares to the players.Players 1 and 2 can extract the secret via a passive transformation (2.7) with the matrix

T12 =1√2

(

1 11 −1

)

. (3.13)

17


The resulting state

|Φ′12〉 =∫

R2

ψ(x1)|x1〉1∣

∣

∣

∣

x2√2

⟩

2

∣

∣

∣

∣

x2√2

⟩

3

dx1 dx2 (3.14)

clearly contains the quantum secret |ψ〉 in mode 1.Players 1 and 3 can extract the secret via an active transformation (2.7) with the matrix

T13 =

( √2 −11 −

√2

)

, (3.15)

which yields the state

|Φ′13〉 =∫

R2

ψ(x1) |x1〉1∣

∣

∣

∣

x1√2− x2

2

⟩

2

∣

∣

∣

∣

x1√2− x2

2

⟩

3

dx1 dx2 (3.16)

= 2

∫

R2

ψ(x1) |x1〉1 |L2〉2 |L2〉3 dx1 dL2 (3.17)

and hence the secret is again reconstructed in mode 1.The secret extraction from components 2 and 3 is almost identical to the extraction from com-

ponents 1 and 3, so we will not discuss it.

3.1.4 Optimizing the secret extraction

On order to realize the k-mode transformation (3.6) for extracting the quantum secret, the collabo-rating players have to employ k active optical elements (squeezers) in general (see Sec. 2.2). However,it would be highly desirable to reduce the number of active elements in some way because of theirhigh experimental cost and difficulty. In out work [38] we have shown that the transformation (3.6) isnot the only one that enables the secret extraction, and by optimizing the extraction procedure onecan reduce the number of active elements down to two, independent of the number of collaboratingplayers k.

To understand this, we return to Eqs. (3.6) and (3.7) and note that even though the variable x1

is still present in the linear combinations Lk+1, . . . , L2k−1, by changing the integration variables tox1, Lk+1, . . . , L2k−1 it was possible formally eliminate it. In this way the quantum secret in the firstmode has been disentangled from all the other modes. The same would be achieved, however, if thematrix T from Eq. (3.5) was replaced by a matrix T ′ that satisfies

T ′

L1

L2...Lk

=

x1

M2(Lk+1, . . . , L2k−1)...

Mk(Lk+1, . . . , L2k−1)

, (3.18)

where M2, . . . ,Mk are linear combinations of the vectors Lk+1, . . . , L2k−1. Also in this case the firstmode is disentangled from all the other modes and yields the quantum secret. The only difference isthat the modes of the collaborating players 2, . . . , k would no more form EPR pairs with the modesof the non-collaborating players but rather a more complicated entangled state.

Equation (3.18) provides a large freedom thanks to the possibility of choosing the linear com-binations Mi (the only condition is that the matrix T ′ is non-singular). To minimize the numberof squeezing elements, we tried to find the T ′ to be close to some orthogonal matrix. We have shownthat the matrix T ′ can be found in the form

T ′ =

α β 0 . . . 00 γ 0 . . . 0

0 0...

... Ik−2

0 0

O . (3.19)

18


Here O is an orthogonal matrix, Ik−2 is the unit matrix of dimension k − 2, the numbers α and βare determined by vectors L1, . . . , L2k−1, and γ is a free parameter. Hence, one can decompose theextraction transformation into a passive operation corresponding to the matrix O and a two-mode

transformation R ≡(

α β0 γ

)

(as the remaining k− 2 modes are no more transformed). For the two-

mode transformation R2 one needs just two squeezing elements and it can be decomposed accordingto Sec. 2.2 as R = O2DO1 Altogether, the transformation T ′ can be realized in three steps (seeFig. 3.2): first comes the passive operation O1O followed by two single-mode squeezing operationscorresponding to D and the last step is another passive operation O2. Hence no more than two activeoperations are needed to extract the quantum secret. Furthermore, by choosing γ one can minimizethe overall squeezing cost of the two squeezers.

23

1

PISS

PIT

k

Figure 3.2: The optimum extraction of the quantum secret: the k modes are first combined in a passiveinterferometer, then the first two of them are squeezed individually and finally the two modes are combinedin a passive interferometer. One of the outputs is then the extracted secret T.

3.1.5 Finite squeezing in dealer’s encoding procedure

Until now we have assumed that the dealer uses 2k − 2 infinitely squeezed states for his encodingprocedure (see Sec. 3.1.1). In practice this is not possible, however, as the mean number of photonsand mean energy are infinite in an infinitely squeezed state. Nowadays one can achieve the squeezingfactor of the order of several units only, so an important question arises of how the protocol willwork if the squeezing employed by the dealer is finite. In this case the secret will not be extractedperfectly but will be degraded increasingly with decreasing amount of squeezing used by the dealer.At the same time, the protection of the secret against unauthorized groups will no more be perfect,and some information can escape to them.

In order to quantify the quality of the secret extraction, it is useful to express the density matrixρout of the extracted secret with the help of the density matrix ρ of the original secret. In ourwork [38] we have derived the following relation between the two matrices in the x-representation:

ρout(x, x′) =

s√π v

exp

[

−u2(x− x′)2

4s2

] ∫

Rρ(x− y, x′ − y) exp

[

−s2y2

v2

]

dy . (3.20)

Here s denotes the squeezing parameter of the squeezed vacuum states employed by the dealer andu, v are parameters depending on the dealer encoding operation and the choice of the collaboratingplayers. The factor in front of the integral in Eq. (3.20) reduces the magnitude of the non-diagonalelements of the density matrix and the integral itself convolutes the secret density matrix witha Gaussian, which both degrades the secret. It would be even more advantageous to express thisdegradation in terms of the Wigner function W (x, p) that provides description equivalent to thatof the density matrix, but symmetrical with respect to quadratures x and p. It turns out that theWigner function of the extracted secret Wout(x, p) is a two-dimensional convolution of the originalWigner function with a Gaussian function of x, p.

As the parameters u and v differ in general for different groups of collaborating players, thedegradation of the extracted secret differs as well. It this sense, the protocol is “unjust” as some

19


��

��

��

��

��

��

��

Figure 3.3: The encoding of the secret in the (2, 3) threshold scheme: the dealer first creates squeezed ancillastates by squeezing the vacuum states in two optical parametric oscillators (OPA), and combines them ona symmetric beam splitter. One of the outputs is then combined with the quantum secret state on anotherbeam splitter. This way the dealer obtains three shares that he distributes to the players.

groups can extract the secret better than other ones1. I believe that for a large number of playersone cannot design a “just” protocol because the dealer does not have enough parameters to vary inorder to satisfy the large number of conditions requiring equal secret degradation for all authorizedgroups of players.

3.2 Heisenberg picture of continuous-variable quantum state shar-ing

During the period of theoretical preparation of the quantum state sharing experiment at the Aus-tralian National University it turned out that the Heisenberg picture is more advantageous in someaspects than the Schrodinger picture for describing the protocol, in particular for the easier treat-ment of finitely squeezed states in the dealer’s process. In this section we describe the (2, 3) thresholdscheme in the Heisenberg picture that has been realized experimentally. Formally, the transition fromthe Schrodinger to Heisenberg pictures is simple and it is explained in the basic course of quantummechanics. However, in a particular calculation it may not be trivial and we will not perform thistransition here, but rather describe quantum state sharing in the Heisenberg picture directly.

When using the Heisenberg picture, one has to consider both quadratures x and p of the trans-formed modes. This is in contrast with the Schrodinger picture where we used the x-representationand did not consider the momenta at all as the wavefunction provided a complete information abouta pure quantum state.

3.2.1 Encoding the secret

As has been said in Sec. 3.1.3, the dealer owns initially the quantum secret (we will label its quadra-tures by the index S) and two ancilla squeezed states, one squeezed in the quadrature p and theother one squeezed in x. We will label the quadratures of these squeezed states by the indexes sqz1and sqz2. Due to the squeezing, the uncertainties of psqz1 and xsqz2 are lower than would be for thevacuum state |0〉, so ∆psqz1 < 1/

√2 and ∆xsqz2 < 1/

√2 hold. The dealer encodes the secret by the

1For example, in the (2, 3) threshold scheme discussed in Sec. 3.1.3 players 1 and 2 can still extract the secretperfectly even if the dealer employs finite squeezing, while players 1 and 3 or 2 and 3 cannot.

20


�! �" �$#�"

�&% "

')(')(*

*

')(

�&+�"

',(

-/.

0�1 325476

0�1 3254�8

0�1 3254:9

0�1 325476 051 325476

051 325476

051 3254�8

051 3254�80�1 3254�8

0�1 3254:9 051 3254:9

051 3254:90 4;%<254<=

>@?@A 0 =B C=�4

>@?@A 0 =B D=�4

>@?@A 0 =E D=�4

FHG +JIJKL D= G 2

=NM GCOPFHG +Q4 05R IQ4<4;S<4;2

0 4 %B2�4B=

0 4;%<254<=

0 4 %B2�4B=

T I FUTWVYXQZ

Figure 3.4: The extraction of the secret by the authorized groups {1, 2} a {1, 3} in the (2, 3) threshold scheme.(a) Players 1 and 2 simply combine their shares on a beam splitter [BS]; players 1 and 3 (or similarly 2 and 3)have several options: they can employ (b) a two-mode squeezer to transform their shares, (c) a combinationof two beam splitters and two single-mode squeezers, or (d) a non-symmetric beam splitter combined witha homodyne detector [HD] and an electro-optical modulator.

transformation (2.6) with the matrix (3.11) to obtain the shares with the following quadratures:

x1 = xS/√2 + (xsqz1 + xsqz2)/2, p1 = pS/

√2 + (psqz1 + psqz2)/2

x2 = xS/√2− (xsqz1 + xsqz2)/2, p2 = pS/

√2− (psqz1 + psqz2)/2 (3.21)

x3 = (xsqz1 − xsqz2)/√2, p3 = (psqz1 − psqz2)/

√2.

This transformation can be realized passively in two steps (see Fig. 3.3). First, the squeezed ancillasare combined on a symmetric beam splitter, thus forming an approximate EPR pair (see Eq. (3.8)),that is, a pair of entangled beams with correlated quadratures x and anticorrelated quadratures p.One of the beam splitter outputs is then combined with the secret state on another symmetric beamsplitter whose outputs yield the first two shares; the last share is the second beam of the EPR pair.The three shares are then distributed to the players.

3.2.2 Extraction of the secret state by players 1 and 2

If players 1 and 2 wish to extract the secret, they simply combine their shares on a 1:1 beam splitter(see Fig. 3.4 (a)). This way, the dealer’s and players’ operations effectively form a Mach-Zehnderinterferometer whose output replicates the input and thus yields the quantum secret. The quadraturesof the beam splitter outputs are

x1out = (x1 + x2)/√2 = xS, p1out = (p1 + p2)/

√2 = pS, (3.22)

x2out = (x1 − x2)/√2 = (xsqz1 + xsqz2)/

√2, p2out = (psqz1 + psqz2)/

√2.

21


Eqs. (3.22) show that the quadratures of the first output are identical to the quadratures of theoriginal secret, so the secret is extracted. It is also important that the quadratures xsqz1,2 a psqz1,2

are not contained in the output quadratures x1out, p1out and hence players 1 and 2 can extract thesecret state with an arbitrary precision, independent of the amount of squeezing employed by thedealer.

3.2.3 Extraction of the secret state by players 1 and 3

Extraction of the secret by players 1 and 3 is more complicated than for players 1 and 2 becauseof the asymmetry of shares 1 and 3 with respect to the content of the anti-squeezed quadratures xsqz1

a psqz2 in the quadratures x1,3 a p1,3 (see Eqs. (3.21)). These anti-squeezed quadratures have to beeliminated, which can be achieved in several ways.

Ideally, players 1 and 3 perform the two-mode active operation

x1out =√2 x1 − x3, p1out =

√2 p1 + p3,

x2out = −x1 +√2 x2, p2out = p1 +

√2 p2, (3.23)

which yields the following quadratures of the first output:

x1out = xS +√2 xsqz2, p1out = pS +

√2 psqz1 . (3.24)

If the squeezing of the quadratures psqz1 and xsqz2 is infinite, the state of all shares is an eigenstateof these quadratures with the eigenvalue zero. Then one can omit psqz1 and xsqz2 in Eqs. (3.24),which yields x1out = xS and p1out = pS. This means that for infinite squeezing of the quadraturespsqz1 and xsqz2, the secret is exactly replicated at the first output. However, for finite squeezing theextraction is not perfect and the quantum noise (uncertainty) of the quadratures psqz1 and xsqz2 istransfered to the extracted secret.

In principle, the transformation (3.23) could be achieved directly by employing a two-modesqueezer realized by a phase insensitive amplifier [43] (see Fig. 3.4 (b)), which is, however, infea-sible experimentally. Another option is to use a pair of symmetric beam splitters and two single-mode squeezers realized by phase-sensitive parametric amplifiers (see Fig. 3.4 (c)), which is also verychallenging experimentally. Therefore we have proposed an alternative extraction method that isexperimentally feasible; it employs linear optical elements, homodyne detection and electro-opticalmodulation.

Secret extraction via electro-optical modulation

The electro-optical modulation method for extracting the quantum secret has the major advantageof being feasible experimentally. However, the disadvantage is that even for infinite squeezing in thedealer setup, the secret is not extracted in its original form but is subject to a unitary squeezingtransformation.

In this scheme, shares 1 and 3 are first interfered on a beam splitter with transmissivity 2/3 andreflectivity 1/3 (see Fig. 3.4 (d)). The quadrature x of one output is then measured via homodynedetection (see Chapter 4) and the detected signal is imparted onto the x quadrature of the secondbeam splitter output via an electro-optic modulator. The beam splitter reflectivity and other param-eters are chosen such that the anti-squeezed quadratures xsqz1 and psqz2 of the dealer’s ancilla statescancel in the output. The quadratures of the output beam are then

xout =√3 (xS +

√2 xsqz2), pout =

1√3(pS +

√2 psqz1). (3.25)

For infinite squeezing in the dealer procedure, these equations become xout = 31/2 xS and pout =3−1/2 pS, which means that this method reconstructs the secret up to a squeezing transformation with

22


the factor s = 1/√3 (see Eq. (2.10)). To obtain the secret in its original form, it would be necessary

to invert this squeezing transformation, which would require additional quantum resources. On theother hand, from the perspective of quantum information theory the reconstructed secret contains allthe information that was in the original secret as both the secrets differ by a unitary transformation.From this point of view, the electro-optical modulation method can be considered as an adequatemethod for the quantum secret extraction.

3.3 Experimental realization of the (2, 3) threshold scheme

Shortly after the idea of continuous-variable quantum state sharing arose, we started to discussa possible experimental realization of this protocol with our colleagues from the Australian NationalUniversity in Canberra. The university has an excellent experimental background in continuous-variable quantum information and several important results [44] have been achieved there. Wedecided to work on the (2, 3) threshold scheme that is not trivial and is feasible at the same time,and we have chosen the electro-optical modulation method for the secret extraction by players 1 and3 (see Sec. 3.2.3). Two years later the scheme was realized successfully [41, 40], which was the firstrealization of quantum state sharing.

The light source for the experimental setup is a Nd:YAG laser at 1064 nm wavelength that pumpsa second harmonic generator based on a non-linear crystal MgO : LiNbO3. The resulting frequency-doubled light is used to pump two MgO : LiNbO3 optical parametric amplifiers that produce twobeams squeezed 4.5±0.2 dB below the vacuum noise limit. The squeezed beams are mixed on a beamsplitter to produce a pair of approximate EPR beams. The quantum secret state is represented bya coherent state at the sideband separated by 6.12 MHz from the carrier wave. The secret is mixedwith one beam of the EPR entangled pair, which yields the first two shares, and the last share is theremaining beam of the EPR pair (see Fig. 3.3). To increase the security of the scheme, additionalGaussian noise is added onto the three shares using electro-optic modulation techniques. This noisedoes not degrade the secret extracted by the authorized groups while it reduces the information thatcan escape to adversary players if the dealer uses finite squeezing.

For quantifying the quality of the extraction, we defined the extraction fidelity (overlap) for a puresecret state |ψ〉 as F = 〈ψ|ρout|ψ〉 [45], where ρout is the density matrix of the extracted secret. IfF = 1, the secret is perfectly extracted. We also used criterions of added noise and signal transfer.

It was relatively easy to extract the secret from shares 1 and 2 by simply combining them at thebeam splitter with the phase set properly (See Fig. 3.4 (a)). The best fidelity achieved for players 1and 2 was F{1,2} = 0.95± 0.05, which is a value fairly close to unity and a very good result.

To extract the secret from shares 1 and 3, we employed the electro-optical modulator method(See Fig. 3.4 (d)). In order to determine the overlap of the extracted and original secrets, ana posteriori symplectic transform was applied to the extracted state. The calculated fidelity was upto F{1,3} = 0.62± 0.02. If the quantum secret had been shared classically, i.e., without squeezing inthe dealer procedure, the highest achievable fidelity would have been 1/2. As the fidelity achieved inthe experiment exceeded this value, the quantum nature of our protocol was proved.

3.4 Conclusion

With my Australian colleagues from Macquarie University in Sydney I have introduced a generalprotocol for sharing quantum states in continuous variables and optimized it with respect to thenumber of active (squeezing) operations. We have shown that for extracting the quantum secret byany number of collaborating players, only two active elements are needed. Further, together with thecolleagues from Australian National University in Canberra I have proposed the experimental real-ization of the (2, 3) threshold scheme and the experiment was later completed successfully. This way,

23


the collection of practically feasible quantum-information protocols has been increased by another el-ement. Even though quantum state sharing is nowadays interesting mainly from the theoretical pointof view, it will probably become an important tool for protecting data in future quantum-informationtechnologies.

24

Chapter 4

Homodyne detection

Homodyne detection is an important detection method in modern quantum-optical experiments [46,47] that is used especially when working with continuous variables. It is based on interference of thedetected field with a coherent beam of the local oscillator and measuring the intensity differenceof the resulting beams. At proper conditions, homodyne detection effectively measures the fieldquadratures and thus it is a phase-sensitive method.

The standard theory of homodyne detection [15, 16, 17, 18, 19, 20] is developed based on severalapproaches mostly using characteristic functions and quasiprobabilistic distributions in the phasespace. The standard theory clearly showed the connection between the field quadrature and thequantities directly measured by a homodyne detector; however, it did not provide a complete de-scription of homodyne detection, in particular the explicit derivation of the corresponding POVM(see Sec. 2.4). I was attracted by this problem in 2000 and after discussions with Barry C. SandersI started working on it. During my stay at Macquarie University in Sydney I managed to find thePOVM of homodyne detection by two different methods. Both of these methods are based on a di-rect calculation of the probability distribution of the detection outcome and they differ by the wayof calculation. We will explain both methods in Sections 4.3 and 4.4 of this chapter but first weintroduce homodyne detection in mode detail and say a few words about the idea of the standarddescription of homodyne detection.

4.1 Homodyne detection as a phase-sensitive method

There are situations, especially in modern quantum optics, where one needs to measure the intensityE of the electric field associated with a certain electromagnetic wave. At low frequencies, it is possibleto detect E directly (e.g. from the force that the field acts on electrons in an antenna) and so onecan determine both the amplitude and phase of the field. However, for a number of reasons sucha direct measurement is not possible at optical frequencies because of the impossibility of processingan electronic signal of an optical frequency, problems with the reference time etc. Moreover, atoptical frequencies the quantum nature of light presents itself significantly, i.e., the fact that lightinteracts with matter in the form of quanta (photons). A detector of light has to absorb a quantumof energy in order to report a detection event and so the most common method of light detection isbased on absorption of photons at photodetectors. As the operators describing such photodetectionare diagonal in the Fock basis, photodetection alone cannot provide information about the phaseof the field but only about its intensity. Indeed, it follows from the uncertainty relations that if oneknows the number of photons in the field, its phase is completely unknown.

However, the phase information can be accessed by interfering the measured field with a referencefield with known properties, which is typically done in holographic imaging. Homodyne detectionis based on the same principle – the measured field (called signal field) is interfered with a localoscillator beam on a beam splitter (usually a half-silvered mirror). The resulting two output modes

25


are then subject to a photodetection that measures the photon numbers in the ideal case. If thelocal oscillator is in a coherent state with a large amplitude (for the exact condition see Sec. 4.5),then the photon number difference at the two outputs is closely related to field quadrature operatorxϕ (see Eq. (2.4)) with ϕ being the phase of the local oscillator. More precisely, the probabilitydistribution of the scaled photon number difference ∆ approaches the distribution of the quadraturexϕ. By detecting ∆ one can thus measure the field quadrature just as if one measured the positionof the harmonic oscillator that represents the mode of the field. Optical homodyne detection hasbeen taken over from electronics where it is quite common – a homodyne detector can be foundalmost in every radio or television receiver.

4.2 Why homodyne detection measures the field quadrature

It is not trivial to show the connection between the photon number difference at a homodyne detectorwith the field quadrature. Now we explain one way how one can see this connection, which is norigorous proof, however. The standard description of homodyne detection is based on this idea butit is much more elaborated.

We will consider here the balanced homodyne detection that uses a symmetric beam splitter withtransmissivity and reflectivity equal to 50%. The annihilation operators of the output modes a′1, a

′2

of a symmetrical beam splitter are connected with the input modes operators a1, a2 by the relations

a′1 =a1 − a2√

2, a′2 =

a1 + a2√2

. (4.1)

The photon number difference operator ∆ at the two beam splitter outputs can then be expressedin terms of the input operators as

∆ = n′1 − n′2 = a′1†a′1 − a′2†a′2 = −a†1a2 − a†2a1. (4.2)

Now, assume the first input mode to be in a coherent state | − Aeiϕ〉 with A > 0 (the minus sign isconvenient for further calculations). The expectation value of the operator ∆ is then

〈∆〉 = Ae−iϕ〈a2〉+Aeiϕ〈a†2〉 =√2A 〈xϕ〉. (4.3)

Hence 〈∆〉 is, up to a multiplicative factor, equal to the expectation value of the quadrature xϕ.Similarly, one can calculate the second moment of the operator ∆:

〈∆2〉 = A2(

e−2iϕ〈a22〉+ 2〈a†2a2〉+ e2iϕ〈a†22〉+ 1

)

+ 〈a†2a2〉 = 2A2 〈x2ϕ〉+ 〈a†2a2〉. (4.4)

This equation shows that for a large amplitude A, it holds approximately 〈∆2〉 = 〈(√2A xϕ)

2〉.Expressing higher moments as well, one can show that for A→∞,

〈∆n〉 → 〈(√2A xϕ)

n〉. (4.5)

holds. Now, if all the moments of quantities ∆ and√2A xϕ are the same for A → ∞, then also

their statistical distributions should be the same. This means that by measuring Xϕ ≡ ∆/√2A we

effectively measure the quadrature xϕ. One could make an objection here that the quantity ∆ hasa discrete spectrum while xϕ has a continuous spectrum so the two quantities cannot have the sameprobability distribution. However, for A→∞ the step of Xϕ goes to zero (as the step of ∆ is unity)so the spectra of both quantities become practically equal in the limit of large A.

The method we have just explained shows connection between the field quadrature and the photonnumber difference at the beam splitter outputs, but it is not fully sufficient for describing homodynedetection. For a more precise description one would have to show how the difference of the nth

26

Chapter 4. Homodyne detection

moments of Xϕ and xϕ depends on the amplitudeA as n grows and in what sense the convergence (4.5)occurs. Moreover, in practice is is not possible to increase the local oscillator amplitude arbitrarily,so for practical use of homodyne detection it is important to know the connection between Xϕ and xϕfor finite A. Most importantly, the explained method does not allow to find the POVM of homodynedetection that would show the direct correspondence of the probability of finding a given photonnumber difference ∆ and the probability for the quadrature to have a given value x.

4.3 POVM calculation using the SU(2) Wigner functions

ψα

(a)

j−m’

j−m j+m(b)

j+m’

Figure 4.1: Balanced homodyne detection scheme: (a) the input state |ψ〉 is mixed with a local oscillator incoherent state |α〉, and photodetection is performed at the two output ports; (b) the probability amplitudeof finding j ± m photons at the beam splitter outputs provided there were j ± m′ photons at the inputs isgiven by the Wigner function dj

mm′ .

The first method of finding the POVM of homodyne detection that we have developed [48] workswith the photon number (Fock) basis and it can only be used for large amplitudes A of the localoscillator. The matrix elements of a beam splitter in this basis are given by so-called Wigner SU(2)functions that were originally defined in the angular momentum theory [49]. The Hilbert space HFof a pair of modes of the electromagnetic field is isomorphic with the Hilbert space HJ of a quantumsystem described by the operators Jx, Jy, Jz satisfying the usual angular momentum commutation

relations [Ji, Jj ] = iεijkJk1. The operators Jx, Jy, Jz are related to the field operators a1,2, a

†1,2 of the

pair of the modes by the Schwinger boson representation [50]

Jx =1

2(a†1a2 + a†2a1), Jy = −

i

2(a†1a2 − a†2a1), Jz =

1

2(a†1a1 − a†2a2) (4.6)

and the commutation relations mentioned above follow from the commutation relations of the fieldoperators. The basis of the Hilbert space HJ is given by the states {|jm〉} with 2j = 0, 1, 2, . . . andm = −j,−j + 1, . . . , j that are the eigenstates of J2 and Jz with the eigenvalues j(j + 1) and m,respectively. In the state |jm〉 there are j +m photons in the first mode and j −m in the secondmode. The beam splitter operator is given by

B(θ) = e−iθJy , (4.7)

where for the symmetric beam splitter θ = π/2 holds. The state |jm〉 is transformed on the beamsplitter as

B(θ)|jm〉 =∑

m′

djm′m(θ) |jm′〉, (4.8)

and djm′m(θ) = 〈jm′|e−iθJy |jm〉 are the SU(2) Wigner functions, that is, the matrix elements of thebeam splitter transformation (4.7) in the basis |jm〉.

1εijk denotes the Levi-Civita symbol that is equal to 1 and −1 for ijk an even and odd permutation of the numbers1,2,3, respectively, and equal to zero if some of the numbers i, j, k coincide

27


If the signal state is |ψ〉 and the amplitude of the local oscillator is −A (we set the phase ϕ tozero for simplicity), then the beam splitter input state is | − A〉|ψ〉. We can express this state inthe basis {|jm〉} and by applying the transformation (4.8) to it we obtain the output state in thesame basis. The probability amplitude M j

m of finding j +m and j −m photons at the beam splitteroutputs is then equal to the coefficient at the state |jm〉 in this decomposition, that is,

M jm = 〈jm|B(π/2)| −A〉1|ψ〉2 = e−A

2/22j∑

n=0

ψn(−A)2j−n√

(2j − n)!djm,j−n(π/2) , (4.9)

with ψn being the nth coefficient in the Fock basis expansion of the signal state |ψ〉. The correspondingprobability P j

m is equal to the square of the modulus of the amplitude M jm.

The key step in the calculation of the the probability P jm is using the asymptotic form of the

Wigner functions for large j that was derived in [51]:

djm,j−n(π/2) ≈ (−1)nj−1/4 un(m/√

j). (4.10)

Here un(x) = 〈x|n〉 denotes the x-representation of the nth stationary state of the harmonic oscillatorwith the Hamiltonian H = (x2 + p2)ω/2, i.e.,

un(x) =e−x

2/2

4√π√2nn!

Hn(x), (4.11)

and Hn(x) is the Hermite polynomial.Using various expansions we have arrived at the following result for the probability P j

m that holdsfor large A:

P jm =

e−(2j−A2)2/2A2

√π A2

〈x|ρ|x〉. (4.12)

Here |x〉 denotes the eigenstate of the quadrature x with the eigenvalue x = m/√j, and ρ = |ψ〉〈ψ|

is the density matrix of the signal state. The corresponding POVM is then

Πjm =

e−(2j−A2)2/2A2

√π A2

|x〉〈x|. (4.13)

The projection operator |x〉〈x| is of a key importance here. As the POVM is proportional to itmeans that homodyne detection really measures the field quadrature x. The factor in front of |x〉〈x|in Eq. (4.13) is connected with the normalization of the POVM and the fact that the probability P j

m

is related not only to the photon number difference but also to the photon number sum. Eq. (4.13)shows that the total photon number 2j has the Gaussian distribution with both the mean value anddispersion equal to A2. This is not surprising as the Poissonian distribution of the photon number inthe local oscillator converges to such Gaussian distribution for A→∞; even though there are somephotons from the signal state among the 2j photons total, they do not influence the distributionof 2j much as there is a negligible minority of them for very large A.

The probability Pm of finding the photon number difference 2m at the beam splitter outputsregardless of the photon number sum 2j is equal to the sum of P j

m over all possible j:

Pm =∞∑

j=|m|,|m|+1,...

P jm (4.14)

In the limit of large A this sum can be evaluated by replacing the summation by integration. Also,for large A the width of the distribution of 2j has a very small relative width, so the eigenvaluex = m/

√j can be replaced by x =

√2m/A. The result is then

Pm =1√2A〈x|ρ|x〉. (4.15)

28


The factor 1/√2A in Eq. (4.15) is connected by the Jacobian

√2m/A of the map m→ x =

√2m/A

and with the fact that m changes in half-integer steps. The probability Pm is normalized properlyas for large A, Eq. (4.15) yields

∑

m

Pm =

∫

R〈x|ρ|x〉 dx = Tr ρ = 1. (4.16)

We also mention the situation of a general phase of the local oscillator. If the local oscillator isin a coherent state |−Aeiϕ〉, then 〈x| and |x〉 in Eqs. (4.12), (4.13) and (4.15) have to be replaced by

ϕ〈x| and |x〉ϕ, which are the left- and right-eigenstates of the quadrature xϕ, respectively, with theeigenvalue m/

√j. Hence, by setting the phase of the local oscillator one can choose what quadrature

will be measured by the homodyne detector.

In this way, we have shown for the first time by a direct calculation that the POVM of homodynedetection is proportional to the projector |x〉〈x|. However, this method did not allow us to findcorrection terms to Eqs. (4.12) and (4.13) for small amplitudes A, the main reason being the absenceof correction terms in Eq. (4.10). This problem can be overcome by using a different method thatworks with coherent states instead of Fock states.

4.4 POVM calculation using the Glauber-SudarshanP -representation

The advantage of working in the coherent state basis is the extremely simple description of thebeam splitter transformation in this basis. This optical element transforms a pair of coherent states|α1〉, |α2〉 into another pair of coherent states as follows:

B(θ)|α1〉 ⊗ |α2〉 = |α1 cosθ

2− α2 sin

θ

2〉 ⊗ |α1 sin

θ

2+ α2 cos

θ

2〉, (4.17)

and θ = π/2 for a symmetric beam splitter. The transformation of the coherent state amplitudes isthe same as the corresponding transformation of annihilation operators would be in the Heisenbergpicture.

Let the (generally mixed) signal state ρ be represented by the Glauber-Sudarshan P -functionP (β) (see Eq. (2.18)) and let the local oscillator coherent state be again | − A〉. The beam splitterinput state is then

ρin = | −A〉〈−A| ⊗∫

P (β) |β〉〈β| d2β (4.18)

and the output state is

ρout =

∫

P (β)

∣

∣

∣

∣

−A− β√2

⟩

1

⟨−A− β√2

∣

∣

∣

∣

⊗∣

∣

∣

∣

−A+ β√2

⟩

2

⟨−A+ β√2

∣

∣

∣

∣

d2β. (4.19)

The probability P jm of finding j +m and j −m photons at the beam splitter outputs is then

P jm = 2〈j −m| 1〈j +m|ρout|j +m〉1|j −m〉2 (4.20)

and we have used Eq. (2.13) for its calculation. To be able to simplify the resulting expressions

containing powers such as(

1± βA

)j±m, we have used the expansion

(1 + x)n = exp[n ln(1 + x)] = exp

[

n

∞∑

k=1

(−1)k−1xk

k

]

, (4.21)

29


that is based on the Taylor expansion of the logarithm and is of key importance for the calculation.The radius of convergence of the series in the exponent in Eq. (4.21) is equal to unity. Thereforewe had to make sure that the expansion was not used for |x| ≥ 1 and hence for |β| ≥ A. However,when performing the integral in Eq. (4.19), the variable β runs over the whole complex plane andthe expansion can therefore be used only if

P (β) = 0 for |β| ≥ A (4.22)

holds. We will come back to this condition later.With the help of Eq. (4.21) and after some algebra we have arrived at the following expression

for the probability P jm:

P jm =

√π 2−2j e−A

2A4j e2m

2/A2

(j +m)! (j −m)!Tr

{

ρ :

[

|x〉〈x| exp(

−2j −A2

2A2{a2 + (a†)2}

+ 2m∞∑

k=2

1

2k − 1

a2k−1 + (a†)2k−1

A2k−1− j

∞∑

k=2

1

k

a2k + (a†)2k

A2k

)]

:

}

, (4.23)

where the eigenvalue of the quadrature is x =√

2mA . The normal-ordering symbol : : should be

understood such as all the creation operators stand to the left of the projector |x〉〈x| and all theannihilation operators stand to the right of it, that is,

: |x〉〈x|ar(a†)s : = (a†)s|x〉〈x|ar. (4.24)

The exponential in Eq. (4.23) can be expanded as a series with an increasing number of the creationand annihilation operators, which yields the probability P j

m in the form of the following series:

P jm =

√π 2−2j e−A

2A4j e2m

2/A2

(j +m)! (j −m)!

{

〈x|ρ|x〉 − 2j −A2

2A2[〈x|a2ρ|x〉+ 〈x|ρ(a†)2|x〉]

+2m

3A3[〈x|a3ρ|x〉+ 〈x|ρ(a†)3|x〉] + . . .

}

, (4.25)

and the corresponding POVM is

Πjm =

√π 2−2j e−A

2A4j e2m

2/A2

(j +m)! (j −m)!

{

|x〉〈x| − 2j −A2

2A2[|x〉〈x|a2 + (a†)2|x〉〈x|]

+2m

3A3[|x〉〈x|a3 + (a†)3|x〉〈x|] + . . .

}

. (4.26)

It is not hard to show that for large A the first term in the parentheses in Eq. 4.25 dominates. As thecorresponding POVM is proportional to the projector |x〉〈x|, homodyne detection clearly measuresthe field quadrature for a strong local oscillator. Furthermore, for large A the fraction before theparentheses in Eq. (4.25) approaches the fraction in Eq. (4.12), which makes the results (4.12)and (4.25) in the limit A→∞ equal. However, Eq. (4.25) yields, in contrast to Eq. (4.12), correctionterms that express how much the real homodyne detection POVM differs from the ideal quadraturePOVM and that can be used for describing homodyne detection with a weak local oscillator. Alsothese correction terms were derived for the first time.

4.4.1 Properties of the series expressing the probability P jm

Before we analyze the result (4.25) for the probability P jm, let us first return to the validity condition

of the calculation. As we have mentioned, the use of expansion (4.21) is the key step in the derivation

30


of P jm and it is allowed only if the condition (4.22) holds. In other words, the support of the P -

function of the signal state has to lie inside the circle with the radius A, which is clearly a very strongcondition. We have shown that this condition is satisfied by the class of so-called z-regular states(with z < A) that we defined as states whose coefficients in the Fock basis decrease asymptoticallyat least as quickly as the coefficients of the coherent state |z〉. Examples of z–regular states includecoherent states with the amplitude smaller than z, all Fock states and superpositions or mixturesof a finite number of such states. At the same time, many states that are typically subject tohomodyne detection do not satisfy the criterion (4.22), e.g. squeezed states or thermal states witha non-zero mean number of photons. This might be a serious problem as one always comes acrosssome thermal noise in real experiments and our calculation would hence not be useful for describingsuch experiments. However, we will show now that this problem can be avoided due to the interestingproperties of the P -representation and one can use the series (4.25) also for the states that do notsatisfy the condition (4.22).

First consider the situation when the density matrix ρ has just a finite number of terms in theFock basis, for example for the signal state in a superposition of a finite number of Fock states.Such a state is z-regular for any z > 0 and the calculation is hence correct. Moreover, there existsa number N such that the density matrix satisfies ρmn ≡ 〈m|ρ|n〉 = 0 for all m > N,n > N . Now, inthe terms in the parentheses in Eq. (4.25) all the annihilation operators are to the left from ρ and allthe creation operators are to the right of ρ. It then follows that the series has only a finite numberof elements as the number of the field operators increases gradually.

Next consider the situation when the P -function of the state ρ does not satisfy the condition (4.22),for example for a squeezed or thermal state. We will show that even in this case one can use thePOVM by the procedure of the state truncation. We define the state ρ[N ] for a given N ∈ N asfollows:

ρ[N ]mn =

{

(∑N

i=1 ρii)−1

ρmn for m ≤ N,n ≤ N0 else

(4.27)

This definition truncates the state ρ in the Fock basis and normalizes the resulting state. Clearly,for N → ∞ the state ρ[N ] converges to ρ. Therefore the probability P j

m(ρ[N ]) calculated for thestate ρ[N ] converges to the probability P j

m(ρ) for the state ρ. Hence, for any ρ one can find N0 largeenough such that the probabilities P j

m(ρ) and P jm(ρ[N0]) differ at an arbitrarily small level. Then

the probability P jm(ρ) is approximated by a finite series for P j

m(ρ[N0]) to a very high precision, eventhough the series expressing P j

m(ρ) itself possibly diverges. From the practical point of view thismeans that the the series (4.25) can be used for evaluating the POVM even for the state for whichit diverges.

These properties of the series (4.25) may seem quite odd and they are connected with the followingproperties of the P -function. The P -function of the Fock state |n〉 is equal to zero for all β 6= 0 2.The same applies to the P -function of an arbitrary truncated state. Of course, for a general (non-truncated) state P (β) can be nonzero also for β 6= 0. If we define a sequence {ρ[N ], N = 0, 1, . . . } fora general state ρ according to Eq. (4.27), then the P -function of each ρ[N ] from this sequence is zerooutside the origin, which does not apply to the P -function of the limit of this sequence.

It is also interesting to note that the convergence of the series (4.25) is not directly related tothe behavior of the initial terms. It can happen (e.g. for a weak thermal state or a weakly squeezedvacuum state) that the initial subsequent terms decrease quickly but after some time, they startto grow and the series diverges. At the same time, for weak signal states (compared to the LO)these first terms provide an increasingly good approximation to the photon counting probability P j

m.The situation is thus similar to the one in perturbation theory: even though a perturbation seriesdiverges, its several (or many) initial terms may give a good approximation.

2as we have mentioned, the P -function of the state |n〉 is equal to the nth derivative of the Dirac δ-function, whichis very singular in the origin; however, outside the origin it is equal to zero in the whole complex plane

31


To verify the result (4.23), we have performed a number of numerical simulations in which fora given state ρ we have calculated the probabilities P j

m for fixed j and all possiblem in two ways – oneused the exact expression in terms of the Wigner functions djmm′ and the other used our result (4.25).We have truncated the series after one, three and five terms, respectively, and have observed whetherthe increasing number of terms approximates the exact probability P j

m with an increasing accuracy.Indeed, it was really so even for the squeezed state for which the series (4.25) does not converge. Theresults of the simulation can be seen in Fig. 4.2.

4.5 Strong local oscillator

It remains to say what the conditions are under which homodyne detection measures the field quadra-ture “well”. As we have seen in Eqs. (4.4) and (4.5) already, this happens for large local oscillatoramplitudes A. To see what this means in a given situation can be seen from analyzing the series (4.25)where the first term 〈x|ρ|x〉 should be dominant as it corresponds to the ideal quadrature measure-ment. It has turned out that the condition is as follows: the mean number of photons n in the signalstate should be much less than the amplitude of the local oscillator A. It is thus not enough if n ismuch smaller than the mean photon number in the local oscillator (which is A2). In fact the conditionis stronger – is can also be expressed in that the mean photon number in the signal state is much lessthan the fluctuation of the mean photon number in the local oscillator. If this condition were notsatisfied, by measuring the total photon number 2j one could approximately determine how manyphotons originate from the signal state. However, this would necessarily disturb the measurementof the quadrature xϕ as the photon number operator does not commute with the quadrature. Thisway, the strong local oscillator condition follows from the complementarity principle.

4.6 Conclusion

Together with Barry Sanders I have analyzed balanced homodyne detection via calculating the prob-ability of detecting given numbers of photons at homodyne detector beam splitter outputs. We havederived the POVM by a direct calculation for the first time by two different methods. We have shownthat for a strong local oscillator the homodyne detection provides a projective measurement of thefield quadrature and the corresponding POVM is proportional to the projector |x〉〈x|. In addition,the calculation that employs the Glauber-Sudarshan P -representation yields correction terms usefulif the local oscillator is not too strong. We have performed numerical simulations that confirmed thetheoretical results including the correction terms.

32


0

5e-05

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0.0004

0.00045

10 20 30 40 50 60 70

P

hoto

n co

untin

g pr

obab

ility

m

(a)

-5e-05

0

5e-05

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0.0004

0.00045

0.0005

-20 -15 -10 -5 0 5 10 15 20

P

hoto

n co

untin

g pr

obab

ility

m

(b)

-1e-05

0

1e-05

2e-05

3e-05

4e-05

5e-05

6e-05

7e-05

-80 -60 -40 -20 0 20 40 60 80

P

hoto

n co

untin

g pr

obab

ility

m

(c)

Figure 4.2: Simulations of the homodyne detection probability distribution P jm for (a) coherent state |γ〉 with

γ = 2 for j = 190, (b) squeezed state with squeezing parameter s = 4.5 for j = 219.5, and (c) number state|6〉 for j = 183.5. The exact probabilities are shown in black, and the truncated ones are shown in green, blueand red, respectively, according to the increasing number of terms in Eq. (4.25) taken into account. The redcurves are so close to the black ones in (b) and (c) that they almost cover them in the plots.

33

Chapter 5

Fermion coherent states

5.1 Introduction

As we have mentioned in Chapter 2, coherent states of light occupy an important position in quantumoptics for their useful physical and mathematical properties, and they provide important represen-tations such as the Glauber-Sudarshan P -representation. These properties of coherent states areconnected with the boson nature of the electromagnetic field – the fact that the quanta of the fieldare subject to Bose-Einstein statistics.

There are many similarities and analogies between the fields of bosons and those of fermions,that is, particles subject to the Fermi-Dirac statistics and the Pauli exclusion principle. One canperform similar interference experiments with the same results with both types of particles, definecoherence for both of them etc. Therefore a natural question arises if it is possible to extend thedefinition of coherent states to fermion fields. Indeed, there is a method of introducing fermioncoherent states that employs so-called Grassmann variables [22, 23, 24]. The resulting states haveformally all the desired properties of coherent states; however, they lack any physical interpretationand do not satisfy the basic axioms for vectors in the Hilbert space of physical states. This way, theGrassmann coherent states are an interesting mathematical structure rather than a physical objectwith a direct relation to reality.

For this reasons my Australian colleagues and I were thinking of defining coherent states withoutusing the Grassmann numbers. We have analyzed this question in the work [52] and we have shownthat generalizing coherent states to fermion fields is very problematic and that it is not possibleto define fermion coherent states analogous to their boson counterpart in the Hilbert space. Inaddition, we have proved several theorems that hold for fermion correlation functions and have nodirect analogy for bosons. These theorems are valid due to the Pauli exclusion principle and showthe distinctive properties of fermion fields regarding multi-particle correlations. Our results from [52]are presented and explained in the following sections.

5.2 The options for introducing coherent states of light

In this section, we have a closer look at some properties of boson coherent states and will show whytheir generalization to fermions is problematic. For concreteness, we will talk about electromagneticfield, but the definitions that follow can be applied to other boson fields as well.

Coherent states of boson fields can be defined in several equivalent ways and we will consider herefour of them, the equivalence of which was shown in our work [52]. As we have mentioned, coherentstates of light are close to states of the classical field with a well-defined amplitude and phase. Inthe classical theory of the electromagnetic field one can define a so-called phasor, which is a complexamplitude of the field that has a well-defined value for classical states of light. As the phasor isreplaced by the annihilation operator a in the quantum theory, it is natural to define coherent states

34

Chapter 5. Fermion coherent states

as those states for which the value of a is well-defined, that is, as the eigenstates of the annihilationoperator. This leads to the following definition:

Definition 1 Coherent state of a given mode of the electromagnetic field is an eigenstate of theannihilation operator a of the mode. A multimode coherent state is the eigenstate of all annihilationoperators (that are linear combinations of the single-mode annihilation operators ak).

The second way of defining coherent states is related to their more general conception as resultsof some group action on a fixed state. This way, coherent states of light emerge by the actionof elements of the Heisenberg-Weyl group HW(1) = {eαa†−α∗a+iϕ |α ∈ C, ϕ ∈ R} on the vacuumstate |0〉. As the physical operation corresponding to the elements of the Heisenberg-Weyl groupis displacement in the phase space, we can understand coherent states as displaced vacuum statesaccording to the following definition:

Definition 2 Coherent state is the vacuum state displaced in the phase space, that is, the resultof the action of the displacement operator D(α) = eαa

†−α∗a on the vacuum |0〉.The third definition is related to the behavior of coherent states on a beam splitter. If a general

state of light is mixed with the vacuum on a beam splitter, the output states will in general beentangled or at least correlated. For example, for a single-photon input state |1〉, the output two-mode state is t|1〉|0〉 + r|0〉|1〉 with t and r being the transmissivity and reflectivity of the beamsplitter, respectively. This state is entangled because it cannot be expressed as a product of statesof the two output modes. However, if the input state is a coherent state, the beam splitter outputswill be unentangled coherent states as we have mentioned in Chapter 4, Eq. (4.17). This propertymakes coherent states useful for complicated optical experiments in which many beams are derivedfrom the same coherent source of light. This way we arrive at the following definition:

Definition 3 Coherent state is a pure state that produces unentangled outputs when mixed with thevacuum on a beam splitter.

In 1963 R. J. Glauber developed the theory of coherent states of light based on the propertiesof normally-ordered correlation functions [53]. These correlation functions (or shortly correlators)are defined by

G(n)(x1, . . . , xn, yn, . . . , y1) ≡ 〈ψ†(x1) · · · ψ†(xn)ψ(yn) · · · ψ(y1)〉 , (5.1)

where ψ†(x) and ψ(x) is the creation and annihilation operator at the space-time point x, respectively.The normal ordering means that all the creation operators are to the left of the annihilation operatorsin Eq. (5.1).

The normally-ordered correlators describe coherence properties of the field related to photode-tection in which photons are absorbed in the detector. Of a particular importance are correlatorswith repeated arguments for which yi = xi. The correlator

G(n)(x1, . . . , xn) ≡ G(n)(x1, . . . , xn, xn, . . . , x1)

expresses the probability density of finding a particle at the point x1, another particle at x2, etc., upto the nth particle at xn. It is possible to measure these correlators directly using detectors placedin the field. One can show that for states of the classical electromagnetic field with a well-definedphase and amplitude the correlators factorize – it is possible to express them as products of functionsof their arguments. For example, the correlator (5.1) factorizes if there exists a function f(x) suchthat

G(n)(x1, . . . , xn, yn, . . . , y1) = f∗(x1) · · · f∗(xn)f(yn) · · · f(y1) (5.2)

holds. Then it is natural to define coherent states of a quantized field as those factorizing thecorrelators, which leads to this definition:

Definition 4 Coherent state is a state for which the normally-ordered correlators factorize.

35


5.3 Fermion analogy of the boson coherent state

Before attempting to define fermion coherent states, we briefly mention fermion creation and annihi-lation operators c†i , ci. Similarly as in the case of bosons, these operators raise and lower the particlenumber in the ith mode by one. However, in contrast to the boson commutation relations (2.2), thefermion field operators satisfy the following anticommutation relations:

{ci, c†j} = δij 1, {ci, cj} = {c†i , c†j} = 0 . (5.3)

Here the anticommutator is defined as {A, B} ≡ AB + BA. One consequence of these relations isthat (c†)2 = 0, that is, it is not possible to create more than one fermion in a given mode, whichis a possible way of expressing the Pauli exclusion principle. Hence, for each mode there are onlytwo states with a definite number of particles: the vacuum |0〉 and the occupied state |1〉. The fieldoperator action on these states is as follows,

c†|0〉 = |1〉, c†|1〉 = 0 (5.4)

c|0〉 = 0, c|1〉 = |0〉. (5.5)

When attempting to generalize coherent states to fermions using the definitions from the previoussection, one meets serious difficulties. Consider Definition 1 first. As can be verified easily using therelations (5.5), the only eigenstate of the fermion annihilation operator in the Hilbert space is thevacuum |0〉. Indeed, when acting by the annihilation operator on a general pure state |ψ〉 = γ|0〉+δ|1〉,one obtains δ|0〉, which is a multiple of |ψ〉 only if δ = 0, which means that |ψ〉 = |0〉. According tosuch definition, the only fermion coherent state would be the vacuum, which is not very interesting.One obtains the same result also from Definition 3 related to the behavior of coherent states ona beam splitter. As can be shown easily, when any state other than vacuum is split on a beamsplitter, the output states will always be entangled. Again, the vacuum would be the only fermioncoherent state according to such definition.

One could also use the generalization of the boson displacement operator to fermion fields. Itturns out that the action of this operator,

D(α) = eαc†−α∗c, (5.6)

on the vacuum produces an arbitrary pure single-mode state. This would not yield a reasonabledefinition of fermion coherent states as all single-mode states would be coherent. However, if themagnitude of the displacement |α| is small, one can define an approximate fermion coherent stateas |α〉 = D(α)|0〉 that also approximately satisfies Definitions 1 and 3. Multimode approximatecoherent states can then be obtained by a consequent action of single-mode displacement operatorson the vacuum. However, at this point one meets other difficulties. The reason is that displacementoperators of different modes do not commute and hence it matters in what order the displacements inthe individual modes are performed. By changing the order of the displacement operators one obtainsa different state in general, which is illustrated in Fig. 5.1. One could try to solve this ambiguityby averaging the multimode state over all possible orderings (permutations) of the displacementoperators, which would yield, however, a state with no more than one fermion and so one could notobtain in this way multi-particle states at all.

When we tried to define fermion coherent states by generalizing Definition 4, i.e., as states thatfactorize the correlation functions, we found out that the result is similar as in the case of Definitions1 – 3, that is, that the fermion coherent states cannot be reasonably defined in an analogous way asfor bosons. We have also found a number of interesting properties of fermion correlation functionsthat are connected with the Pauli exclusion principle. Some of them can be expected intuitivelywhile others may be surprising. In the next section we explain the statements that we proved in [52]and we will show that fermion correlation functions cannot be factorized up to some exceptions.

36


(a) (b)line 1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

-2-1.5

-1-0.5

0 0.5

1 1.5

2

-2-1.5

-1-0.5

0 0.5

1

x

y

1.5 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

line 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

-2-1.5

-1-0.5

0 0.5

1 1.5

2

-2-1.5

-1-0.5

0 0.5

1

x

y

1.5 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Figure 5.1: The square of modulus of the complex degree of coherence, |γ(x, y)|2 defined in Eq. (5.7), for twoapproximate complex fermion coherent states that were obtained from the vacuum by the consequent actionof displacement operators of forty modes. The modes and the corresponding amplitudes αk are the samefor both figures (a) and (b) and the only difference is the ordering of the displacement operators. As theseoperators do not commute, one obtains physically different states with different observable properties.

5.4 Properties of fermion correlators

According to the first proposition we proved, the probability of finding n fermions at n points, twoof which approach each other, goes to zero. This is a direct consequence of the fermion anticommu-tation relations and the Pauli principle. The proposition can be formulated as follows:

Proposition 1 For any fermion field state, the fermion correlator G(n)(x1, . . . , xn) tends to zerowhenever two points xi, xj approach each other.

This proposition is demonstrated in Fig. 5.2 for the approximate fermion coherent state.

One consequence of Proposition 1 is that a correlation function of order higher than one (i.e.,for n > 1) cannot factorize except the case in which it is identically equal to zero. Indeed, if thereexisted a function f(x) such that Eq. (5.2) holds, then f(x) would have to be zero because it holdsG(n)(x1, . . . , xn)→ 0 for xi → xj . The following proposition is hence valid:

Proposition 2 If the fermion correlator G(n)(x1, . . . , xn, yn, . . . , y1) factorizes for some for n > 1,then it is identically equal to zero.

As we can see, the definition of coherent states based on correlator factorization does not have muchsense for fermions as the correlators of the second or higher order cannot be factored non-trivially.

One of the important characteristics of the coherence properties of boson and fermion fields isthe normalized first-order correlator with mixed arguments. It is called complex degree of coherenceand is defined as

γ(x, y) =G(1)(x, y)

√

G(1)(x, x)G(1)(y, y). (5.7)

The complex degree of coherence has a direct physical meaning: in a double-slit experiment in whichequal slits are placed at the points x and y the visibility of the interference fringes is given by |γ(x, y)|(see Fig. 5.3). If the field has a large coherence of the first order (or second order coherence accordingto the terminology in [21]), then the fringe visibility is high and |γ(x, y)| is close to unity. We havefound an interesting consequence of such coherence for multi-particle correlators. If |γ(x, y)| = 1 forsome x, y, it is not possible to find a fermion at the point x and another one at y. The followingproposition formulates this even more generally:

37


line 1 600 500 400 300 200 100

-2-1.5

-1-0.5

0 0.5

1 1.5

2

-2-1.5

-1-0.5

0 0.5

1

x y

1.5 2

0

100

200

300

400

500

600

700

Figure 5.2: The second-order correlator |G(2)(x, y)|2 for a multimode approximate complex fermion coherentstate, i.e., the vacuum displaced in the individual modes with a small displacement |α|. The dip along theline x = y is a common feature of this correlator regardless of the state and is a consequence of the fact thatG(2)(x, y) → 0 when x → y. For this reason, it is not possible to factorize this correlator into a productof a function of a and a function of y.

y

xdetector

Figure 5.3: Fringe visibility in a double-slit experiment with equal slits located at the points x, y is givenby the magnitude of the complex degree of coherence γ(x, y) when a quasimonochromatic beam of bosons orfermions incides on the slits.

Proposition 3 Let |γ(x, y)| = 1 holds for the fermion state |ψ〉 and a pair of points x, y. ThenG(n)(x, y, x3, . . . , xn, yn, yn−1, . . . , y1) = 0 holds for any n > 1 and arbitrary points x3, . . . , xn,y1, . . . , yn.

There is an interesting corollary of Proposition 3 for fields that have a full first-order coherence,e.g., that satisfy |γ(x, y)| = 1 for all x, y. According to Proposition 3 it is then not possible to find twofermions at two different points and hence the field contains one fermion at most. This is expressedby the following proposition:

Proposition 4 Let |γ(x, y)| = 1 for some state |ψ〉 for all x, y. Then the state has support over onlythe vacuum and single-particle states. That is, the probability of finding two fermions in the field isidentically zero.

Proposition 4 imposes a strong condition on the coherence of fermion fields: a field that containsmore than one fermion cannot exhibit a full first-order coherence. Moreover, for such fields eventhe first-order correlator cannot factorize as from G(1)(x, y) = f(x)g(y) it follows that |γ(x, y)| = 1,which is not possible according to Proposition 4.

38


None of Propositions 1 – 4 is valid for bosons. Hence it is clear that there is a fundamentaldifference between both types of particles that does not vanish even when considering fields withvery low occupation numbers. If the field contains more than one fermion and correlators of orderlarger than one are in question, the difference will always be present.

We have demonstrated some of the propositions mentioned above on the example of approximatefermion coherent states introduced in Sec. 5.3. It has turned out that even for very small values|α| the second-order correlators do not factorize, which is in agreement with Proposition 1 and isillustrated in Fig. 5.2.

5.4.1 Correlators of chaotic states

We illustrate the general properties of fermion correlators on the example of chaotic states [54]. Thesestates have the maximum entropy of all states satisfying certain conditions, e.g. having a given energyor mean particle numbers in the individual modes. The best-known example of a chaotic state is thethermal state that has the maximum entropy for a given energy.

For noninteracting bosons and fermions, the single-mode density matrix of a chaotic state can beexpressed as

ρB =1

1 +M

∞∑

n=0

(

M

1 +M

)n

|n〉〈n|, ρF = (1−M)

1∑

n=0

(

M

1−M

)n

|n〉〈n| , (5.8)

respectively.

Now consider a multi-mode chaotic state of bosons or fermions. Let N denote the numberof occupied modes that will be labeled by 1, . . . , N , and let Mi be the mean number of particles inthe ith mode. We have shown that the first-order correlator is the same for both bosons and fermions:

G(1)B,F(x, y) =

N∑

i=1

Miϕ∗i (x)ϕi(y) . (5.9)

Here ϕi(x) are the spatial mode functions that connect the mode and point annihilation operatorsvia the relation ψ(xi) =

∑

k ϕk(xi)ak. Eq. (5.9) shows that the coherence properties of the firstorder are not influenced by the boson of fermion nature of the particles. This can be expected as thefirst-order coherence is not connected with multi-particle correlations and therefore it should not beinfluenced by the exchange interaction of identical particles.

In contrast, higher-order correlators do depend on the type of particles. The nth-order correlatorsof the boson and fermion chaotic state are

G(n)B,F(x1, . . . , xn, yn, . . . , y1) =

∑

P

χpar(P )B,F G

(1)B,F(x1, yP (1))G

(1)B,F(x2, yP (2)) · · ·G(1)

B,F(xn, yP (n)) . (5.10)

For fermions, this is an exact result while for bosons it is valid to a high precision if Mi ¿ 1 for alli. The sum runs over all permutations P of the indexes 1, 2, . . . , n, par(P ) denotes the parity of Pand χB = 1 and χF = −1 is the boson and fermion sign factor, respectively.

Eq. (5.10) shows that multiparticle correlators are different for fermions and bosons. If xi →xj , i 6= j, the correlator GF goes to zero due to the factor (−1)par(P ), which is in a full agreementwith Proposition 1.

5.5 Conclusion

As we have shown, introducing fermion coherent states is connected with large difficulties. It is notpossible to define states having analogous properties as boson coherent states even in a single mode.

39


When attempting to generalize approximate fermion coherent states to multi-mode situation, incon-sistencies arise due to the non-invariance of the states with respect to changing the mode ordering.Further, we have shown that approximate fermion coherent states cannot factorize correlation func-tions. All these difficulties are connected with the Pauli exclusion principle and the anticommutingproperties of the field operators.

Thus it seems that the only reasonable option for defining fermion coherent states is to use theGrassmann variables. However, then the physically significant quantities such as amplitude andphase as well as the inner product of such states loose their meaning. Is may happen that wheninvestigating the Grassmann coherent states further, a closer relationship of these states with thephysical reality will be discovered. However, it is probably not reasonable to try to introduce fermioncoherent states without the Grassmann variables.

40

Bibliography

[1] C. Kittel and H. Kroemer, Thermal Physics (W. H. Freeman and Company, New York, 1980);C. Cohen-Tannoudji, B. Diu and F. Laloe, Quantum mechanics (John Wiley, New York, 1977).

[2] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (CambridgeUniversity Press, Cambridge, 2000).

[3] L. Vaidman, Phys. Rev. A 49, 1473 (1994).

[4] S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett. 80, 869 (1998).

[5] A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble and E. S. Polzik, Science282, 706 (1998).

[6] D. Bouwmeester, A. K. Ekert and A. Zeilinger (eds.), The Physics of Quantum Information(Springer, New York, 2000).

[7] C. H. Bennett and G. Brassard, Quantum cryptography, public key distribution and coin tossing,in Proc. Int. Conf. Computers, Systems and Signal Processing (Bangalore, 1984).

[8] P. W. Shor and J. Preskill, Phys. Rev. Lett 85, 441 (2000).

[9] D. Deutsch and R. Jozsa, Proc. Royal Soc. London, 439A (1992).

[10] P. Shor, Proc. 35th Ann. Symp. on Foundations of Computer Science, 124 (1994).

[11] S. D. Bartlett and B. C. Sanders, J. Mod. Opt. 50, 2331 (2003).

[12] L. K. Grover, Proc. of the 28th Ann. ACM Symp. on the Theory of Computing, 212 (1996).

[13] W. K. Wootters and W. H. Zurek, Nature 299, 802 (1982).

[14] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Ams-terdam, 1982).

[15] H. P. Yuen and J. H. Shapiro, in Coherence and Quantum Optics IV, L. Mandel and E. Wolfeds. (Plenum, New York, 1978).

[16] H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-25, 179 (1979); IT-26, 78 (1980);H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).

[17] H. P. Yuen and V. W. Chan, Opt. Lett. 8, 177 and 8, 345 (1983).

[18] P. M. Radmore and S. M. Barnett, Methods in Theoretical Quantum Optics (Oxford UniversityPress, 1997).

[19] S. L. Braunstein, Phys. Rev. A 42, 474 (1990).

41


[20] K. Banaszek and K. Wodkiewicz, Phys. Rev. A 55, 3117 (1997).

[21] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press,Cambridge, 1995).

[22] J. L. Martin, Proc. Roy. Soc. Lond. A 251, 543 (1959); Y. Ohnuki and T. Kashiwa, Prog. Theor.Phys. 60, 548 (1978); J. R. Klauder and B.-S. Skagerstam, Coherent States (World Scientific,Singapore, 1985).

[23] F. A. Berezin, The Method of Second Quantization (Academic Press, New York, 1966).

[24] K. E. Cahill, R. J. Glauber, Phys. 59, 1538 (1999).

[25] I. M. Gelfand and N. Y. Vilenkin, Generalized functions, Vol. 4 – Applications of HarmonicAnalysis (Academic Press, New York, 1964).

[26] F. Gieres, Rep. Prog. Phys. 63, 1893 (2000).

[27] S. L. Braunstein, arXiv: quant-ph/9904002.

[28] R. J. Glauber, Phys. Rev. 131, 2766 (1963).

[29] E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).

[30] J. R. Klauder, Phys. Rev. Lett. 16, 534 (1966).

[31] K. Mølmer, Phys. Rev. A 55, 3195 (1997).

[32] T. Rudolph and B. C. Sanders, Phys. Rev. Lett. 87, 077903 (2001).

[33] C. W. Helstorm, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

[34] A. D. Smith, arXiv: quant-ph/0001087.

[35] R. Cleve, D. Gottesman and H.-K. Lo, Phys. Rev. Lett. 83, 648 (1999).

[36] D. Gottesman, Phys. Rev. A 61, 042311 (2000).

[37] T. Tyc and B. C. Sanders, Phys. Rev. A 65, 042310 (2002).

[38] T. Tyc, D. Rowe and B. C. Sanders, J. Phys. A 36, 7625 (2003).

[39] A. M. Lance, T. Symul, W. P. Bowen, T. Tyc, B. C. Sanders and P. K. Lam, New J. Phys. 5,4 (2003).

[40] A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders, T. Tyc, T. C. Ralph and P. K. Lam, Phys.Rev. A 71, 33814 (2005).

[41] A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders and P. K. Lam, Phys. Rev. Lett. 92, 177903(2004).

[42] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935).

[43] C. M. Caves, Phys. Rev. D 26, 1817 (1982).

[44] C. Weedbrook et al., Phys. Rev. Lett. 93, 170504 (2004); N. Treps et al., Science 301, 940(2003); N. Treps et al., Phys. Rev. Lett. 88, 203601 (2002).

[45] B. Schumacher, Phys. Rev. A 51, 2738 (1995).

42

Bibliography

[46] S. Reynaud, A. Heidmann, E. Giacobono and C. Fabre, Quantum fluctuations in optical systems,in Progress in Optics XXX, E. Wolf ed. (Elsevier, 1992)

[47] U. Leonhardt, Measuring the quantum state of light (Cambridge University Press, Cambridge,1997).

[48] T. Tyc and B. C. Sanders, J. Phys. A 37, 7341 (2004).

[49] J. J. Sakurai, Modern quantum mechanics (Addison-Wesley, 1985); L. D. Landau and E. M.Lifshitz, Quantum mechanics (Pergamon Press, 1977).

[50] B. Yurke, S. L. McCall and J. R. Klauder, Phys. Rev. A 33, 4033 (1986).

[51] D. J. Rowe, B. C. Sanders and H. de Guise, J. Math. Phys. 42, 2315 (2001).

[52] T. Tyc and B. C. Sanders, in the review process in New Journal of Physics.

[53] R. J. Glauber, Phys. Rev. 130, 2529 (1963).

[54] R. Glauber, in Quantum Optics, edited by S. Kay and A. Maitland. (Academic Press, New York,1970).

43

Three problems from quantum opticstomtyc/habilitation.pdf · basic unit of quantum information is a...

Documents

Transcript of Three problems from quantum opticstomtyc/habilitation.pdf · basic unit of quantum information is a...