detection of quantum entanglement in physical systems · detection of quantum entanglement in...

Detection of quantum entanglementin physical systems

Carolina Moura Alves

Merton College

University of Oxford

A thesis submitted for the degree of

Doctor of Philosophy

Trinity 2005

Abstract

Quantum entanglement is a fundamental concept both in quantum mechanics and inquantum information science. It encapsulates the shift in paradigm, for the descrip-tion of the physical reality, brought by quantum physics. It has therefore been a keyelement in the debates surrounding the foundations of quantum theory. Entangle-ment is also a physical resource of great practical importance, instrumental in thecomputational advantages offered by quantum information processors. However, theproperties of entanglement are still to be completely understood. In particular, thedevelopment of methods to efficiently identify entangled states, both theoreticallyand experimentally, has proved to be very challenging. This dissertation addressesthis topic by investigating the detection of entanglement in physical systems.

Multipartite interferometry is used as a tool to directly estimate nonlinear propertiesof quantum states. A quantum network where a qubit undergoes single-particleinterferometry and acts as a control on a swap operation between k copies of thequantum state ρ is presented. This network is then extended to a more generalquantum information scenario, known as LOCC. This scenario considers two distantparties A and B that share several copies of a given bipartite quantum state.

The construction of entanglement criteria based on nonlinear properties of quantumstates is investigated. A method to implement these criteria in a simple, experimen-tally feasible way is presented. The method is based of particle statistics’ effectsand its extension to the detection of multipartite entanglement is analyzed. Finally,the experimental realization of the nonlinear entanglement test in photonic systemsis investigated. The realistic experimental scenario where the source of entangledphotons is imperfect is analyzed.

Acknowledgements

Contents

Abstract i

Acknowledgements ii

1 Introduction 11.1 Entanglement as a property of quantum systems . . . . . . . . . . . . . . . . . . 11.2 Entanglement as a physical resource . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Detection and characterization of entanglement . . . . . . . . . . . . . . . . . . . 21.4 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Chapter outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Basic concepts 52.1 State Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Density Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Mathematical properties of density operators . . . . . . . . . . . . . . . . 82.2.2 Ensemble interpretation of density operators . . . . . . . . . . . . . . . . 9

2.3 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Superoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Mathematical properties of superoperators . . . . . . . . . . . . . . . . . 102.4.2 Jamiolkowski isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Mathematical characterization of bipartite entanglement . . . . . . . . . . . . . . 112.5.1 Mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Experimental detection of entanglement . . . . . . . . . . . . . . . . . . . . . . . 132.6.1 Bell’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6.2 Entanglement witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.7 Multipartite entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7.1 Maximally entangled state . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.7.2 W State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.7.3 Cluster state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.8 Quantum networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.8.1 Universal set of gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.8.2 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

iii

CONTENTS iv

3 Direct estimation of density operators 213.1 Modified interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Multiple target states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Spectrum estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 Quantum communication . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 Extremal eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.4 State estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.5 Arbitrary observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Quantum channel estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Direct estimation of density operators using LOCC 284.1 LOCC estimation of nonlinear functionals . . . . . . . . . . . . . . . . . . . . . . 284.2 Structural Physical Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 SPA using only LOCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Entanglement detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4 Channel capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Entanglement Detection in Bosons 335.1 Nonlinear entanglement inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Estimation of the purities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2.1 Bipartite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2.2 Multipartite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3 Realization of the entanglement detection network . . . . . . . . . . . . . . . . . 365.4 Detection of entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.5 Degree of macroscopicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.5.1 Determination of ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Entropic inequalities 416.1 Entropic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.1.1 Graphical comparison between Bell-CHSH and entropic inequalities . . . 426.2 Experimental proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.2.1 Realistic sources of entangled photons . . . . . . . . . . . . . . . . . . . . 46

7 Conclusion 49

Bibliography 51

List of Figures

2.1 The controlled-U gate. The top line represents the control qubit and the bottomline represents the target qubit. U acts on the target qubit iff the control qubitis in the logical state |1〉. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 The Mach-Zender interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 The quantum network corresponding to the Mach-Zender interferometer (ϕ =

θ1 − θ0). The visibility of the interference pattern associated with p0 varies as afunction of ϕ according to Eq.(2.70). . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 A modified Mach-Zender interferometer with coupling to an ancilla by a controlled-U gate. The interference pattern is modified by the factor veiα = Tr [Uρ]. . . . 22

3.2 Quantum network for direct estimations of both linear and non-linear functionsof a quantum state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 A quantum channel Λ acting on one of the subsystems of a bipartite maxi-mally entangled state of the form |ψ+〉 =

∑k |k〉|k〉/

√d. The output state

%Λ = 1d

∑kl |k〉〈l| ⊗ Λ (|k〉〈l|), contains a complete information about the channel. 26

4.1 Network for remote estimation of non-linear functionals of bipartite density op-erators. Since Tr[V (k)%⊗k] is real, Alice and Bob can omit their respective phaseshifters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 Network of BS acting on pairs of identical bosons. The two rows of N atoms,labelled I and II respectively, are identical, and the state of each of the rows isρ123...N . The total state of the system is ρ123...N ⊗ ρ123...N . . . . . . . . . . . . . . 34

5.2 In Fig. 4.2(a), we plot the violation V of the inequalities Eq. (5.2), V1 = Tr(ρ2123)−

Tr(ρ212) (dashed), V2 = Tr(ρ2

12)−Tr(ρ21) (grey) and V3 = Tr(ρ2

12)−Tr(ρ22) (solid),

as a function of the phase φ, for N = 3 atoms. Whenever V > 0, entanglement isdetected by our network. In Fig. 4.2(b) we plot different purities associated witha cluster state of size N , as a function of φ. B is any one atom not at an end(dotted), any two atoms not at ends and with at least two others between them(dashed), any two or more consecutive atoms not including an end (dash-dotted),any one or more consecutive atoms including one end (solid). The plotted puritiesare independent of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

v

LIST OF FIGURES vi

5.3 Plot of the purity ΠN−m for m = 1 (solid black), m = 7 (dashed black), m = 14(solid grey) and m = 20 (dashed grey), as a function of ε, for N = 300 atoms. . . 40

6.1 A graphical comparison of the Bell-CHSH inequalities with the entropic inequali-ties (6.2). All points inside the ball satisfy the entropic inequalities and all pointswithin the Steinmetz solid satisfy all possible Bell-CHSH inequalities. NB not allthe points in the outlining cube represent quantum states. . . . . . . . . . . . . . 43

6.2 In a special case of locally depolarized states, represented by points within thetetrahedron, the set of separable states can be characterized exactly as an octahe-dron. All states in the ball but not in the octahedron are entangled states whichare not detectable by the entropic inequalities. . . . . . . . . . . . . . . . . . . . 44

6.3 An outline of our experimental set-up which allows to test for the violation of theentropic inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.4 Possible emissions leading to four-photons coincidences. The central diagramshows the desired emission of two independent entangled pairs – one by sourceS1 and one by source S2. The top and the bottom diagrams show unwelcomeemissions of four photons by one of the two sources. . . . . . . . . . . . . . . . . 47

CHAPTER 1

Introduction

The subject of this dissertation is the detection of quantum entanglement in physical systems.Quantum entanglement was singled out by Erwin Schrodinger as “...the characteristic trait ofquantum mechanics, the one that enforces its entire departure from classical lines of thought.” [1].Indeed, after playing a significant role in the development of the foundations of quantum me-chanics [1, 2, 3], quantum entanglement has been recently rediscovered as a physical resource inthe context of quantum information science [4, 5, 6, 7]. This set of correlations, to which a clas-sical counterpart does not exist, arises from the interaction between distinct quantum systems.Entanglement is instrumental in the improvements of classical computation and classical com-munication results, of which two particularly important examples are the exponential speedupof certain classes of algorithms [8, 9] and physically secure cryptographic protocols [4].

1.1 Entanglement as a property of quantum systems

Entanglement was first used by Einstein, Podolski and Rosen (EPR) [2] to illustrate the con-ceptual differences between quantum and classical physics. In their seminal paper publishedin 1935, EPR argued that quantum mechanics is not a complete theory of Nature, i.e. it doesnot include a full description of the physical reality, by presenting an example of an entangledquantum state to which it was not possible to ascribe definite elements of reality. EPR definedan element of reality as a physical property, the value of which can be predicted with certainty,before the actual property measurement. This condition is straightforwardly obeyed in the con-text of classical physics, but not in the context of quantum mechanics. The predictive powerof quantum mechanics is limited to, given a quantum state and an observable, the probabilitiesof the different measurement outcomes. This feature led EPR to deem quantum mechanics asincomplete. The incompleteness of quantum mechanics, as understood by EPR, was to plaguephysicists for decades.

On one hand the quantum mechanical formalism explained the behaviour of microscopicalsystems to a great degree of accuracy. On the other hand, it was conceptually unsatisfactoryas a fundamental theory of Nature and the EPR argument seemed a valid one. It was notuntil John Bell published his seminal paper in 1964 [3], where he discussed the validity of theEPR assumptions, that light was shed into the matter. In his paper Bell does not make anyassumption about quantum mechanics. It does, however, assume that our classical common senseview of the world is true. He considered a thought experiment where two causally disconnected

1

CHAPTER 1. INTRODUCTION 2

observers share many identical pairs of physical systems and are allowed to perform two differenttypes of measurements on their respective systems. The measurements performed in each pairare chosen at random and correspond to elements of reality. The expectation values of theseobservables depend of the probability associated with a given outcome and the actual value ofthe outcome. Bell then derived a set of inequalities that bound the expectation value of a linearcombination of the observables. It turns out that certain entangled states theoretically violatethese inequalities, which means that either quantum mechanics is an incomplete description ofNature or the EPR assumptions are incorrect. The only way to decide which is the case wasby performing an experimental test of Bell’s inequalities. This test was realized with entangledpairs of photons in 1982 [10] and it shown the violation of the Bell’s inequalities, as predictedby quantum mechanics. This type of experimental test has subsequently been used to detectentanglement experimentally in physical systems [11].

1.2 Entanglement as a physical resource

Fundamental quantum effects, such as quantum tunnelling or stimulated emission, have yieldedover the last century important technological breakthroughs, of which semiconductors or lasersare two examples. Entanglement too has proved to be a physical resource capable of revolution-izing the theories of computation and information. Within quantum information science, thelogical unit of information is the qubit, a two-level quantum system. The qubit differs from thebit in that is can be any superposition of ”0” and ”1”. In particular, a set of qubits can be inan entangled state. The possibility of exploiting these quantum correlations between qubits, forrealizing computations faster than it would be possible classically, was first realized by Deutschin 1985 [12]. The development of quantum algorithms that ensued culminated with a result byShor for the efficient factoring the primes of a number [8]. The best classical algorithms forthis task scale exponentially with the size of the number to be factored, which means that it iseffectively impossible to factor large numbers. However, Shor’s algorithm can factor the primesin a time that scales polynomially with the number size, i.e. efficiently. This result is particularlyrelevant since the security of currently used cryptographic protocols is based on the difficulty offactoring large numbers. Therefore a quantum factoring machine would render these protocolsuseless.

Ironically, entanglement turns out to be the key resource in one of the possible solutionsto the security of cryptographic protocols. This solution, proposed by Ekert in 1991 [4], usesentangled states as the carrier of protected information. The security of the protocol comes fromthe fact that any attempt to gain access to the encrypted information, via a measurement on thestate, will necessarily disturb the quantum correlations. As mentioned earlier, the amount ofentanglement in a given state can be measured by checking for the violation of Bell’s inequalities.Therefore, any tampering of the carriers of information can be detected and the protocol aborted.

1.3 Detection and characterization of entanglement

We have seen how entanglement is not only a key concept in quantum mechanics, but alsoa physical resource of great practical importance. It is therefore no wonder that it has beenextensively researched, both as a mathematical concept and as a property of physical systems.In particular the experimental detection of entanglement is of paramount relevance for bothprobing the limits of validity of quantum mechanics, as a physical theory, and for the monitoringof quantum information processes. Its success is intimately related to the successful developmentof theoretical tools that not only help us to further understand the properties of entanglement,


but also provide practical experimental methods of detection.There have been so far two different approaches to investigating the concept of entanglement.

One approach, the mathematical one, treats quantum states as mathematical objects and tries todefine entanglement as a mathematical property. It considers the density matrix representationof quantum states and attempts to derive conditions that the matrices must obey in order torepresent an entangled state. This approach enabled the derivation of necessary and sufficientconditions for entanglement in systems of two or three qubits. These results were obtained byPeres [13] and the Horodeckis [14]. They pointed the way to a more general strategy of identifyingthe mathematical properties of entanglement, based on the theory of positive maps. I will returnto this statement in more detail in the next chapter. However, a full characterization of the setof entangled states for high-dimensional bipartite systems is yet to be found. In particular theunderstanding of entanglement between more than two systems, multipartite entanglement, isat present quite limited. Here, additional problems arise in the classification of entanglement,since it is possible for states to exhibit multipartite entanglement while being separable withrespect to some of the subsystems. A general framework for the classification of entanglementis yet to be developed and researchers have so far concentrated in studying specific classes ofmultipartite entangled states. I will present some examples of these classes in the next chapterthat we believe illustrate simultaneously the complexity of multipartite entanglement and itsgreat potential for quantum information processing.

The second approach to entanglement research, the physical one, treats quantum states asproperties of physical systems, that either exist in Nature or can be experimentally generated inthe laboratory. This approach differs fundamentally from the mathematical one in that it focuseson the types of states actually generated in a given physical setting. The characterization ordetection of entanglement in this case is accomplished by via tests that are tailored for the specificclass of states considered. In the next chapter we will present the two most commonly usedexperimental entanglement tests. Rather than aiming at a full characterization of entanglement,this approach aims at developing techniques and methods for entanglement detection that areexperimentally accessible. In particular, it tries to identify which properties of a given quantumsystem are relevant for entanglement detection. Providing a solution for this question will haveimportant consequences on the realization of experiments in quantum information processing,since it will direct the experimentalists to a more efficient, and possibly easier, detection ofentanglement in the laboratory.

Despite all the effort devoted in recent years to the characterization of entanglement, the fullunderstanding of entanglement’s properties still eludes researchers. My doctoral research aimedto contribute to our knowledge about entanglement by pursuing the physical approach. I havedeveloped new methods for not only the detection of both bipartite and multipartite entangle-ment but also the characterization of certain properties of quantum states. These methods areexperimentally realistic and one of them was in particular realized experimentally.

1.4 Outline of thesis

When writing this thesis, I was faced with the difficult choice of which of my doctoral researchresults to include. I decided to include the results that were not only the most directly relevantto the subject of the dissertation, entanglement detection, but also the results that formedthe most chronologically coherent set. It will become apparent that these results were obtainedsequentially and that they are different instances of one research program. This program startedfrom a rather abstract setting of quantum networks, specifically designed to measure stateproperties, and ended in the development of tailor-made experimental methods for the detection


of entanglement in photons. However, I also pursued other research projects, such as the studyof the computational complexity of quantum languages [15], the development of methods togenerate classes of bound entangled states [16] and the investigation of methods to efficientlygenerate graph states [17].

1.5 Chapter outline

I will now present the outline of remainder chapters of the thesis. Chapter 2 introduces the basicconcepts underlying the research results of the thesis. In particular it provides a mathematicaldescription of entanglement and discusses in more detail the general methods to detect andcharacterize entanglement. Chapter 3 addresses the problem of estimating nonlinear functionalsTrρk, k = 1, 2, ... of a general density operator ρ. The estimation method we proposed allows thedirect estimation of these nonlinear functionals. Our method uses an interferometric networkwhere a qubit undergoes single-particle interferometry and acts as a control on a swap operationbetween k copies of ρ. Chapter 4 extends the above result to a more general quantum informationscenario, known as LOCC. In this scenario we consider two distant parties A and B that shareseveral copies of a given bipartite quantum state ρAB and are only allowed to perform localoperations and communicate classically. Chapter 5 investigates entanglement criteria basedon nonlinear functionals of ρ that could be implemented in a simple, experimentally feasibleway. Our method is based of particle statistics’ effects and uses the fact that measuring thepurity of ρ is tantamount to measuring the probability of projecting the state of two copies ofρ in its symmetric or antisymmetric subspaces. We extend of the nonlinear inequalities to thedetection of multipartite entanglement. Chapter 6 investigates the experimental realization ofthe nonlinear entanglement test. We consider two copies of a polarization entangled pair ofphotons ρAB. We also analyze the realistic experimental scenario where the source of entangledphotons is imperfect. Chapter 7 presents a conclusion to the thesis, with a summary of the mainresearch results presented.

CHAPTER 2

Basic concepts

2.1 State Vectors

Statistical predictions of quantum mechanics are based on two main concepts, quantum statesand quantum observables. With every isolated physical system S, we associate a complex Hilbertspace HS of a suitable dimension, so that quantum states are represented by time-dependentunit vectors |ψ(t)〉 ∈ HS , and quantum observables by Hermitian operators acting in this space.Given a observable represented by the operator A, there is a set of vectors {|ψi〉} such that

A|ψi〉 = ai|ψi〉 , ai ∈ R. (2.1)

The vectors |ψi〉 are called the eigenvectors of A, with respective eigenvalues ai. The set ofvalues {ai} is called the spectrum of A. The time evolution of state vectors is unitary, i.e.

|ψ(t)〉 = U(t, t0) |ψ(t0)〉 , (2.2)

where U(t, t0) is a unitary operator, UU † = 11.Given a quantum system described by a state vector |ψ〉 and any observable A, represented

by a Hermitian operator, we can calculate all statistical properties of A from the relation

〈A〉 = 〈ψ |A |ψ〉 , (2.3)

where 〈A〉 stands for the average value of A. In particular, when A is a projection operator,projecting on a one dimensional subspace spanned by vector |ϕ〉, A = |ϕ〉〈ϕ|. In this case〈A〉 = |〈ψ|ϕ〉|2 represents the probability, for a system in state |ψ〉, to pass a test for being inthe state |ϕ〉.

Quantum states can be equally well represented by projectors on the state vectors. Namely,if instead of states |ψ〉 we consider the corresponding projectors |ψ〉〈ψ|, then the time evolutionof the state of the system will be given by

|ψ(t)〉〈ψ(t) | = U |ψ(t0)〉〈ψ(t0) |U †, (2.4)

and the average value of observable observable A will be written as

〈A〉 = Tr ρA, (2.5)

5

CHAPTER 2. BASIC CONCEPTS 6

where ρ = |ψ〉〈ψ| and the trace Tr ρA stands for the sum of the diagonal elements of ρA. Thetrace operation is linear, Tr (αA+βB) = αTrA+βTrB, and is basis-independent. The operatorρ is called density operator.

2.1.1 Subsystems

Consider a quantum system S composed of two subsystems A and B. The Hilbert space asso-ciated with system S is the tensor product of the Hilbert spaces of sub-system A and B

HS = HA ⊗HB. (2.6)

The dimension of HS is dimHS = dimHA · dimHB and any state |ψS〉 of the system S can beexpressed as a linear superposition of elements of the type |a〉⊗|b〉, where |a〉 ∈ HA and |b〉 ∈ HB.Whenever convenient, we’ll also write |a〉⊗ |b〉 as |a〉|b〉 or as |a, b〉. If we introduce orthonormalbases, i.e. maximal sets of vectors {|ak〉} in HA and {|bm〉} in HB, such that 〈ak| al〉 = δkl,〈bm | bn〉 = δmn, then any vector in HS can be written as,

|ψS〉 =∑

k,l

ckl|ak〉|bl〉 ,∑

kl

|ckl|2 = 1. (2.7)

A particular subset of the states in HS can be written as a tensor product of state vectorsof HA and HB,

|ψS〉 = |ψA〉 ⊗ |ψB〉 =

(∑

k

αk|ak〉)⊗

(∑

l

βl|bl〉)

(2.8)

=∑

kl

αlβl|ak〉|bl〉, (2.9)

where∑

k |αk|2 =∑

l |βl|2 = 1. This requires (comparing Eq.(2.8) and Eq.(2.7)) that

ckl = αkβl. (2.10)

The states for which this holds are called separable states. Note that this decomposition isbasis-independent. Thus, if |ψS〉 is separable, we can associate state |ψA〉 with the subsystem Aand state |ψB〉 with the subsystem B. Otherwise we need to resort to density operators in orderto represent quantum states in subsystems A and B.

2.2 Density Operators

Any linear operator S acting in HS can be written as a superposition of operators of the typeA⊗B, where A acts on HA and B acts on HB. We can choose operators bases, {Ak} acting onHA, {Bk} acting on HB, such that

S =∑

k,l

SklAk ⊗Bl. (2.11)

The most common operator bases are formed from operators of the type |ϕi〉〈ϕj |. In our casewe have | ak〉〈al |, for operators acting on HA, and | bm〉〈bn |, for operators acting on HB (recallthat |ai〉 and |bj〉 are, respectively, orthonormal bases in HA and HB). This means that S canbe expressed as


Tr B(ρAB) =∑

k,l,m,n

ckmc∗ln | ak〉〈al |Tr | bm〉〈bn | (2.20)

=∑

k,l,m,n

ckmc∗ln | ak〉〈al | δmn

=∑

k,l,m

ckmc∗lm | ak〉〈al |

= ρA.

2.2.1 Mathematical properties of density operators

Density operators provide a description of quantum states. They can be defined as such withoutany reference to state vectors. Let H be a finite-dimensional Hilbert space. A density operatorρ, on H, is a linear operator such that

• ρ is positive semi-definite, that is 〈φ|ρ|φ〉 ≥ 0, for any |φ〉 ∈ H.

• Trρ = 1.

Any linear positive semi-definite operator X on H is always Hermitian, with non-negativeeigenvalues, and can be written as X = Y †Y for some Y [18]. Many inequalities regarding pos-itive operators can be derived directly from 〈φ|X|φ〉 ≥ 0 by special choices of |φ〉. In particular,if |φ〉 has only two non-zero components, labelled by i and j, then the submatrix of X withthe elements labelled by the indices i and j is also positive semi-definite. More generally, anysubmatrix of a positive semi-definite matrix, obtained by keeping only the rows and columnslabelled by a subset of the original indices, is itself a positive semi-definite matrix and as suchmust have a nonnegative determinant (because all its eigenvalues are nonnegative).

To make a connection with the state vectors, let us consider a particular state (a pure state)which can be described by a state vector |Ψ〉 ∈ H. The density operator of any pure statecorresponds to a projection operator on that particular state, defined as

ρ = |Ψ〉〈Ψ|, (2.21)

which, like any projection operator, is idempotent:

ρ2 = ρ. (2.22)

For example, the state of a qubit α|0〉+ β|1〉 is described by the density operator

ρ = (α|0〉+ β|1〉) (〈1|β∗ + 〈0|α∗) = |α|2 |0〉〈0|+ αβ∗|0〉〈1|+ α∗β|1〉〈0|+ |β|2 |1〉〈1|, (2.23)

or, in the matrix form,

ρ =( |α|2 αβ∗

α∗β |β|2)

. (2.24)

The diagonal elements ρ00 = |α|2 and ρ11 = |β|2 correspond, respectively, to the expectationvalues 〈0|ρ|0〉 and 〈1|ρ|1〉, giving the probabilities of observing bit values 0 and 1 respectively.


2.2.2 Ensemble interpretation of density operators

Consider a quantum source which emits particles in states |Ψ1〉, |Ψ2〉... |Ψn〉 with a priori prob-abilities p1,p2...pn. We will write it as an ensemble {pi,|Ψi〉} . In this case

〈S〉 =n∑

i=1

pi〈Ψi|S|Ψi〉 =n∑

i=1

piTrS|Ψi〉〈Ψi| = TrS

(n∑

i=1

pi|Ψi〉〈Ψi|)

= TrSρ. (2.25)

The result depends on the observable S and on the quantum state, which appears in the expres-sion above only as the combination

ρ =n∑

i=1

pi|Ψi〉〈Ψi|. (2.26)

We call this operator the density operator that describes a mixture of pure states |Ψ1〉, |Ψ2〉...|Ψn〉 with weights p1,p2...pn. The operator ρ is not a projector any more, ρ2 6= ρ, but it hasall the properties we require for density operators (self-adjoint, semi-positive, unit-trace). If werefer to a single particle, we are uncertain as to which particular pure state |Ψi〉 it is preparedin. However, it makes perfect sense to say that the particle is in the state ρ. Please note thatmany different mixtures may lead to the same density operator:

ρ =n∑

i=1

pi|Ψi〉〈Ψi| =n∑

i=1

qi|Φi〉〈Φi|. (2.27)

Note the sets of pure states {|Ψi〉, |}, {|Φi〉, |} are not in general orthonormal. In fact, unlessthere is any degeneracy in the values pi, only one such set can be orthonormal.

Now take, for example, this particular density operator of a qubit:

ρ =(

34 00 1

4

). (2.28)

It can be viewed as the mixtures of |0〉 and |1〉 with the probabilities 34 and 1

4 , or as a mixtureof |Ψ1〉 =

√3

2 |0〉 + 12 |1〉 and |Ψ2〉 =

√3

2 |0〉 − 12 |1〉 with probabilities p1 = 1

2 and p2 = 12 . Even

though states |0〉 and |1〉 are clearly different from states |Ψ1〉 and |Ψ2〉, according to Eq.(2.25),these mixtures behave identically under any any physical investigation, i.e. we are not able todistinguish between different mixtures described by the same density operator.

2.3 Entanglement

We have previously introduced the concept of separable sates. However, there are states in HSwhich are not separable, i.e. they cannot be written as a simple tensor product of two states|ψA〉 and |ψB〉 (states for which ckl 6= αkβl). These states are referred to as entangled states.Entanglement is a set of quantum correlations arising from the interaction between two or morequantum systems that does not have a classical counterpart. An example of an entangled stateis the singlet state of two spin-half particles

|Ψ−〉 =1√2

(|↑〉|↓〉 − |↓〉|↑〉) , (2.29)

where |↑〉 and |↓〉 denote respectively spin up and spin down with respect to a chosen quantizationaxis.


As we mentioned in the previous chapter, entanglement is a very important physical resourcein quantum information science and both its mathematical characterization and experimentaldetection have been subjected to extensive research. Unfortunately, the only general mathe-matical definition of an entangled state is a negative one. A state is entangled if it cannot bewritten as a convex sum of product states [19]

ρ123...N =∑

`

C`ρ`1 ⊗ ρ`

2 ⊗ ρ`3 ⊗ . . .⊗ ρ`

N , (2.30)

where ρ`j is a state of subsystem j, and

∑` C` = 1. This fact means that in order to test

whether a given unknown state ρ is entangled, we have in principle to check whether the statecan be decomposed in any of all the possible convex sums of product states. We will discussin a later section the most important results concerning the characterization and detection ofentanglement. But first, we will introduce the concept of superoperators, since they have provedparticularly relevant in the construction of entanglement criteria.

2.4 Superoperators

As we pointed out before, the time evolution of a state ρ of system S is unitary and obeysEq.(2.4). Suppose now that S is composed of two sub-systems, A and B, and that we areinterested in the time evolution of sub-system A only. We can, without loss of generality, choosethe state of A to be ρA and the state of B to be the pure state |0〉. The time evolution of thestate of system S, ρA ⊗ |0〉〈0|, is given by

ρ′ = UρA ⊗ |0〉〈0|U †, (2.31)

which is still a density operator describing system S. The time evolution of the state of sub-system A is then obtained by performing the partial trace, on sub-system B, of the state ρ′ ofsystem S:

ρ′A = Tr B(UρA ⊗ |0〉〈0|U †). (2.32)

If we now consider an orthonormal basis |i〉, i = 0, 1, ..., for sub-system B, Eq.(2.32) becomes

ρ′A =∑

i

〈i|U |0〉ρA〈0|U †|i〉 ≡∑

i

EiρAE†i , (2.33)

where Ei = 〈i|U |0〉 are operators, acting on sub-system A, and are trace-preserving:

∑

i

E†i Ei =

∑

i

〈0|U †|i〉〈i|U |0〉 = 〈0|U †U |0〉 = 11. (2.34)

Eq.(2.33) defines a linear map L that takes linear operators ρA to linear operators ρ′A. Sucha map, if the property in Eq.(2.34) is satisfied, is called a superoperator. The representation ofthe superoperator given in Eq.(2.33) is called the operator-sum representation.

2.4.1 Mathematical properties of superoperators

A superoperator L : ρ → ρ′ that takes density operators to density operators has the followingproperties [18]:

• L is trace-preserving, that is Trρ′ = TrL(ρ) = 1.


• L is linear, that is L(α1ρ1 + α2ρ2) = α1L(ρ1) + α2L(ρ2), α1 + α2 = 1.

• L is a is completely positive map, that is, if ρ is positive, then ρ′ = L(ρ) is positive andthe extension of L to a larger sub-system (11⊗ L)ρ is also positive.

All the mathematical properties originate from physical requirements. The first and thirdproperties originate from the requirement that, assuming ρ to be a density operator, ρ′ will alsobe a density operator. The second property originates from our desire to reconcile the densityoperator time evolution and its ensemble interpretation.

The first property of superoperators is quite straightforward to accept, since any densityoperator has, by definition, trace equal to one. The third property is perhaps less obvious.Clearly, L must be a positive map to assure that ρ′ will be a positive operator (necessarycondition for ρ′ to be a density operator). But why must L be completely positive? The answeris: in order to assure that, if we decide to consider the action of the superoperator on an extendedsystem, ρext ⊗ ρ, the resulting operator ρ′′ = ρext ⊗ L(ρ) will still be a density operator.

2.4.2 Jamiolkowski isomorphism

The Jamiolkowski isomorphism [20] establishes an equivalence between quantum states andsuperoperators. Consider the action of a superoperator Λ on half of the maximally entangledstate |ψJ〉 = 1√

N

∑Ni |i〉|i〉:

1⊗ Λ|ψJ〉〈ψJ | →∑

ij

|i〉〈j|Λ (|i〉〈j|) = ρ′. (2.35)

The bipartite state ρ′ encodes all the properties of the superoperator Λ, as from it we can learnhow each density matrix element is transformed by Λ

|i〉〈j| → Λ (|i〉〈j|) . (2.36)

This establishes the equivalence between a completely positive map acting on density operatorspertaining to a Hilbert space H of dimension d2 − 1 and a density operator pertaining to aHilbert space H⊗H of dimension 4d2 − 1.

2.5 Mathematical characterization of bipartite entanglement

When studying the existence of entanglement in bipartite states, it is very useful to distinguishbetween pure states of the form Eq.(2.7) and mixed states. Pure bipartite states are entangled iffthe number number of terms of their Schmidt decomposition is greater than one. The Schmidtdecomposition of |ψS〉 is defined as:

|ψS〉 =∑

k,l

ckl|ak〉|bl〉 =∑

i

λi|a′i〉|b′i〉, (2.37)

where |a′i〉 and |b′i〉 are orthonormal bases for HA and HB, respectively, and λi are non-negativereal coefficients such that

∑i λ

2i = 1. Any state of the form Eq.(2.7) admits a Schmidt de-

composition [18]. Hence, given a pure bipartite state, the computation of the coefficients λi inthe Schmidt decomposition is sufficient for entanglement detection. However, there are not anyknown efficient methods to determine experimentally the Schmidt coefficients of an unknownstate Eq.(2.7). Therefore, other more accessible entanglement criteria were developed. An ex-ample is the entropic inequalities. Entropy measures uncertainty or our lack of information


about a particular physical property. Entropic inequalities, which quantify relations betweenthe information content of a composite quantum system and its parts, are of the form

S(%A) ≤ S(%AB) , S(%B) ≤ S(%AB), (2.38)

where %AB is a density operator of a composite quantum system and %A and %B are the reduceddensity operators pertaining to individual subsystems. They indicate that no matter whichphysical property is measured there is more uncertainty in the composite system than in anyof its parts. Here S stands for several different types of entropies, including the regular vonNeumann entropy S(%) = −Tr% log % and the Renyi entropy S(%) = − log Tr%2 [21]. Theseinequalities depend on the spectrum of both the state of the composite system and the statesof each individual subsystem, and provide necessary conditions for separability of bipartite purestates. We will introduce in a later chapter of the thesis an efficient method for the determinationof the spectrum of unknown density operators.

2.5.1 Mixed states

However, not all bipartite states are of the form Eq.(2.7). In fact, for more general bipartitestates such as Eq.(2.26), the Schmidt decomposition is no longer valid [18]. Therefore newmethods to identify entangled states were developed. These methods are based on the theoryof positive maps.

Positive, but not completely positive maps are the most powerful tool in the detection ofentanglement. These maps are not physical, that is, they cannot be directly implemented in thelaboratory, but they provide the best mathematical criteria for the existence of entanglementin a given state. In fact, they provide a necessary and sufficient condition for the existence ofentanglement [22]: a bipartite state ρAB ∈ HA ⊗ HB is entangled iff (11 ⊗ L)ρAB ≥ 0, for allL ∈ HB

2. Unfortunately, very little is known about the structure of positive maps, even forsmall dimensional spaces like C⊗3. It is therefore very difficult to extract practical entanglementcriteria from the above condition.

Still, Peres [13] and the Horodeckis [14] have shown that the positive partial-transpositionmap provides a necessary and sufficient condition for systems of two or three qubits. This mappreserves the eigenvalues of ρ, so it’s clearly positive and trace preserving. For example, letconsider a generic density operator of a qubit. This is a 2× 2 matrix of the form

(α γ∗

γ β

), (2.39)

where the coefficients α, β, γ are chosen such that Eq.(2.39) is a valid density operator. It issometimes convenient to represent the density operators of qubits as

ρ =1 +−→r · −→σ

2=

1 +∑

i=x,y,z riσi

2, (2.40)

where 1 is the identity operator, −→r is a three dimensional vector of length smaller or equal toone and

σx =(

0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

), (2.41)

are the Pauli operators. The action of the transposition map on the density operator of thequbit is

2Or conversely, (L⊗ 11)ρAB ≥ 0.


(α γ∗

γ β

)→T

(α γγ∗ β

). (2.42)

Suppose now that we consider a qubit, part of a larger system in the entangled state

|φ+〉 =1√2

(|0〉|0〉+ |1〉|1〉) . (2.43)

If we now apply the transposition map to the second qubit, which corresponds to a situation inwhich we consider the extension of transposition to a larger system (11⊗T ), the density operatorwill suffer a partial transpose of its matrix elements:

12

1 0 0 10 0 0 00 0 0 01 0 0 1

→T 1

2

1 0 0 00 0 1 00 1 0 00 0 0 1

. (2.44)

The resulting density matrix has eigenvalues 12 , 1

2 , 12 and −1

2 , so it’s not a valid density operator.The negativity under partial transposition is a signature of entanglement, even for more generalcases. It is in fact a sufficient condition for the existence of entanglement.

2.6 Experimental detection of entanglement

Entanglement tests based on positive maps are not physical, since positive maps cannot bedirectly implemented in the laboratory. While this problem can be circumvented, by mathemat-ically constructing completely positive maps out of the positive maps relevant for entanglementdetection [23], the actual implementation of these tests in the laboratory is yet to be achieved.Instead researchers have focussed on experimental tests that, albeit less powerful than positivemaps, are within reach of current technology.

2.6.1 Bell’s inequalities

Bell’s inequalities [3] were introduced as an attempt to encapsulate the non-locality of quantummechanics. While this is a completely different goal from the detection of entanglement, thefact that they were designed to capture the quantum essence of physical systems meant thatthey were also an entanglement test. In fact, they are the most widely used experimentalentanglement test. We will next briefly present the derivation of the Bell-CHSH inequality [24]and show that it is violated by the maximally entangled singlet state introduced in Eq.(2.29).

If we remember the thought experiment mentioned in the introduction, we have the followingscenario: two distant observers A and B share many identical pairs of particles; A and B canperform two different types of measurements on their respective particles, XA, YA and XB, YB,respectively; Each measurement is chosen randomly and has two possible outcomes: +1 and −1.Let us consider the quantity Q = XAXB + YAXB + YAYB −XAYB. Note that

XAXB + YAXB + YAYB −XAYB = (XA + YA)XB + (XA − YA)YB. (2.45)

Since XA, YA = ±1, it follows that either XA + YA = 0 or XA − YA = 0, which in turn meansXAXB + YAXB + YAYB −XAYB = ±2. Hence, the expectation value of Q is


E(Q) =∑

xAyAxByB

p(xA, yA, xB, yB)(xAxB + yAxB + yAyB − xAyB) (2.46)

≤∑

xAyAxByB

p(xA, yA, xB, yB)× 2 = 2, (2.47)

where p(xA, yA, xB, yB) is the probability that, before the measurements are performed, XA =xA, YA = yA, XB = xB, YB = yB. If we further notice that E(Q) = E(XAXB) + E(YAXB) +E(YAYB)−E(XAYB), we obtain the Bell inequality

E(XAXB) + E(YAXB) + E(YAYB)−E(XAYB) ≤ 2. (2.48)

However, if we now compute the expectation value of Q, with

XA = σAz , (2.49)

YA = σAx , (2.50)

XB = −σBz + σB

x√2

, (2.51)

YB =σB

z − σBx√

2, (2.52)

on the singlet state |Ψ−〉, we obtain that

〈XAXB〉|Ψ−〉 = 〈YAXB〉|Ψ−〉 = 〈YAYB〉|Ψ−〉 = −〈XAYB〉|Ψ−〉 =1√2. (2.53)

Thus, 〈Q〉|Ψ−〉 = 2√

2, which is in clear violation of Eq.(2.47) and implies that the state isentangled.

The violation of this and other Bell’s inequalities has been extensively observed experimen-tally [10, 11], mostly in systems of photons. While being a very convenient entanglement test,that requires only the computation of expectation values of linear operators on the state of thecomposite system, these inequalities fail to detect many entangled states currently produced inthe laboratory. Hence, researchers have actively looked for other types of experimental entan-glement tests.

2.6.2 Entanglement witnesses

Entanglement witnesses W were recently introduced as a tool for experimental entanglement de-tection [25, 26]. They are particularly well suited to the experimental detection of entanglement,where quite often the type of entangled state generated is known. They are linear operatorsacting on the composite Hilbert space HA ⊗HB that obey the following properties:

• W is Hermitian, that is W † = W .

• Tr(W |a, b〉〈a, b|) ≥ 0, for all states |a, b〉 in HA ⊗HB, that is, the expectation value of W

on any separable state is greater or equal to zero.

• W is not a positive operator, that is, it has at least one negative eigenvalue.

• Tr(W)=1.


Thus, if we have Tr(Wρ) < 0 for some ρ, then ρ is entangled. In that case we say that Wdetects ρ. Every entanglement witness detects something [26], since it detects in particular theprojector on the subspace corresponding to the negative eigenvalues of W. We will next give anexample of an entanglement witness that detects bipartite entangled states.

Consider an experimental setup that, due to the imperfections, produces the mixed ratherthan pure bipartite state of two qubits [27]

ρ = p|ψ〉〈ψ|+ (1− p)14, (2.54)

where |ψ〉〈ψ| is the pure state generated under ideal experimental circumstances, 0 ≤ p ≤ 1 and1/4 is the completely mixed state (white noise).

The witness is constructed by first computing the eigenvector corresponding to the negativeeigenvalue of the partially transposed density operator ρTB . The witness is given by the partiallytransposed projector onto this eigenvector. If the Schmidt decomposition of |ψ〉 is |ψ〉 = a|01〉+b|10〉, with a, b ≥ 0, the spectrum of ρTB is given by

{1− p

4+ pa2,

1− p

4+ pb2,

1− p

4+ pab,

1− p

4− pab}. (2.55)

Therefore ρ is entangled iff p > 1/(1 + 4ab). The eigenvector corresponding to the minimaleigenvalue λ− is given by

|φ−〉 =1√2(|00〉 − |11〉). (2.56)

Hence the witness W is given by

W = |φ−〉〈φ−|TB =12

1 0 0 00 0 −1 00 −1 0 00 0 0 1

. (2.57)

Note that this witness does neither depend on p, nor on the Schmidt coefficients a, b. It detectsρ iff it is entangled, since we have that

Tr(|φ−〉〈φ−|TBρ) = Tr(|φ−〉〈φ−|ρTB ) = λ−. (2.58)

Note also that in this particular case we just considered, if Tr(Wρ) ≥ 0, ρ is separable. This isnot a general property of witnesses, and indeed if the noise is not white this is not true anymore.

2.7 Multipartite entanglement

Multipartite entanglement, as a set of quantum correlations, is much more complex than bi-partite entanglement. Hence, we know considerably less about its mathematical structure andexperimental detection. Still, the general approach of the methods described in the previous sec-tion is equally suited to detect multipartite entanglement. In fact, Bell’s inequalities have beenderived for multipartite entangled states [28] and so have entanglement witnesses [29]. How-ever their experimental implementation has proved to be too challenging so far. The approachto multipartite entanglement detection is similar to the bipartite case. Therefore we will usethis section to try to capture the complexity of multipartite entanglement by presenting threeexamples of multipartite entangled states. These states were all introduced in the context ofquantum information and have proved to be useful resources for quantum information tasks.


The classification of multipartite entanglement differs from the bipartite case in that itis difficult to compare the different types of multipartite entanglement that are possible in agiven composite system. For example, multipartite states of N subsystems can be biseparable,i.e. admit the decomposition

ρ =∑

i

ciρiA ⊗ ρi

B, (2.59)

where A, B are two disjunct partitions of the composite system. How does one compare thistype of state with a state that is triseparable or non-separable with respect to any partition?This question is still open and considerable research is being currently devoted to it.

We will next present three classes of states that are representative of different features ofmultipartite entanglement. We will also briefly discuss their application to quantum information.

2.7.1 Maximally entangled state

Just as we introduced the concept of maximally entangled state for the case of two qubits, wewill equally define the maximally entangled state of N qubits:

|ψN 〉 =1√2

(|0000....0〉N + |1111....1〉N ) , (2.60)

where |iiii....i〉N = |i〉⊗N , i = 0, 1. In this case all the qubits are entangled with one another,but the state of any subset m of qubits is separable

ρm = TrN−m(|ψN 〉〈ψN |) =12

(|00...0〉m〈00...0|m + |11...1〉m〈11...1|m) . (2.61)

These states are particularly useful for multi-party quantum communication protocols, suchas multiparty quantum coin flipping [30].

2.7.2 W State

This class of symmetric states is, after the maximally entangled state, the most widely usedexample of multipartite entanglement. Unfortunately, a practical application in the context ofquantum information is yet to be found. The W state is defined as

|WN 〉 =1√N

(|1000....0〉N + |0100....0〉N + |0010....0〉N + ... + |0000....1〉N ) . (2.62)

In this case all the qubits are again entangled with one another, but interestingly enough thestate of any subset m of qubits is not separable. In fact, for the case of three qubits, the W stateretains maximally bipartite entanglement when any one of the three qubits is traced out [31].

2.7.3 Cluster state

The cluster state is perhaps the best example of the computational advantage of multipartiteover bipartite entanglement. This class of pure states is represented by a connected subset of asimple cubic lattice of qubits [32]. The cluster state is defined as the set of states |φ{k}〉C thatobey the set of eigenvalue equations

K(a)|φ{k}〉C = (−1)ka |φ{k}〉C , (2.63)

with the correlation operators


K(a) = σ(a)x

⊗

b∈nghb(a)

σ(b)z . (2.64)

Therein, {ka ∈ {0, 1}|a ∈ C} is a set of binary parameters which specify the cluster state andnghb(a) is the set of all neighboring lattice sites of a. This class of states in cubic lattices withtwo or more dimensions is, together with single qubit measurements, sufficient for universalquantum computation [32]. It is remarkable how a multipartite entangled state is alone thecomputational resource required for quantum computation.

2.8 Quantum networks

A quantum computation is nothing but changing the logical values of a set of qubits through aseries of operations, such that the final result has logical meaning. Similarly to classical com-putations, quantum computations are described through quantum circuits or networks. Thesenetworks are a sequence of quantum gates, unitary operations that change the logical values ofthe qubits, acting on one or more qubits at a time. They are a very useful paradigm to describethe dynamical evolution of systems of qubits, where the emphasis is on the state of the systemafter the implementation of the quantum gate, rather than on the actual physical interactionthat realizes the gate. Deutsch [33] showed the existence of a universal set of quantum gates,i.e. a set of gates that can approximate any unitary evolution of a set of qubits with arbitraryaccuracy. It was later shown that this set is finite [34].

2.8.1 Universal set of gates

The universal set of quantum gates is constituted by the set of all possible single qubit unitariesplus an entangling two-qubit gate [18]. Any single qubit unitary operator can be written in theform

U = exp(iα)Rn(θ) = exp(iα) exp(−iθn · −→σ ), (2.65)

where α, θ are real numbers and Rn(θ) denotes a rotation by θ about the n axis. However,the actual implementation of arbitrary rotations in a given physical qubit can be experimentallyvery challenging. Therefore, researchers have instead concentrated in finding a finite set of singlequbit gates that can approximate an any unitary operation U to arbitrary accuracy δ, i.e.

ε(U, V ) = max|ψ〉

||(U − V )|ψ〉|| ≤ δ, (2.66)

where V is the unitary implemented instead of U , ε(U, V ) is the unitary error and the maximumis taken over all normalized states |ψ〉. A possible such set of gates is constituted by theHadamard, π/4 and π/8 gates [18]:

H =1√2

(1 11 −1

), π/4 =

(1 00 i

), π/8 =

(1 00 eiπ/4

). (2.67)

As for the two-qubit gate, it is a controlled operation, i.e. it is a quantum gate where theinputs have different roles. One of the inputs is the control qubit while the other is the targetqubit. The gate acts on the target qubit iff the control qubit is in state |1〉. A generic controlled-U gate is depicted in Fig. 2.1. The prototypical example of the entangling two-qubit gate is thecontrolled-NOT gate. It has the following matrix representation in the |control, target〉 basis:


U

Truth tableC T0 0 0 00 1 0 11 0 1 U(0)1 1 1 U(1)

C TC

T

Figure 2.1: The controlled-U gate. The top line represents the control qubit and the bottomline represents the target qubit. U acts on the target qubit iff the control qubit is in the logicalstate |1〉.

C −NOT =

1 0 0 00 1 0 00 0 0 10 0 1 0

. (2.68)

The C −NOT gate flips the target qubit iff the control qubit is in state |1〉, otherwise it acts asthe identity gate. Let us now understand why is it that this gate is an entangling gate. Considerthe case where the control qubit is in state (|0〉 + |1〉)/√2 and the target qubit is in state |0〉.The composite state of the two qubits is clearly separable. After the C −NOT ,

1√2(|0〉+ |1〉)|0〉 =

1√2(|00〉+ |10〉) →CNOT

1√2(|00〉+ |11〉), (2.69)

which is no longer separable and is in fact the maximally entangled state |φ+〉 mentioned earlier.

2.8.2 Interferometry

As we mentioned earlier, the quantum network formalism provides us with a very useful set oftools to describe the dynamical evolution of physical systems of qubits. A particularly simpleand relevant example that of quantum interferometry. Consider a single particle going througha Mach-Zender interferometer (Fig. 2.2).

The incoming particle enters the interferometer from the lower left, in the path labelled|0〉. It encounters a 50:50 beam-splitter that deflects the particle into arm |1〉 with probabilityp = 0.5. If the particle goes through arm |0〉, it acquires a phase θ0, while if it goes througharm |1〉 it acquires the phase θ1. The two paths are then recombined in a second 50:50 beam-splitter. If we place particle detectors at each of the output ports of the second beam-splitter,and repeated this experiment many times, we would observe that the probability of the particlebeing detected in port |0〉 or |1〉 after the interferometer is given by


1q

0q

0

1

2 1 0

0cos

2P

q q-æ ö= ç ÷

è ø

2 1 0

1sin

2P

q q-æ ö= ç ÷

è ø

50/50 Beam Splitter

50/50 Beam Splitter

Figure 2.2: The Mach-Zender interferometer.

p0 = cos2(

θ1 − θ0

2

), (2.70)

p1 = sin2

(θ1 − θ0

2

). (2.71)

This result can be easily understood if we translate the Mach-Zender interferometer into thelanguage of quantum networks (Fig. 2.3). Let us encode our qubit in the two arms of theinterferometer. The qubit is initially is state |0〉. After the beam-splitter, which is nothing buta Hadamard gate, the state of the qubit becomes (|0〉 + |1〉)/√2. The qubit then acquires thephases θ0, θ1: (eiθ0 |0〉+eiθ1 |1〉)/√2, which is equivalent the the action of a phase gate 2(θ1−θ0).After the second beam-splitter the state of the qubit is

|0〉 →BS |0〉+ |1〉√2

→θ |0〉+ ei(θ1−θ0)|1〉√2

→BS (1 + ei(θ1−θ0))|0〉+ (1− ei(θ1−θ0))|1〉2

, (2.72)

hence the probability of finding the qubit in state |0〉 or |1〉 after the interferometer is simplygiven by Eq.(2.70) and Eq.(2.71), respectively.

2.9 Summary

We have now presented the basic concepts underlying this thesis: mixed states, superoperators,entanglement and quantum circuits. We have discussed the ambiguity in the definition of anymixed quantum state, which is due to the indistinguishability of state preparations. We haveintroduced superoperators, completely positive maps acting on quantum states, as the mostgeneral evolution of quantum systems. We have mentioned the Jamiolkowski isomorphism be-tween superoperators and quantum states. Entanglement, and in particular its detection, is themain object of research of this thesis. We gave an overview of the main results concerning the


VIS

IBIL

ITY

Figure 2.3: The quantum network corresponding to the Mach-Zender interferometer (ϕ = θ1 −θ0). The visibility of the interference pattern associated with p0 varies as a function of ϕaccording to Eq.(2.70).

mathematical characterization and experimental detection of entanglement. Finally, we intro-duced the circuit model of quantum computation and we shown its suitability to describe thedynamical evolution of quantum systems, and in particular interferometric effects.

CHAPTER 3

Direct estimation of density operators

This chapter presents the results that were published in an article written in collaboration withA. K. Ekert, D. K. L. Oi, M. Horodecki, P. Horodecki, L. C. Kwek: A. K. Ekert, C. MouraAlves, D. K. L. Oi, M. Horodecki, P. Horodecki, L. C. Kwek, Phys. Rev. Lett. 88, 217901(2002).

Certain properties of a quantum state %, such as its purity, degree of entanglement, or itsspectrum, are of significant importance in quantum information science. They can be quantifiedin terms of linear or non-linear functionals of %. Linear functionals, such as average values ofobservables {A}, given by TrA%, are quite common as they correspond to directly measurablequantities. Non-linear functionals of state, such as the von Neumann entropy −Tr% ln %, eigen-values, or a measure of purity Tr%2, are usually extracted from % by classical means i.e. % is firstestimated and once a sufficiently precise classical description of % is available, classical evalu-ations of the required functionals can be made. However, if only a limited supply of physicalobjects in state % is available, then a direct estimation of a specific quantity may be both moreefficient and more desirable [35]. For example, the estimation of purity of % does not requireknowledge of all matrix elements of %, thus any prior state estimation procedure followed byclassical calculations is, in this case, inefficient. However, in order to bypass tomography and toestimate non-linear functionals of % more directly, we need quantum networks [33, 36] performingquantum computations that supersede classical evaluations.

In this chapter, we shall present and examine a simple quantum network that can be used asa basic building block for direct quantum estimations of both linear and non-linear functionalsof any %. The network can be realized as multiparticle interferometry. While conventionalquantum measurements, represented as quantum networks or otherwise, allow the estimation ofTrA% for some Hermitian operator A, our network can also provide a direct estimation of theoverlap of any two unknown quantum states %a and %b, i.e. Tr%a%b.

3.1 Modified interferometry

In order to explain how the network works, let us start with a general observation related tomodifications of visibility in interferometry. Consider a typical interferometric set-up for a singlequbit: Hadamard gate, phase shift ϕ

21

CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 22

Figure 3.1: A modified Mach-Zender interferometer with coupling to an ancilla by a controlled-Ugate. The interference pattern is modified by the factor veiα = Tr [Uρ].

ϕ =(

1 00 eiϕ

), (3.1)

Hadamard gate, followed by a measurement in the computational basis. We modify the interfer-ometer by inserting a controlled-U operation between the Hadamard gates, with its control onthe qubit and with U acting on a quantum system described by some unknown density operatorρ, as shown in Fig. 3.1. The action of the quantum network is given by

|0〉|ψ〉 H−→ 1√2

(|0〉+ |1〉) |ψ〉c−U−→ 1√

2(|0〉|ψ〉+ |1〉U(|ψ〉))

φ−→ 1√2(|0〉|ψ〉+ eiφ|1〉U(|ψ〉))

H−→ 12

[|0〉

(|ψ〉+ eiφU(|ψ〉)

)+ |1〉

(|ψ〉 − eiφU(|ψ〉)

)]. (3.2)

The action of the controlled-U on ρ modifies the interference pattern:

P0(φ) =14(1 + veiαeiφ + ve−iαe−iφ + 1) =

12

(1 + v cos (φ + α)) , (3.3)

by the factor Tr(|ψ〉〈ψ|U) = TrρU = veiα [37], where v is the new visibility and α is the shiftof the interference fringes, also known as the Pancharatnam phase [38]. Thus, the observedmodification of the visibility gives an estimate of TrUρ, i.e. the average value of the unitaryoperator U in state ρ. Let us mention in passing that this property, among other things, allowsthe estimation of an unknown quantum state ρ as long as we can estimate TrUkρ for a set ofunitary operators Uk which form a basis in the vector space of density operators.

Let us now consider a quantum state ρ of two separable subsystems, such that ρ = %a ⊗ %b.We choose our controlled-U to be the controlled-V , where V is the swap operator, defined as,V |α〉A|β〉B = |β〉A|α〉B, for any pure states |α〉A and |β〉B. In this case, the modification of theinterference pattern given by Eq. (3.3) can be written as,


· ·Va

b

··

a b

Figure 3.2: Quantum network for direct estimations of both linear and non-linear functions ofa quantum state.

v = TrV (%a ⊗ %b)

=∑

ij

∑rs

λrλs〈fj |〈ei| (|fs〉|er〉〈fs|〈er|) |ei〉|fj〉

=∑

ij

∑rs

λrλsδjsδir〈 ei | fs 〉〈 fj | er 〉

=∑

ij

λiλj |〈 ei | fj 〉|2

= Tr%a%b. (3.4)

Since Tr%a%b is real, we can fix ϕ = 0 and the probability of finding the qubit in state |0〉 at theoutput, P0, is related to the visibility by v = 2P0 − 1. This construction, shown in Fig. 3.2,provides a direct way to measure Tr%a%b (c.f. [39] for a related idea).

3.2 Multiple target states

There are many possible ways of utilizing this result. For pure states %a = |α〉〈α| and %b = |β〉〈β|the formula above gives Tr%a%b = |〈α |β 〉|2 i.e. a direct measure of orthogonality of |α〉 and|β〉. If we put %a = %b = % then we obtain an estimation of Tr%2. In the single qubit case,this measurement allows us to estimate the length of the Bloch vector, leaving its directioncompletely undetermined. For qubits Tr%2 gives the sum of squares of the two eigenvalues whichallows to estimate the spectrum of %.

3.2.1 Spectrum estimation

In general, the evaluation of the spectrum of any d× d density matrix % requires the estimationof d−1 parameters Tr%2, Tr%3,... Tr%d. We can do so directly via the controlled-shift operation,which is a generalization of the controlled-swap gate. Given k systems of dimension d we definethe shift V (k) as

V (k)|φ1〉|φ2〉...|φk〉 = |φk〉|φ1〉...|φk−1〉, (3.5)


for any pure states |φ〉. Such an operation can be easily constructed by cascading k − 1 swapsV . If we extend the network and prepare ρ = %⊗k at the input then the interference will bemodified by the visibility factor,

v = TrV (k)%⊗k = Tr %k =k∑

i=1

λik. (3.6)

Thus measuring the average values of V (k) for k = 2, 3...d allows us to evaluate the spectrumof % [35]. Although we have not eliminated classical evaluations, we have reduced them by asignificant amount. The average values of V (k) for k = 2, 3...d provide enough information toevaluate the spectrum of %, but certainly not enough to estimate the whole density matrix.

It should be mentioned that other spectrum estimation methods, relying on single collectivemeasurements of several copies of %, have been proposed [40]. These methods essentially projectthe initial state ρ = %⊗n, which forms an operator on the n-fold tensor product space, ontoorthogonal subspaces corresponding to irreducible representations of the permutation group ofn points. This decomposition is labelled by Young frames, the arrangement of n boxes into d

rows of decreasing length. The normalized row lengths of each tableau are taken as estimatesof the ordered sequence of eigenvalues of %. The probability that the error in the spectrumestimation is greater than some fixed ε decreases exponentially with n [40].

3.2.2 Quantum communication

So far we have treated the two inputs %a and %b in a symmetric way. However, there are severalinteresting applications in which one of the inputs, say %a, is predetermined and the other isunknown. For example, projections on a prescribed vector |ψ〉, or on the subspace perpendicularto it, can be implemented by choosing %a = |ψ〉〈ψ|. By changing the input state |ψ〉 we effectively“reprogram” the action of the network which then performs different projections. This propertycan be used for quantum communication, in a scenario where one carrier of information, in state|ψ〉, determines the type of detection measurement performed on the second carrier. Note thatas the state |ψ〉 of a single carrier cannot be determined, the information about the type of themeasurement to be performed by the detector remains secret until the moment of detection.

3.2.3 Extremal eigenvalues

Another interesting application is the estimation of the extremal eigenvalues and eigenvectorsof %b without reconstructing the entire spectrum. In this case, the input states are of the form|ψ〉〈ψ|⊗%b and we vary |ψ〉 searching for the minimum and the maximum of v = 〈ψ|%b|ψ〉. This,at first sight, seems to be a complicated task as it involves scanning 2(d − 1) parameters of ψ.The visibility is related to the overlap of the reference state |ψ〉 and %b by,

vψ = Tr

(|ψ〉〈ψ|

∑

i

λi|ηi〉〈ηi|)

=∑

i

λi |〈ψ | ηi 〉|2 =∑

i

λipi, (3.7)

where∑

i pi = 1. This is a convex sum of the eigenvalues of %b and is minimized (maximized)when |ψ〉 = |ηmin〉 (|ηmax〉). For any |ψ〉 6= |ηmin〉 (|ηmax〉), there exists a state, |ψ′〉, in theneighbourhood of |ψ〉 such that vψ′ < vψ (vψ′ > vψ). Thus this global optimization problemcan be solved using standard iterative methods, of which steepest decent [41] is an example.


Estimation of extremal eigenvalues plays a significant role in the direct detection [35] anddistillation [22] of quantum entanglement. As an example, consider two qubits described by thedensity operator %b, such that the reduced density operator of one of the qubits is maximallymixed. We can test for the separability of %b by checking whether the maximal eigenvalue of %b

does not exceed 12 [42].

3.2.4 State estimation

Finally, we may want to estimate an unknown state, say a d× d density operator, %b. Such anoperator is determined by d2−1 real parameters. In order to estimate matrix elements 〈ψ|%b|ψ〉,we run the network as many times as possible (limited by the number of copies of %b at ourdisposal) on the input |ψ〉〈ψ| ⊗ %b, where |ψ〉 is a pure state of our choice. For a fixed |ψ〉, afterseveral runs, we obtain an estimation of,

v = 〈ψ|%b|ψ〉. (3.8)

In some chosen basis {|n〉} the diagonal elements 〈n|%b|n〉 can be determined using the inputstates |n〉〈n|⊗%b. The real part of the off-diagonal element 〈n|%b|k〉 can be estimated by choosing|ψ〉 = (|n〉+ |k〉)/√2, and the imaginary part by choosing |ψ〉 = (|n〉+ i|k〉)/√2. In particular,if we want to estimate the density operator of a qubit, we can choose the pure states, |0〉 (spin+z), (|0〉+ |1〉) /

√2 (spin +x) and (|0〉+ i|1〉) /

√2 (spin +y), i.e. the three components of the

Bloch vector.Quantum tomography can also be performed in many other ways, the practicalities of which

depend on technologies involved. However, it is worth stressing that the strength of our scheme isthe use of a fixed architecture network, controlled only by input data, to perform the estimationof properties of ρ.

3.2.5 Arbitrary observables

We can extend the procedure above to cover estimations of expectation values of arbitraryobservables A. This can be done with the network shown in Fig. 3.2, since estimations of meanvalues of any observable can always be reduced to estimations of a binary two-output POVMs.We shall apply the technique developed in Refs. [23, 35]. As A′ = γ1+A is positive if −γ is theminimum negative eigenvalue of A, we can construct the state %a = %A′ = A′

Tr(A′) and apply ourinterference scheme to the pair %A′ ⊗ %b. The visibility gives us the mean value of V,

v = 〈V 〉%A′⊗%b= Tr

(A′

Tr(A′)%b

), (3.9)

which leads us to the desired value,

〈A〉%b≡ Tr(%bA) = vTrA + γ(vd− 1), (3.10)

where Tr1 = d.

3.3 Quantum channel estimation

Any technique that allows direct estimations of properties of quantum states can be also usedto estimate certain properties of quantum channels. Recall that, from a mathematical pointof view, a quantum channel is a superoperator, % → Λ(%), which maps density operators intodensity operators, and whose trivial extensions, 1k⊗Λ do the same, i.e. Λ is a completely positive


Figure 3.3: A quantum channel Λ acting on one of the subsystems of a bipartite maximallyentangled state of the form |ψ+〉 =

∑k |k〉|k〉/

√d. The output state %Λ = 1

d

∑kl |k〉〈l|⊗Λ (|k〉〈l|),

contains a complete information about the channel.

map. In a chosen basis the action of the channel on a density operator % =∑

kl %kl|k〉〈l| can bewritten as

Λ(%) = Λ

(∑

kl

%kl|k〉〈l|)

=∑

kl

%klΛ (|k〉〈l|) . (3.11)

Thus the channel is completely characterized by operators Λ (|k〉〈l|). In fact, with every channelΛ we can associate a quantum state %Λ that provides a complete characterization of the channel.If we prepare a maximally entangled states of two particles, described by the density operatorP+ = 1

d

∑kl |k〉〈l| ⊗ |k〉〈l|, and we send only one particle through the channel, as shown in

Fig. 3.3, we obtainP+ → [1⊗ Λ]P+ = %Λ, (3.12)

where%Λ =

1d

∑

kl

|k〉〈l| ⊗ Λ (|k〉〈l|) . (3.13)

We may interpret this as mapping the |k〉〈l|th-element of an input density matrix to theoutput matrix, Λ (|k〉〈l|). Thus, knowledge of %Λ allows us to determine the action of Λ onan arbitrary state, % → Λ(%). If we perform a state tomography on %Λ we effectively performa quantum channel tomography. If we choose to estimate directly some functions of %Λ thenwe gain some knowledge about specific properties of the channel without performing the fulltomography of the channel.

For example, consider a single qubit channel. Suppose we are interested in the maximalrate of a reliable transmission of qubits per use of the channel, which can be quantified as thechannel capacity. Unlike in the classical case, quantum channels admit several capacities [43, 44],because users of quantum channels can also exchange classical information. We have then thecapacities QC where C = ø,←,→,↔, stands for zero way, one way and two way classicalcommunication. In general, it is very difficult to calculate the capacity of a given channel.However, our extremal eigenvalue estimation scheme provides a simple necessary and sufficientcondition for a one qubit channel to have non-zero two-way capacity. Namely, Q↔ > 0 iff %Λ


has maximal eigenvalue greater than 12 . Note that this condition is also necessary for the other

three capacities to be non-zero.This result becomes apparent by noticing that if we trace %Λ over the qubit that went through

the channel Λ (particle 2 in Fig. 3.3), we obtain the maximally mixed state. Furthermore, thetwo qubit state, %Λ, is two-way distillable iff the operator 1

2 ⊗ 1− %Λ has a negative eigenvalue(see [42] for details), or equivalently when %Λ has the maximal eigenvalue greater than 1

2 . Thisimplies Q↔(Λ) > 0 because two-way distillable entanglement, which is non-zero iff given stateis two way distillable, is the lower bound for Q↔(Λ) [44].

3.4 Summary

In summary, we have described a simple quantum network which can be used as a basic buildingblock for direct quantum estimations of both linear and non-linear functionals of any densityoperator %. It provides a direct estimation of the overlap of any two unknown quantum states%a and %b, i.e. Tr%a%b. Its straightforward extension can be employed to estimate functionalsof any powers of density operators. The network has many potential applications ranging frompurity tests and eigenvalue estimations to direct characterization of some properties of quantumchannels.

Finally let us also mention that the controlled-SWAP operation is a direct generalizationof a Fredkin gate [45] and can be constructed out of simple quantum logic gates [36]. Thismeans that experimental realizations of the proposed network are within the reach of quantumtechnology that is currently being developed (for an overview see, for example, [46]).

CHAPTER 4

Direct estimation of density operators using LOCC

This chapter presents the results that were published in an article written in collaboration withD. K. L Oi, P. Horodecki, A. K. Ekert, L. C. Kwek: C. Moura Alves, D. K. L Oi, P. Horodecki,A. K. Ekert, L. C. Kwek, Phys. Rev. A 68, 32306 (2003).

In the previous chapter we presented a family of quantum networks that directly estimatemulti-copy observables, Tr[%k], of an unknown state % [47, 35, 23]. As mentioned before, thesenonlinear functionals quantify important properties of %, such as the degree of entanglement orthe spectrum. Therefore it would be very useful to be able to estimate them even when % is abipartite state %AB shared by two distant parties, Alice and Bob, who can perform only localoperations and communicate classically (LOCC). In this chapter we show that the estimationof non-linear functionals of quantum states admit LOCC implementation. We also show thatStructural Physical Approximations [35, 23], an important tool for entanglement detection, canbe constructed locally. This opens the possibility of the direct estimation of entanglement andsome channel capacities using only LOCC.

As a general remark, let us recall that a quantum operation Λ can be implemented usingLOCC if it can be written as a convex sum

Λ =∑

k

pk Ak ⊗Bk, (4.1)

where Ak acts on the subsystem at Alice’s location and Bk on the subsystem at Bob’s location,and pk represent the respective probabilities.

4.1 LOCC estimation of nonlinear functionals

The direct estimation method is extended to the LOCC scenario by constructing two localnetworks, one for Alice and one for Bob, in such a way that the global network is similar tothe network with the controlled-shift. Unfortunately, the global shift operation V (k) cannotbe implemented directly using only LOCC, since it does not admit local decomposition (4.1).Hence, we will implement it indirectly, using the global network shown in Fig. 4.1. Alice andBob share a number of copies of the state %AB ∈ Hd. They group them respectively into setsof k elements, and run the local interferometric network on their respective halves of the stateρAB = %⊗k

AB. For each run of the experiment, they record and communicate their result.

28

CHAPTER 4. DIRECT ESTIMATION OF DENSITY OPERATORS USING LOCC 29

Figure 4.1: Network for remote estimation of non-linear functionals of bipartite density opera-tors. Since Tr[V (k)%⊗k] is real, Alice and Bob can omit their respective phase shifters.


The individual interference patterns Alice and Bob record will depend only on their respectivereduced density operators. Alice will observe the visibility vA = Tr[%k

A] and Bob will observethe visibility vB = Tr[%k

B]. However, if they compare their individual observations, they will beable to extract information about the global density operator %AB, e.g. about

Tr[%kAB] = Tr

[%⊗k

AB

(V

(k)A ⊗ V

(k)B

)]. (4.2)

This is because Alice and Bob can estimate the probabilities Pij that in the measurement Alice’sinterfering qubit is found in state |i〉A and Bob’s in state |j〉A for i, j = 0, 1. These probabilitiescan be conveniently expressed as

Pij =14Tr

[%⊗k

AB

(1 + (−1)iV

(k)A

)⊗ (1 + (−1)jV

(k)B

)], (4.3)

hence the formula for the basic non-linear functional of %AB reads

Tr[%kAB] = P00 − P01 − P10 + P11. (4.4)

In fact, the expression above is the expectation value 〈σz ⊗ σz〉, measured on Alice’s and Bob’squbits (the two qubits that undergo interference). Given that we are able to directly estimateTr[%k

AB] for any integer value of k, we can estimate the spectrum of %AB without resorting to afull state tomography.

4.2 Structural Physical Approximations

We next show how to implement Structural Physical Approximations within the LOCC scenario.Structural Physical Approximations (SPAs) were introduced recently as tools for determiningrelevant parameters of density operators (see [23, 35] for more details). Basically the SPA of amathematical operation Λ, denoted as Λ, is a physical operation, a process that can be carriedout in a laboratory, that emulates the character of Λ. More precisely, suppose Λ : Hd 7→ Hd is atrace preserving map which does not represent any physical process, for example, an anti-unitaryoperation such as transposition. Then a convex sum

Λ = αD + (1− α)Λ, (4.5)

where D is the depolarizing map which sends any density operator into the maximally mixedstate, represents a physical process, i.e. a completely positive map, as long as α is sufficientlylarge. On top of this D, with its trivial structure, does not mask the structure of Λ. TheStructural Physical Approximation to Λ is obtained by selecting, in the expression above, thethreshold value α = (d2λ)/(d2λ + 1), where −λ is the lowest eigenvalue of (1⊗Λ)P (d)

+ and P(d)+

is a maximally entangled state of a d×d system 1. Note that we impose the positivity conditionon the map 1⊗ Λ to ensure that Λ is a completely positive map.

Please note that the physical implementation of SPAs is not a trivial problem as the for-mula (4.5), which explicitly contains the physically impossible map Λ, is of little guidance here.Let us also mention in passing that if Λ is not trace preserving then Λ may be implementablebut only in a probabilistic sense e.g. using experimental post-selection.

There are many examples of mathematical operations which, though important in the quan-tum information contexts, do not represent a physical process. For example, mathematical cri-teria for entanglement involve positive but not completely positive maps [21] and as such theyare not directly implementable in a laboratory — they tacitly assume that a precise descriptionof a quantum state of a physical system is given and that such operations are mathematicaltransformations on the matrix describing the quantum state.

1The threshold value for α is obtained from the requirement of complete positivity of Λ, which in this case canbe reduced to ΛP

(d)+ ≥ 0


4.2.1 SPA using only LOCC

If Λ does not represent any physical process then its trivial extension to a bipartite case, 1⊗Λ,does not represent a physical process either. Still, its SPA, 1⊗ Λ, does describe a physicaloperation. But can it be implemented with LOCC?

The positive answer is obtained by putting 1⊗ Λ into the tensor product form (4.1). Let usstart by writing it as

1⊗ Λ = αD ⊗D + (1− α)1⊗ Λ

= (1− α + β)1⊗(

1− α

1− α + βΛ +

β

1− α + βD

)

+ (α− β)(

α

α− βD +

−β

α− β1)⊗D

= (1− α + β)1⊗ Λ + (α− β)Θ⊗D, (4.6)

where

Λ =1− α

1− α + βΛ +

β

1− α + βD, (4.7)

Θ =α

α− βD +

β

α− β(−1). (4.8)

Equation (4.6) does not represent a convex sum of physically implementable maps for any valuesof α and β but if we choose

β ≥ (1− α)λd2 (4.9)

α ≥ βd2, (4.10)

where −λ is the minimum eigenvalue of 1 ⊗ Λ(P d+), then indeed 1⊗ Λ is completely positive

map in the LOCC form. Note, however, that the map Θ is not trace preserving and as such itcan be implemented only with a certain probability of success. The minimal parameters α andβ that satisfy inequalities Eqs. (4.9) and(4.10) are

α =λd4

λd4 + 1, (4.11)

β =λd2

λd4 + 1. (4.12)

Hence, the SPA 1⊗ Λ can be implemented, by Alice and Bob, using only only LOCC.

4.3 Entanglement detection

One of the applications of the methods presented above is LOCC entanglement detection. Forexample, in Bell diagonal states (i.e. two-qubit states with maximally entangled eigenvectors)the entanglement of formation (or negativity, see below) can be inferred from its spectrum [44].Hence, our method allows Alice and Bob to determine this entanglement property using onlyLOCC. An important subclass of Bell diagonal states are the maximally correlated states, whichrank two states equivalent (up to UA ⊗ UB transformations) to mixtures of two pure states,|ψ+〉 = 1√

2(|0〉|0〉+ |1〉|1〉) and |ψ−〉 = 1√

2(|0〉|0〉 − |1〉|1〉). The one-way distillable entanglement

can be calculated for such states as D→ = log 2−S(%), which is a function solely of the spectrum.Thus, instead of estimating the seven parameters required to describe maximally correlatedstates, we need to estimate only three parameters.


SPAs have also been employed to test for quantum entanglement [35]. Recall that a necessaryand sufficient condition for a bi-partite state %AB to be separable is 1 ⊗ Λ(%AB) ≥ 0, for allpositive maps Λ [21]. This condition, when considering the SPA 1⊗ Λ on %AB, is equivalent to

[1⊗ Λ

]%AB ≥ d2λ

d4λ + 1, (4.13)

where −λ is the minimal eigenvalue of the state [(1⊗ 1)⊗ (1⊗ Λ)] (P d2

+ ) [35]. Thus, by esti-

mating the spectrum (or the lowest eigenvalue) of the state[1⊗ Λ

]%AB, we can directly detect

quantum entanglement.In particular, if we choose Λ = T , where T is the partial transposition map, we are also able

to estimate the measure of entanglement [48], N (%AB) ≡ log ||%TBAB|| = log(

∑i |λi|). Note that

the computation of entanglement measures (see [49] for review) is known only for very particularcases. The measure introduced in [48] is valid for any shared bipartite state with a maximallymixed reduced density operator of at least one sub-system, and it is a function of the spectrum{λi} of the partially transposed matrix %TB

AB ≡ 1⊗ T (%AB). It is worth mentioning that for theparticular case of partial-transposition, a method that bypasses the implementation of a SPAwas developed by Carteret [50]. This method simulates the action of the partial-transpositionmap on nonlinear functionals of ρ, allowing the direct estimation of Tr[(ρTA)k], k = 2, 3, ....

4.4 Channel capacities

Another potential application of the methods presented above is the LOCC estimation of channelcapacities. Let a completely positive map Λ : Hd 7→ Hd represent a quantum channel shared byAlice and Bob. Estimating the channel capacity can involve either channel tomography or directestimation. In the case of tomography Alice prepares a maximally entangled pair of particles instate P d

+ and sends one half of the pair to Bob. They now share the state

%Λ = [1⊗ Λ]P d+. (4.14)

From the JamioÃlkowski isomorphism [20], this bi-partite state encodes all properties of thechannel Λ, so state tomography on %Λ is effectively channel tomography on Λ. However, givena bi-partite state %Λ, Alice and Bob can also use the LOCC techniques to directly estimate itsdesired properties. For example, we have previously shown that a single qubit channel Λ hasnon-zero channel capacity if and only if the maximal eigenvalue of %Λ is strictly greater than 1

2

(see [47] for details). This can be estimated directly via spectrum estimation, which in the caseof two qubits requires three measurements of the type σz ⊗ σz as opposed to the 15 parametersrequired for the state estimation.

4.5 Summary

In summary, we have demonstrated that both direct spectrum estimations and the structuralphysical approximations can be implemented in the case of bi-partite states using only localoperations and classical communication. This leads to more direct, LOCC type, detectionsand estimations of quantum entanglement and of some properties of quantum channels. Directestimations of specific properties have the natural advantage over the state tomography becausethey avoid estimating superfluous parameters.

CHAPTER 5

Entanglement Detection in Bosons

This chapter presents the results that were published in an article written in collaboration withD. Jaksch: C. Moura Alves and D. Jaksch, Phys. Rev. Lett. 93, 110501 (2004).

The implementation of almost any quantum information tasks requires precise knowledge onthe entangled states being used. Hence, the development of “measurement tools” for the char-acterization and detection of entanglement in physical systems is of great practical importance.The usual experimental methods to detect entanglement are based on the violation of Bell typeinequalities [3], which are known to be quite inefficient, in the sense that they leave many entan-gled states undetected [19]. Alternatively, one can perform a complete state tomography of thesystem [51], but this method requires the preparation of an exponentially large number of copiesof the state and it is redundant, since not all parameters of the density operator are relevant forthe entanglement detection.

In this chapter we present a simple quantum network to detect multipartite entangled statesthrough an entanglement test more powerful than the Bell-CHSH inequalities for all possiblesettings [52], albeit less powerful than full state tomography. The network is realized by couplingtwo identically prepared 1D rows of N previously entangled qubits via pairwise beam splitters(BS), as shown in Fig. 5.1. We also show how to implement this network in an optical latticeor array of magnetic microtraps loaded with atoms in a Mott insulating state with filling factorone [53, 54]. Each of the atoms has two long lived internal states a and b which represent thequbit. The pairwise BS can be implemented by decreasing the horizontal barrier between thetwo rows of atoms.

5.1 Nonlinear entanglement inequalities

We start by introducing the set of inequalities used by our network for the detection of multi-partite entanglement. The information-theoretic approach to separability of bipartite quantumsystems leads to a set of entropic inequalities satisfied by all separable bipartite states [52]. Weextend these inequalities to separable multipartite states by considering a state ρ123...N of N

subsystems. If ρ123...N is separable then we can write it as

ρ123...N =∑

`

C`ρ`1 ⊗ ρ`

2 ⊗ ρ`3 ⊗ . . .⊗ ρ`

N , (5.1)

33

CHAPTER 5. ENTANGLEMENT DETECTION IN BOSONS 34

... ...

... ...

r

r

j

j

BS BS BS BS BS BS BS

1 2 3 4 j N-1 N

x

y

I

II

z

1 2 3 4 j N-1 N

Figure 5.1: Network of BS acting on pairs of identical bosons. The two rows of N atoms, labelledI and II respectively, are identical, and the state of each of the rows is ρ123...N . The total stateof the system is ρ123...N ⊗ ρ123...N .

where ρ`j is a state of subsystem j, and

∑` C` = 1. The purity Tr(ρ2

123...n) of ρ123...n, wheren ∈ 1, 2, ..., N , is smaller or equal than the purity of any of its reduced density operators. Forexample,

Tr(ρ2123...n) ≤ Tr(ρ2

123...n−1) ≤ Tr(ρ2123...n−2) . . . ≤ Tr(ρ2

12) ≤ Tr(ρ21) ≤ 1. (5.2)

This set of nonlinear inequalities provides a set of necessary conditions for separability, i.e. if forany state % any of these inequalities is violated then % is entangled. For the case where ρ123...n

is separable and pure we have that Tr(ρ2123...n) = 1 and the inequalities become equalities.

In order to test Eq. (5.2) we need to be able to determine the non-linear functional Tr(%2),where % is any of the different reduced density operators of ρ123...n. The direct estimation of thisfunctional has already been addressed both in chapter 2 and in [47].

5.2 Estimation of the purities

Let us consider again the network depicted in Fig. 3.2, with input target state % ⊗ %. Afterimplementing the network, we measuring the state of the control qubit. This measurementprojects the state of the target qubits onto its symmetric subspace S%⊗%S†, if the control qubitis found in state |0〉, or onto its antisymmetric subspace A%⊗ %A†, if the control qubit is foundin state |1〉. Here, S = (1 + V )/2 and A = (1 − V )/2 are the symmetric and antisymmetricprojectors, respectively, with V the swap operator previously introduced and 1 the identityoperator. Hence the value of Tr(%2) is determined from the measurement of the expectationvalue, on state % ⊗ %, of the symmetric and antisymmetric projectors. We will next show howa BS transformation effectively projects a pairs of bosons on its symmetric and antisymmetricsubspaces.

5.2.1 Bipartite case

Let us first consider the simple scenario of one pair of identical bosons, in state ρj⊗ρj , impingingon the BS (as depicted in Fig. 5.1). In order to better grasp the action of the BS, it is convenientto consider the purification of ρj ⊗ ρj . Writing ρj in its spectral decomposition


ρj = λ0|ψ0〉〈ψ0|+ λ1|ψ1〉〈ψ1|, (5.3)

with λ0 +λ1 = 1, |ψi〉 = αia(j)†k |vac〉+βib

(j)†k |vac〉, where a

(j)k and b

(j)k , k = I, II, are the bosonic

destruction operators for particles in site number j = 1 · · ·N , internal state a and b, and rowI, II respectively. Setting |αi|2 + |βi|2, i = 0, 1, the purification of ρj ⊗ ρj is simply

|ψ〉 ⊗ |ψ〉 =1∑

i,j=0

√λi

√λj |ψi〉|ψj〉, (5.4)

After the BS, the resulting state ρ′j = UBSρj ⊗ ρjU†BS, where

UBS : a(j)I,II →

a(j)I,II − ia

(j)II,I√

2(5.5)

UBS : b(j)I,II →

b(j)I,II − ib

(j)II,I√

2(5.6)

is the unitary time evolution operator of the BS, becomes

ρ′j = λ21|φ1〉〈φ1|+ λ2

2|φ2〉〈φ2|+ λ1λ2|φ3〉〈φ3|+ λ1λ2|φ4〉〈φ4|, (5.7)

where |φi〉 = (α(j)†I β

(j)†I +α

(j)†II β

(j)†II )|vac〉, α, β ∈ a, b, i = 1, 2, 3 are states with double occupancy

spatial modes, i.e are states where the two bosons occupy the same site. As for |φ4〉 = (a(j)†I b

(j)†II −

a(j)†II b

(j)†I )|vac〉, it is a single occupancy spatial mode, i.e each boson occupies a different site.

The double occupancy states originate from the symmetric component of ρj ⊗ ρj , while thesingle occupancy state originates from the antisymmetric component. Hence the BS effectivelyprojects ρj ⊗ ρj onto the symmetric (doubly occupied spatial modes) and the antisymmetric(singly occupied spatial modes) subspaces, with probabilities

P j+ = 1− λ1λ2 =

12Tr[(1 + V (2))ρj ⊗ ρj ] =

12

+12Tr(ρ2

j ), (5.8)

P j− = λ1λ2 =

12Tr[(1− V (2))ρj ⊗ ρj ] =

12− 1

2Tr(ρ2

j ). (5.9)

5.2.2 Multipartite case

We now extend the above two-boson scenario to the general situation (see Fig. 5.1) and considertwo copies of a state of N bosons, undergoing pairwise BS. By correlating the probabilities ofprojecting the state of each pair of identical bosons on the symmetric/antisymmetric subspaces,we can estimate the purity of ρ123...N and of any of its reduced density operators. As a moreconcrete example, let us consider the probabilities for N = 3. We will label the subsystems1, 2, 3, respectively:

P±1±2±3 =123

Tr[3∏

i=1

(1±i Vi)ρ123 ⊗ ρ123] (5.10)

=18[1±1 Tr(ρ2

1)±2 Tr(ρ22)±3 Tr(ρ2

3)±1,2 Tr(ρ212)±1,3 Tr(ρ2

13)±2,3 Tr(ρ223)

±1,2,3Tr(ρ2123)

],


where ±i,i′ = (±i)(±′i), i, i′ = 1, 2, 3, and V1,2,3 stand for the swap operator acting on subsystem1, 2, 3. The purities related to ρ123 are unequivocally determined by the eight probabilitiesP±1±2±3 . For example,

Tr(ρ123)2 = P+1+2+3 + P+1−2−3 + P−1+2−3 + P−1−2+3 − P−1−2−3 (5.11)

−P+1−2+3 − P+1+2−3 − P−1+2+3 ,

Tr(ρ12)2 = P+1+2+3 + P+1+2−3 + P−1−2+3 + P−1−2−3 − P−1+2+3 (5.12)

−P−1+2−3 − P+1−2+3 − P+1−2−3 ,

Tr(ρ1)2 = P+1+2+3 + P+1+2−3 + P+1−2+3 + P+1−2−3 − P−1+2+3 (5.13)

−P−1+2−3 − P−1−2+3 − P−1−2−3 .

Note that the purity of any subset of bosons is given simply by the probabilities of even numberantisymmetric projections (”-”) minus the probabilities of odd number in the subset, varied overall projections in the remaining bosons. The expression for the probabilities in Eq. (6.8) can bestraightforwardly extended to states of N bosons, where we consider the expectation values ofthe projector

∏Ni=1(1±i Vi)/2, on ρ123...N ⊗ ρ123...N . In the N boson case, the 2N − 1 unknown

purities will be determined by the 2N − 1 independent probabilities.

5.3 Realization of the entanglement detection network

The implementation of this entanglement detection scheme in optical lattices and magneticmicrotraps follows four steps:

(i) Creation of two identical copies of the entangled state ρ123...N : Each of the two rowsof bosons shown in Fig. 5.1 is realized by a 1D chain of entangled atoms. The entanglementcan e.g. be created by spin selective movement and controlled interactions between atoms asdescribed in [55, 56] or by entangling beam splitters as investigated in [57]. We assume that anyhopping of atoms between the lattice sites is initially turned off and that the two chains consistof exactly one atom per lattice site [53, 54].

(ii) Implementation of the pairwise BS: This is achieved by decreasing the potential barrierbetween the two rows of atoms. In an optical lattice one can decrease the corresponding laserintensities [53] while in an array of magnetic microtraps electric/magnetic fields can be switchedto change the barrier height [58]. The dynamics after lowering the potential barrier is describedby the Hamiltonian H = HBS + Hint where (c.f. [53])

HBS =N∑

j=1

−J(a(j)†I a

(j)II + b

(j)†I b

(j)II + h.c.) (5.14)

Hint =∑

l=I,II

N∑

j=1

Ua

2a

(j)†l a

(j)†l a

(j)l a

(j)l +

Ub

2b(j)†l b

(j)†l b

(j)l b

(j)l + Uabb

(j)†l b

(j)l a

(j)†l a

(j)l . (5.15)

Here HBS describes vertical hopping of particles between the two rows with hopping matrixelement J [59] and Hint gives the on-site interaction of two particles in a lattice site withinternal state dependent interaction strengths Ua, Ub and Uab. For simplicity we assume thatthe interaction terms Hint can be neglected while the hopping is turned on, i.e. J À U , and weassume Ua = Ub = Uab = U 1. Turning on the Hamiltonian HBS for the specific time T = π/(4J)

1When J ≈ U , an extra phase is introduced during the BS, leading to a different time T .


implements the N pairwise BS. This can be seen by solving the Heisenberg equations of motionand calculating the time evolution of modes a and b:

a(b)(j)I (t) = cos(Jt)a(b)(j)I − i sin(Jt)a(b)(j)II

a(b)(j)II (t) = cos(Jt)a(b)(j)II − i sin(Jt)a(b)(j)I . (5.16)

The N pairwise BS are implemented for the specific time T = π/(4J).(iii) Acquisition of a relative phase between the symmetric and antisymmetric parts of the

wave function: After implementing the BS we let the system evolve according to Hint for timeτ . This introduces a phase θ = Uτ in each doubly occupied lattice site while it has no influenceon singly occupied lattice sites. Recently it was demonstrated in an interference experiment [60]that this phase θ allows the double occupancy sites to be distinguished from single occupancyones.

(iv) Turning off the lattice and measuring the resulting interference pattern [60]: After timeτ the particles are released from the trap such that their wave function dominantly spreadsalong the vertical direction x (see Fig. 5.1). The density profile resulting from the pair of atomsj in state ρ′j will exhibit interference terms dependent on λ2

1, λ22, λ1λ2 and θ, so by varying the

interaction phase θ we can determine λ1λ2 = P−j and subsequently P+j . For N pairs of atomsthe density profile will depend on λ2

1, λ22, λ1λ2, as well as on correlations between the density

profiles of different pairs of atoms, according to Eq. (6.8). Measuring the density profile for theN pairs of atoms allows us to determine the different joint probabilities P±1±2...±N , so by solvingEq. (6.8) we can detect multipartite entangled states which violate Eq. (5.2). We note that boththe creation of the two copies of ρ123...N and the network can be implemented with currentexperimental technology and do not introduce any novel or unknown sources of imperfections.

5.4 Detection of entanglement

The multipartite entropic inequalities defined in Eq. (5.2) detect entanglement in different classesof states, such as maximally entangled states |ψ〉, Werner states ρW = p|ψ〉〈ψ| + (1 − p)1 [19],and cluster states, the entanglement resource used in one-way computation [32]. In fact, ourentanglement network does not presuppose any initial knowledge on the state, unlike entangle-ment witnesses or Bell inequalities, and its detection power is not affected by purity-preservinglocal unitaries. For example, our test detects entanglement in Werner states for p > 1/

√3,

irrespective of the actual maximally entangled state |ψ〉 defining the state. It also unequivocallyidentifies any maximally entangled multipartite state, since these state have the property ofbeing pure while all related reduced density operators have purity 1/2.

In the specific case of optical lattices, multipartite entanglement was recently generatedexperimentally [55, 56], via cold controlled collisions between nearest-neighboring atoms. Thecreation procedure worked as follows: (i) Start with a row of N atoms, one atom per latticesite, all in internal state |0〉. (ii) Apply a π/2 pulse to the atoms, putting each atom in state(|0〉+ |1〉)/√2. (iii) Shift the lattice across one lattice site, let them interact for a variable lengthof time (the conditional phase acquired by the atoms depends on the interaction time), andshift the lattice back to its original position. This process generates a class of states |φi〉, wherei = 1, ..., n is the total number of atoms in the row,


0 p

0

0.25

0.5

f

2p(a)

V

(b)

Figure 5.2: In Fig. 4.2(a), we plot the violation V of the inequalities Eq. (5.2), V1 = Tr(ρ2123)−

Tr(ρ212) (dashed), V2 = Tr(ρ2

12)− Tr(ρ21) (grey) and V3 = Tr(ρ2

12)− Tr(ρ22) (solid), as a function

of the phase φ, for N = 3 atoms. Whenever V > 0, entanglement is detected by our network.In Fig. 4.2(b) we plot different purities associated with a cluster state of size N , as a function ofφ. B is any one atom not at an end (dotted), any two atoms not at ends and with at least twoothers between them (dashed), any two or more consecutive atoms not including an end (dash-dotted), any one or more consecutive atoms including one end (solid). The plotted purities areindependent of N .

|φ2〉 =12

(|00〉+ |01〉+ eiφ|10〉+ |11〉

),

|φn〉 = cos(

φ

2

)|000...0〉+ sin

(φ

2

)|Cluster〉, (5.17)

where |Cluster〉 is as defined in [32]. Since the entangled state of the N atoms, generated bythe process described above, is be a pure state, it will violate the inequalities Eq. (5.2), forany reduced density operator of m < n atoms we might consider. In particular, for n = 2, 3,whenever the value of φ is such that the state is entangled, the inequalities in Eq. (5.2) arealways violated, Fig. 5.2(a) [61].

However, if we consider rows of n > 3 atoms, we will not always get violation of Eq. (5.2),even though the state might be entangled. To understand better the type of states generatedby the controlled collisions, it is worth devoting some attention to the process itself. In thisprocess, we have that all atoms in the row will interact only with their two nearest-neighbors;except for the atoms at the extremities, which will interact with only one neighbor. This meansthat the state of any equally numbered sets of adjacent atoms, located in different parts of therow, will be the same, as long as these sets do not include the two extremal atoms. If one of thesets includes and extremal atoms, then its state will be equal to the set that includes the otherextremal atom. So, in order to study the violation of Eq. (5.2) for states of m < n − 2 atoms,it is enough to consider rows of atoms of length m + 2, where the 2 accounts for the extremalatoms. We plotted the different purities associated with a cluster state of size N , as a functionof φ, and we observed that indeed violation of Eq. (5.2) occurs for certain subsets of atoms.


5.5 Degree of macroscopicity

Our network can also be used to study superpositions of distinct quantum macroscopic stateswhich are of great importance for the better understanding of fundamental aspects of quantumtheory [62, 63, 64]. There have been several proposals on how to create macroscopic superposi-tions in systems ranging from superconductors [65], Bose-Einstein condensates (BECs) [66, 67]to opto-mechanical setups [68]. In the case of BECs, the macroscopic superpositions are multi-partite entangled states of the form

|ψ〉 =1√K

(|φ1〉⊗N + |φ2〉⊗N ), (5.18)

where K = 2 + 〈φ1 |φ2 〉N + 〈φ2 |φ1 〉N and we define a parameter ε by the overlap ε2 =1 − |〈φ1 |φ2 〉|2. Recently, a measure based on ε for the effective size S of such superpositionsof distinct macroscopic quantum states was introduced [69]. It compares states of the form |ψ〉with generalized GHZ states of N atoms (|0〉⊗N + |1〉⊗N )/

√2, where ε = 1 for a generalized

GHZ state. The effective size S of the state |ψ〉 is given by S = Nε2 [69].We can determine S from the measurement of the purity of any reduced density operator of

Eq. (5.18). We derive an explicit formula for the purity ΠN−m = Tr(ρ2N−m), where ρN−m is the

density operator ρN = |ψ〉〈ψ| reduced by m subsystems. We find

ΠN−m =1 + γm + γN + 4γN/2 + γN−m

2(1 + γN/2)2, (5.19)

with γ = 1− ε2 = |〈φ1 |φ2 〉|2.

5.5.1 Determination of ε

Suppose we create two identical BECs, each in state ρN , wait for a time tc to let their densityoperators be inelastically reduced via single particle loss processes to ρN−m ⊗ ρN−m′ , and thenlet the two BEC’s go through a BS like transformation. As an aside we note that the reduceddensity operators emerging from multi particle collisions not only depend on ε, but also on|φ1〉, |φ2〉 and thus could be used to gain further insight into the properties of the state. TheBS can be implemented either through collisional interactions between the atoms in two armsof a spatial interferometer [70], or by first turning both BECs into Mott insulator states [53]trapping them in an optical lattice and then switching on HBS. We only consider the lattermethod since it corresponds more directly to the situation of Fig. 5.1. The loss processes whichreduce the density operators ρN are stochastic so in general m 6= m′ which means that onlyN − n, where n = max{m,m′}, pairs of atoms will undergo pairwise BS in the lattice. Sinceonly density profiles of pairs of vertical sites with two atoms contribute to the interferencepattern, measuring the collective density profile will determine Tr(ρ2

N−n). Plots of differentΠN−n for an initial number of N = 300 atoms as a function of ε are presented in Fig. 5.2(b).The dependence of these curves on N is very weak but for constant ε the values of ΠN−n quicklytend towards 1/2 as n increases. Therefore from measuring the density profile the determinationof ε from ΠN−n is best done for small n ∼ 15. For a given particle loss rate the average value of n

after time tc will be known and ε can be found by averaging over several runs of the experimentperformed under identical initial conditions.

We note that this measurement is considerably simpler than those in the previous entangle-ment detection schemes, since we do not require the ability to distinguish between individualpairs of bosons but only need to find the overall probability of projecting on the symmetric andantisymmetric subspaces. If the experimental setup allows to determine the number of pairwise


0 0.2 0.4 0.6 0.8 10.5

0.75

1

Purity (N=300)

e

n=299

n=293

n=286

n=280

Figure 5.3: Plot of the purity ΠN−m for m = 1 (solid black), m = 7 (dashed black), m = 14(solid grey) and m = 20 (dashed grey), as a function of ε, for N = 300 atoms.

beam splitters N −n, the measurement can be performed in one run and inelastic processes arenot necessary if one can measure the collective density profile associated with a subset of thepairs of atoms.

5.6 Summary

In summary, we have presented and investigated a simple quantum network that detects multi-partite entanglement, requiring only two identical copies of the quantum state and pairwise BSbetween the constituents of each copy. We have shown how the network can be implemented inoptical lattices and magnetic microtraps, using current technology. As examples of its power wehave applied the network to detect entanglement and imperfections in cluster states and shownthat it also can be used to characterize macroscopic superposition states.

CHAPTER 6

Entropic inequalities

Research on efficient experimental methods for detection, verification, and estimation of quan-tum entanglement if of great practical importance. However, despite a remarkable progress inthe field, entanglement still eludes both a rigorous mathematical classification and an efficientexperimental detection. In particular, the most popular experimental methods of detecting en-tanglement in photons are based on inefficient tests, such as Bell’s inequalities, which leave manyentangled states undetected. In this chapter we present a simple experimental technique thatallows to test for entanglement of polarized photons. The test is more powerful than all theBell-CHSH inequalities taken together [52]. The experimental implementation employs photonbunching and anti-bunching effects [71].

Consider a source which generates pairs of photons. The photons in each pair fly apart fromeach other to two distant locations A and B. Let us assume that the polarization of each pairis described by some density operator %, which is unknown to us. Our task is to determinewhether % represents an entangled state or not. From a mathematical point of view we need toassert whether % can be written as a convex sum of product states [19],

% =∑

i

pi |αi〉〈αi| ⊗ |βi〉〈βi|, (6.1)

where |αi〉 and |βi〉 are the polarization states of individual photons in the pair, and∑

i pi = 1.If we were given a precise description of % then we could benefit from a number of mathematicaltests that check for the existence of the decomposition Eq. (6.1) [72]. Of course, if we canmeasure polarizations of sufficiently many photons we can estimate the state %, but as long asour sole concern is one particular property of the state, namely whether it is entangled or not,this is a very wasteful procedure.

There are recent proposals for direct tests of quantum entanglement, which are as strong astheir corresponding mathematical tests [35], however, they rely on technology which is not yetavailable. Thus experimentalists are effectively left with the Bell-CHSH [24] inequalities or, moregenerally, with entanglement witnesses [73] as the method of choice. This is, to some extend, aheritage of the Einstein Podolsky Rosen programme [2], where the primary motivation was therefutation of the local hidden variables theories rather than detecting quantum entanglement.In fact, there are many entangled states that cannot be detected by any of the Bell-CHSHinequalities.

41

CHAPTER 6. ENTROPIC INEQUALITIES 42

6.1 Entropic inequalities

Following Schrodinger remarks on relations between the information content of the total systemand its sub-systems [1], a number of entropic inequalities have been derived. These inequalitiesare satisfied by all separable states [52, 74, 75]. The simplest one is based on the purity measureTr(%2) and can be rewritten as

Tr(%2A) ≥ Tr(%2),

Tr(%2B) ≥ Tr(%2), (6.2)

where %A and %B are the reduced density operators pertaining to individual photons. Theinequalities above are non-linear functions of density operators and are known to be strongerthan all Bell-CHSH inequalities [52]. There are entangled states which are not detected by theBell-CHSH inequalities but which are detected by the inequality in Eq. (6.2).

6.1.1 Graphical comparison between Bell-CHSH and entropic inequalities

There is a succinct way to write all possible two-qubit Bell-CHSH inequalities [52]. We start bywriting the density operator % in the basis of the Pauli operators σi, i = 1, 2, 3,

% =14

11⊗ 11 + a · σ ⊗ 11 + 11⊗ b · σ +

∑

i,j

Tij σi ⊗ σj

. (6.3)

We can always choose local axes such that the correlation matrix Tij is diagonal i.e. the state %

is described only by 9 parameters, namely, the local Bloch vectors a and b, and the correlationsti = Tr(% σi ⊗ σi), such that −1 ≤ ti ≤ 1. Please note that we require % to be a positive matrixthus only some of these nine parameters specify a density operator. In this parametrizationthe Bell-CHSH inequalities for all possible settings correspond to the set of the following threeinequalities [76]:

t21 + t22 ≤ 1,

t21 + t23 ≤ 1,

t22 + t23 ≤ 1. (6.4)

In the parameter space spanned by t1, t2 and t3 the points satisfying all three inequalities form asolid common to three right circular cylinders of unit radii intersecting at right angles. The solidis also known as the Steinmetz solid [77], and has volume 8(2−√2) ≈ 4.68629. The Steinmetzsolid contains all separable states but also some entangled states.

Using the same notation we can rewrite the two entropic inequalities Eq. (6.2) as

t21 + t22 + t23 ≤ 1− |a2 − b2|, (6.5)

where a and b are the lengths of the vectors a and b, respectively. They represents a ball of radius√1− |a2 − b2|. The ball contains all separable states and is itself contained in the Steinmetz

solid. It’s volume is at most 4π/3 ≈ 4.18879 (for the unit radius). Thus for all admissible valuesof parameters t1, t2 and t3 whenever the entropic inequalities Eq. (6.2) are satisfied, all of theBell-CHSH inequalities Eq. (6.4) are also satisfied. The reverse does not hold. Thus the entropicinequalities are strictly stronger then all of the Bell-CHSH inequalities. This is illustrated inFig. 6.1.

The ball corresponding to the entropic inequalities contains, apart from all separable states,some entangled states. In a particular case of states with the maximally mixed reduced density


-1

-0.5

0

0.5

-1

-0.5

0

0.5

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

-1

-0.5

0

0.5

-1

0.5

-1

-1

0

-1

0.5

-1

1

t1

t2

t3

Figure 6.1: A graphical comparison of the Bell-CHSH inequalities with the entropic inequali-ties (6.2). All points inside the ball satisfy the entropic inequalities and all points within theSteinmetz solid satisfy all possible Bell-CHSH inequalities. NB not all the points in the outliningcube represent quantum states.


1

-1

-0.5

0

0.5

-1

-0.5

0

0.5

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

-1

-0.5

0

0.5

-1

-0.5

0.5

-1

-1

-0.5

0

0.5

1

-1

-0.5

0.5

-1

t1

t2

t3

Figure 6.2: In a special case of locally depolarized states, represented by points within thetetrahedron, the set of separable states can be characterized exactly as an octahedron. Allstates in the ball but not in the octahedron are entangled states which are not detectable by theentropic inequalities.

operators, also known as locally depolarized states (a = b = 0), we can provide a simplegeometrical relationship between the set of separable states and those detected as entangled bythe entropic inequalities. Following the same notation as above we first represent the class oflocally depolarized states (up to local rotations of the axes) by the tetrahedron spanned by thevertices [-1,-1,-1], [1,1,-1], [1,-1,1], [-1,1,1]. All locally depolarized and separable states form theoctahedron defined by the inequality

|t1|+ |t2|+ |t3| ≤ 1. (6.6)

The octahedron does not contain any entangled states. Thus in the case of locally depolarizedstates we have a clear classification: octahedron corresponds to all separable states, all statesin the ball but not in the octahedron are the entangled states which are not detectable by theentropic inequalities, all states in the Steinmetz solid but not in the ball are entangled statesdetectable by the entropic inequalities but not detectable by any of the Bell-CHSH inequalities.All the points outside the outlining tetrahedron do not represent quantum states. This isillustrated in Fig. 6.2.

It is quite remarkable that while the above reasoning has been performed for a specificbasis, in which with the correlation matrix is diagonal, the experimental test of the entropicinequalities involves only a single setting and provides more information than all the settings ofthe Bell-CHSH inequalities.


Figure 6.3: An outline of our experimental set-up which allows to test for the violation of theentropic inequalities.

6.2 Experimental proposal

Our experimental proposal for testing the entropic inequalities is based on the phenomenonof bunching and anti-bunching of photons. If two identical photons are incident on two dif-ferent input ports of a beam-splitter they will bunch, i.e. they will emerge together in one ofthe two, randomly chosen, output ports. More precisely, all pairs of photons (in general allpairs of bosons) with a symmetric polarization state will bunch and all pairs of photons withan antisymmetric polarization state will anti-bunch i.e. photons will emerge separately in twodifferent output ports of the beam-splitter. A beautiful experimental observation of this effectwas reported by Hong, Ou, and Mandel over fifteen years ago [71], and more recently by DiGiuseppe et al [78].

Consider an experimental set up in outlined in Fig. 6.3. Sources S1 and S2 emit pairs ofpolarization-entangled photons. The entangled pairs are emitted into spatial modes 1 and 3, and2 and 4. One photon from each pair is directed into location A and the other into location B. Atthe two locations photons impinge on beam-splitters and are then detected by photo-detectors.The beam-splitters at A and B, as long as the photons from two different pairs arrive within thecoherence time, effectively project on the symmetric and anti-symmetric subspace in the fourdimensional Hilbert space associated with the polarization degrees of freedom.

Let us consider four possible detections in this experiment: bunching at A and bunchingat B, bunching at A and anti-bunching at B, anti-bunching at A and bunching at B, andfinally, anti-bunching at A and anti-bunching at B. Anti-bunching at A (B) manifest itself in acoincidence detection in the two detectors at A (B). This is a preferable method of detection inall cases where photo-detectors are unable to differentiate between different numbers of photons.However, more recent experiments can handle both bunching and anti-bunching detection [78].Bunching at A (B) generates a “click” in one of the photo-detectors at A (B) but the “click”is due to two photons arriving together at this detector. Let the probabilities associated withthe four outcomes be, respectively, p00, p01, p10, and p11 (0 stands for bunching and 1 for anti-


bunching). They correspond to probabilities of projecting the state %⊗% of two pairs of photonson symmetric and antisymmetric subspaces, e.g. p01 = TrPS ⊗ PA% ⊗ % etc, where PS and PA

are the corresponding projectors. In terms of the entropic inequalities we have that, as shownin the previous chapter,

Tr(%2A) = p00 + p01 − p10 − p11 ≥ p00 + p11 − p01 − p10 = Tr(%2),

Tr(%2B) = p00 + p10 − p01 − p11 ≥ p00 + p11 − p01 − p10 = Tr(%2). (6.7)

Hence the inequality Eq.(6.2) can be rewritten in a new and simple, form,

p01 ≥ p11

p10 ≥ p11 (6.8)

6.2.1 Realistic sources of entangled photons

Currently available sources of entangled photons are probabilistic. Pairs of maximally entangledphotons are generated when a UV laser pulse passes through a BBO crystal. This process,known as parametric down-conversion, is not an ideal source of entangled photons. It generatesa superposition of vacuum, two-entangled photons, four-entangled photons, etc. Hence, a four-photon coincidence in our set-up may be caused by two entangled pairs from two different sourcesbut also by four photons from one source and no photons from the other, as shown in Fig. 6.4,moreover, the three scenarios are equally likely. In order to discriminate unwelcome four-photoncoincidences we can use ”phase marking” - for certain values of the phase difference betweenthe two pumping beams we register only coincidences that were not corrupted by the spuriousemissions.

The description can be made more quantitative by analysing an effective Hamiltonian de-scribing entanglement generation in two coherently pumped BBO crystals,

H = η(K + K†) + η(Le−iφ + L†eiφ). (6.9)

Here η is a coupling constant, proportional to the amplitude of the pumping beams, φ

is the relative phase shift between the beams introduced by the tilted quartz-plate, and K =a1Ha3V −a1V a3H and L = a2Ha4V −a2V a4H are the linear combination of annihilation operatorsdescribing the down-converted modes. The subscripts 1, 2, 3, 4 label the spatial modes and H,V stand for horizontal and vertical polarizations. The four-photon term of a quantum stategenerated by this Hamiltonian can be written as

|Ψ〉 =eiφ

√10

(a†1Ha†3V − a†1V a†3H)(a†2Ha†4V − a†2V a†4H)|vac〉

+1√10

(12a†21Ha†23V − a†1Ha†1V a†3V a†3H +

12a†21V a†23H

)| vac〉 (6.10)

+e2iφ

√10

(12a†22Ha†24V − a†2Ha†2V a†4V a†4H +

12a†22V a†24H

)| vac〉 ,

where the first term describes the desired two polarization-entangled pairs, each in the singletstate |H〉 |V 〉 − |V 〉 |H〉, whereas the last two terms describe unwelcome four-photon statesgenerated by an emission from only one of the two crystals (see Fig. 6.4).

The bunching and anti-bunching coincidences for the state Eq. (6.11) are given by


Figure 6.4: Possible emissions leading to four-photons coincidences. The central diagram showsthe desired emission of two independent entangled pairs – one by source S1 and one by sourceS2. The top and the bottom diagrams show unwelcome emissions of four photons by one of thetwo sources.


pab = pba =320

(1− cos 2φ), paa =14

+320

cos 2φ. (6.11)

In order to recover the coincidences associated with the desired singlet state we notice thatfor φ = 0 and φ = π/2 there are no spurious contributions to pab = pba and paa respectively. Forthese two phase settings the symmetric and antisymmetric superposition of the last two termsin Eq. (6.11)lead to additional symmetries at the input of the beam-splitters and cancels outthe unwelcome outcomes. The use of symmetry properties of photonic states in post-selection ofunwanted states, by letting them impinge in beam-splitter, was first observed by Shih-Alley [79],in an experiment similar in spirit to the experiment performed by Hong, Ou, and Mandel.

Let us stress that these inequalities involve nonlinear functions of a quantum state. Theirpower exceeds all linear tests such as the Bell-CHSH inequalities with all possible settings andentanglement witnesses. In fact, in a different context, our result can be viewed as the firstexperimental proposal of a non-linear entanglement witness. The nonlinear inequalities Eq. (6.8)can be tested with the current state of the art technology. In fact, an experiment following ourproposed setup was recently realized [80].

CHAPTER 7

Conclusion

In this thesis I have presented the main research results obtained during my doctorate. Themain topic of my research was the detection of entanglement in physical systems. Chapter 1 andChapter 2 were the introductory chapters, where I motivated and introduced the main conceptsunderlying my research.

Chapter 3 addressed the problem of estimating nonlinear functionals Trρk, k = 1, 2, ... of ageneral density operator ρ. These functionals, of which the purity Trρ2 is an example, are knownto be relevant quantities both in entanglement detection and in the characterization of ρ. Inthe standard estimation methods the functional is estimated from the classical description ofthe quantum state, i.e full state tomography is initially performed yielding the density operatormatrix from which the functional is calculated. This procedure is both resource demanding andredundant, since a number of state parameters exponential in the dimension of ρ is measuredin the state tomography, and yet we are only interested in estimating one quantity. However,the estimation method I together with collaborators proposed overcomes the redundancy of thestandard methods and in fact allows the direct estimation of each of the nonlinear functionals.Our method uses an interferometric network where a qubit undergoes single-particle interfer-ometry and acts as a control on a swap operation between m copies of ρ, i.e the swap occursconditional on the logical state of the qubit. By measuring the probability of finding the con-trol qubit in either ”0” or ”1”, we directly estimate Trρm. From the knowledge of Trρm form = 1, ..., d, where d is the dimension of the quantum state, we can determine the spectrum ofρ and other important quantities such as the von Neumann entropy.

Chapter 4 extended the above result to a more general quantum information scenario, knownas LOCC. In this scenario we consider two distant parties A and B that share several copiesof a given bipartite quantum state ρAB and are only allowed to perform local operations andcommunicate classically, i.e each party can only act on its part/subsystem of the total state andsend classical information, e.g. the outcome of a measurement. The LOCC setup is particularlyrelevant to information tasks where entangled states shared between the parties are used as acommunication resource. The extension of our estimation method to the LOCC setup works asfollows: each party implements the interferometric network earlier described on their respectiveset of halves ρA, ρB of the quantum state, measures the logical state of the respective controlqubit and then correlates the results using classical communication. From the correlated resultsA and B estimate not only Trρm

A ,TrρmB but also Trρm

AB, i.e from performing just local operationsin the individual subsystems and communicating classically the results, A and B are able to

49

CHAPTER 7. CONCLUSION 50

estimate nonlinear functionals of the bipartite state.Chapter 5 investigated entanglement criteria based on nonlinear functionals of ρ, e.g if a

given bipartite state is separable, then TrρkAB ≤ Trρk

α, α = A,B and k = 2, 3, ..., that could beimplemented in a simple, experimentally feasible way. Even though our interferometric networkdirectly estimates Trρk for any k, implementing it experimentally requires the ability to performcontrolled swap operations on sets of physical copies of ρ, which unfortunately is not withinpractical reach of current technology. Nevertheless we introduced a significant experimentalsimplification, based of particle statistics’ effects, for the simpler inequality involving the purityTrρ2. This inequality is in fact strictly stronger than the currently employed Bell’s inequalities.Our method uses the fact that measuring the purity of ρ is tantamount to measuring the proba-bility of projecting the state of two copies of ρ in its symmetric or antisymmetric subspaces. Forbosons, the projection on these subspaces is simply accomplished by a beam-splitter transforma-tion, after which the symmetric component of ρAB corresponds to two bosons in the same spatialmode while the antisymmetric component corresponds to each boson in a different spatial mode.We extended the nonlinear inequalities and the purity measurement to the multipartite settingand we proposed an experimental realization with neutral atoms stored in optical lattices. Inthis case the experimental setup consists of two identical rows of N atoms each in a multipartiteentangled state ρ123...N that undergo pairwise beam-splitter transformations. The beam-splitteris implemented by lowering the potential barrier between the pairs of atoms, allowing themto tunnel between the two sites. We also proposed a method to evaluate experimentally themacroscopicity ε of a given quantum superposition of states |ψ〉.

Chapter 6 investigated the experimental realization of the nonlinear entanglement test inphotonic systems. We considered two copies of a polarization entangled pair of photons ρAB.The experimental setup for entanglement detection is quite simple: the two respective halvesρA impinge on beam-splitter A and the two halves ρB impinge on beam-splitter B, after whichthe number of photons in each of the four spatial modes is counted. From the probabilities ofdetecting one or two photons at each mode we directly estimate Trρ2

A, Trρ2B and Trρ2

AB andcheck for the violation of the nonlinear inequalities. We analyzed the case where the sourceof entangled photons is imperfect, and we modified the experimental procedure to take theimperfection into account. This experiment was recently realized at Elsag SPA, Italy [80], andit successfully detected the singlet state.

Bibliography

[1] E. Schrodinger, Proceedings of the Cambridge Philosophical Society 31, 555(1935)

[2] A. Einstein, B. Podolsky, and N. Rosen, Physical Review 47, 777 (1935)

[3] J. S. Bell, Physics 1 195, (1964)

[4] A.K. Ekert, Phys. Rev. Lett. 67, 661 (1991)

[5] C.H. Bennet et al., Phys. Rev. Lett. 70, 1895 (1993)

[6] J.J. Bollinger, W.M. Itano, D.J. Wineland, and D.J. Heinzen, Phys. Rev. A 54,4649 (1996)

[7] V. Giovannetti, S. Lloyd, and L. Maccone, Nature, 412, 417 (2001).

[8] P.W. Shor, Proceedings, 37th Anual Symposium on Fundamentals of ComputerScience, 56, IEEE Press (1996)

[9] M. Mosca, A.K. Ekert, quant-ph/9903071

[10] A. Aspect, P. Grangier, G. Roger, Phys. Rev. Lett. 49, 91 (1982)

[11] A. Vaziri, G. Weihs, A. Zeilinger, Phys. Rev. Lett. 89, 240401 (2002)

[12] D. Deutsch, Proc. R. Soc. Lond. A 400 97 (1985)

[13] A. Peres, Phys. Rev. Lett. 77, 1413 (1996)

[14] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223, 1 (1996)

[15] E. Kashefi, C. Moura Alves, quant-ph/0404062

[16] P. Hyllus, C. Moura Alves, D. Bruss, C. Macchiavello, Phys. Rev. A 70, 032316(2004)

[17] S. C. Clark, C. Moura Alves, D. Jaksch, quant-ph/0406150

[18] M. Nielsen, I. Chuang, Quantum Computation and Quantum Information, Cam-bridge University Press (2000)

51

BIBLIOGRAPHY 52

[19] R.F. Werner, Phys. Rev. A 40, 4277 (1989)

[20] A. JamioÃlkowski, Rep. Math. Phys. 3, 275 (1972)

[21] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223, 8 (1996)

[22] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 78, 574 (1997)

[23] P. Horodecki, quant-ph/0111036

[24] J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Physical Review Letters 23,880 (1969)

[25] B. Terhal, Phys. Lett. A 271, 319 (2000)

[26] M. Lewenstein, B. Kraus, I.J. Cirac, P. Horodecki, Phys. Rev. A 62, 052310(2000)

[27] O. Guhne et al., Phys. Rev. A 66, 062305 (2002)

[28] O. Guehne, G. Toth, P. Hyllus, H. J. Briegel, quant-ph/0410059

[29] G. Toth, Physical Review A 71, 010301 (2005)

[30] A. Ambainis, H. Buhrman, Y. Dodis, H. Roehrig, quant-ph/0304112

[31] W. Dur, G. Vidal, J.I. Cirac, Phys. Rev. A 62, 062314 (2000)

[32] H.-J. Briegel, R. Raussendorf, Phys. Rev. Lett., 86, 910 (2001)

[33] D. Deutsch, Proc. R. Soc. Lond. A 425, 73 (1989)

[34] D. Deutsch, A. Barenco, A.K. Ekert, Proc. R. Soc. Lond. A 449, 669 (1995)

[35] P. Horodecki, A. Ekert, Phys. Rev. Lett. 89, 127902 (2002).

[36] A. Barenco, et al., Phys. Rev. A 52, 3457 (1995).

[37] E. Sjoqvist, A. K. Pati, A. K. Ekert, J. S. Anandan, M. Ericsson, D. K. L. Oi,V. Vedral, Phys. Rev. Lett. 85, 2845 (2000)

[38] S. Pancharatnam. Proceedings of the Indian Academy of Science, XLIV(5), 247(1956).

[39] R. Filip, quant-ph/0108119

[40] M. Keyl, R.F. Werner, Phys. Rev. A 64, 52311 (2001)

[41] P. E. Gill,W. Murray, M. H.Wright, Practical Optimization (Academic Press,Inc., London, 1981)

[42] M. Horodecki, P. Horodecki, Phys. Rev. A 59, 4206 (1999)

[43] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 85, 433 (2000)

[44] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wooters, Phys. Rev.A 54, 3824 (1996)

BIBLIOGRAPHY 53

[45] E. Fredkin and T. Toffoli, Int. J. Theor. Phys. 21, 219 (1982)

[46] The physics of quantum information : quantum cryptography, quantum telepor-tation, quantum computation. Springer, New York, 2000. Dirk Bouwmeester,Artur K. Ekert, Anton Zeilinger (eds.).

[47] A. K. Ekert, C. Moura Alves, D. K. L Oi, M. Horodecki, P. Horodecki, L. C.Kwek, Phys. Rev. Lett. 88, 217901 (2002)

[48] R. F. Werner, G. Vidal, Phys. Rev. A 65, 032314 (2002)

[49] M. Horodecki, Quantum Information and Computation 1, 3 (2001)

[50] H. A. Carteret, Phys. Rev. Lett., vol. 94(4), 040502 (2005)

[51] A.S. Holevo, Probabilistic and Statiscal Aspects of Quantum Theory (North Hol-land, Amsterdam, 1982)

[52] M. Horodecki, P. Horodecki, R. Horodecki, Physics Letters A, 210, 377 (1996)

[53] D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, P. Zoller, Phys. Rev. Lett. 81,3108 (1998)

[54] M. Greiner, O. Mandel, T. Esslinger, T.W. Haensch, I. Bloch, Nature 415, 39(2002)

[55] O. Mandel et al., Nature 425, 937 (2003)

[56] D. Jaksch, H.-J. Briegel, J.I. Cirac, C.W. Gardiner, P. Zoller, Phys. Rev. Lett.82, 1975 (1999)

[57] U. Dorner, P. Fedichev, D. Jaksch, M. Lewenstein, P. Zoller, Phys. Rev. Lett.91, 073601 (2003)

[58] T. Calarco et al., quant-ph/9905013

[59] J. Pachos, P.L. Knight, Phys. Rev. Lett. 91, 107902 (2003)

[60] A. Widera et al., cond-mat/0310719

[61] C. Moura Alves, D. Jaksch, Phys. Rev. Lett. 93, 110501 (2004)

[62] C. A. Sackett et al., Nature 404, 256 (2000)

[63] A. Rauschenbeutel et al., Science 288, 2024 (2000)

[64] J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge Univer-sity Press, New York, (1987)

[65] A.J. Leggett, A. Garg, Phys. Rev. Lett. 54, 857 (1985)

[66] J.I. Cirac, M. Lewenstein, K. Molmer, P. Zoller, Phys. Rev. A 57, 1208 (1998)

[67] A. Micheli, D. Jaksch, J.I. Cirac, P. Zoller, Phys. Rev. A 67, 13607 (2003)

[68] W. Marshall, C. Simon, R. Penrose, D. Bouwmeester, Phys. Rev. Lett. 91, 130401(2003)

BIBLIOGRAPHY 54

[69] W. Dur, C. Simon, J.I. Cirac, Phys. Rev. Lett., 89, 210402 (2002)

[70] U.V. Poulsen, K. Molmer, Phys. Rev. A 65, 033613 (2002)

[71] C.K. Hong, Z.Y. Ou, L. Mandel, Phys. Rev. Lett. 59, 2044 (1987)

[72] G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rottler, H. We-infurter, R. F. Werner, and A. Zeilinger. Quantum information: an introductionto basic theoretical concepts and experiments., volume 173 of Springer Tracts inModern Physics

[73] B.M. Terhal, M.M. Wolf, A.C. Doherty, Physics Today 56, 46 (2003)

[74] B.M. Terhal, Journal of Theoretical Computer Science 287, 313 (2002)

[75] K.G. Vollbrecht, M.M. Wolf, quant-ph/0202058

[76] R. Horodecki, P. Horodecki M. Horodecki, Phys. Lett. A 200, 340 (1995)

[77] D. Wells. The Penguin dictionary of curious and interesting geometry, p.118.Penguin, London, (1991)

[78] G. Di Giuseppe, M. Atature, M.D. Shaw, A.V. Sergienko, B.E.A. Saleh, M.C.Teich, A.J. Miller, S.W. Nam, J. Martinis, Phys. Rev. A 68, 063817 (2003)

[79] Y.H. Shih, C.O. Alley, Phys. Rev. Lett. 61, 2921 (1988)

[80] F.A. Bovino, G. Castagnolli, A.K. Ekert, P. Horodecki, C. Moura Alves, A.V.Sergienko, submitted

detection of quantum entanglement in physical systems · detection of quantum entanglement in...

Documents