Detection of Quantum Entanglement in Physical Systems

Detection of quantum entanglement in 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 in quantum information science. It encapsulates the shift in paradigm, for the description of the physical reality, brought by quantum physics. It has therefore been a key element in the debates surrounding the foundations of quantum theory. Entangle- ment is also a physical resource of great practical importance, instrumental in the computational advantages offered by quantum information processors. However, the properties of entanglement are still to be completely understood. In particular, the development of methods to efficiently identify entangled states, both theoretically and experimentally, has proved to be very challenging. This dissertation addresses this topic by investigating the detection of entanglement in physical systems. Multipartite interferometry is used as a tool to directly estimate nonlinear properties of quantum states. A quantum network where a qubit undergoes single-particle interferometry and acts as a control on a swap operation between k copies of the quantum state ρ is presented. This network is then extended to a more general quantum information scenario, known as LOCC. This scenario considers two distant parties A and B that share several copies of a given bipartite quantum state. The construction of entanglement criteria based on nonlinear properties of quantum states is investigated. A method to implement these criteria in a simple, experimentally feasible way is presented. The method is based of particle statistics’ effects and its extension to the detection of multipartite entanglement is analyzed. Finally, the experimental realization of the nonlinear entanglement test in photonic systems is investigated. The realistic experimental scenario where the source of entangled photons is imperfect is analyzed. Acknowledgements Contents

Abstract i

Acknowledgements ii

1 Introduction 1 1.1 Entanglement as a property of quantum systems ...... 1 1.2 Entanglement as a physical resource ...... 2 1.3 Detection and characterization of entanglement ...... 2 1.4 Outline of thesis ...... 3 1.5 Chapter outline ...... 4

2 Basic concepts 5 2.1 State Vectors ...... 5 2.1.1 Subsystems ...... 6 2.2 Density Operators ...... 6 2.2.1 Mathematical properties of density operators ...... 8 2.2.2 Ensemble interpretation of density operators ...... 9 2.3 Entanglement ...... 9 2.4 Superoperators ...... 10 2.4.1 Mathematical properties of superoperators ...... 10 2.4.2 Jamiolkowski isomorphism ...... 11 2.5 Mathematical characterization of bipartite entanglement ...... 11 2.5.1 Mixed states ...... 12 2.6 Experimental detection of entanglement ...... 13 2.6.1 Bell’s inequalities ...... 13 2.6.2 Entanglement witnesses ...... 14 2.7 Multipartite entanglement ...... 15 2.7.1 Maximally entangled state ...... 16 2.7.2 W State ...... 16 2.7.3 Cluster state ...... 16 2.8 Quantum networks ...... 17 2.8.1 Universal set of gates ...... 17 2.8.2 Interferometry ...... 18 2.9 Summary ...... 19

iii CONTENTS iv

3 Direct estimation of density operators 21 3.1 Modiﬁed interferometry ...... 21 3.2 Multiple target states ...... 23 3.2.1 Spectrum estimation ...... 23 3.2.2 Quantum communication ...... 24 3.2.3 Extremal eigenvalues ...... 24 3.2.4 State estimation ...... 25 3.2.5 Arbitrary observables ...... 25 3.3 Quantum channel estimation ...... 25 3.4 Summary ...... 27

4 Direct estimation of density operators using LOCC 28 4.1 LOCC estimation of nonlinear functionals ...... 28 4.2 Structural Physical Approximations ...... 30 4.2.1 SPA using only LOCC ...... 31 4.3 Entanglement detection ...... 31 4.4 Channel capacities ...... 32 4.5 Summary ...... 32

5 Entanglement Detection in Bosons 33 5.1 Nonlinear entanglement inequalities ...... 33 5.2 Estimation of the purities ...... 34 5.2.1 Bipartite case ...... 34 5.2.2 Multipartite case ...... 35 5.3 Realization of the entanglement detection network ...... 36 5.4 Detection of entanglement ...... 37 5.5 Degree of macroscopicity ...... 39 5.5.1 Determination of ² ...... 39 5.6 Summary ...... 40

6 Entropic inequalities 41 6.1 Entropic inequalities ...... 42 6.1.1 Graphical comparison between Bell-CHSH and entropic inequalities . . . 42 6.2 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 bottom line represents the target qubit. U acts on the target qubit iﬀ the control qubit is in the logical state |1i...... 18 2.2 The Mach-Zender interferometer...... 19 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 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 functions of a quantum state...... 23 3.3 A quantum channel Λ acting on one of the subsystems of a bipartite maxi- P √ mally entangled state of the form |ψ+i = k |ki|ki/ d. The output state 1 P %Λ = d kl |kihl| ⊗ Λ(|kihl|), contains a complete information about the channel. 26 4.1 Network for remote estimation of non-linear functionals of bipartite density operators. Since Tr[V (k)%⊗k] is real, Alice and Bob can omit their respective phase shifters...... 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 is

ρ123...N . The total state of the system is ρ123...N ⊗ ρ123...N ...... 34 2 5.2 In Fig. 4.2(a), we plot the violation V of the inequalities Eq. (5.2), V1 = Tr(ρ123)− 2 2 2 2 2 Tr(ρ12) (dashed), V2 = Tr(ρ12) − Tr(ρ1) (grey) and V3 = Tr(ρ12) − Tr(ρ2) (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 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 purities are 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 ², for N = 300 atoms. . . 40

6.1 A graphical comparison of the Bell-CHSH inequalities with the entropic inequalities (6.2). All points inside the ball satisfy the entropic inequalities and all points within the Steinmetz solid satisfy all possible Bell-CHSH inequalities. NB not all the points in the outlining cube represent quantum states...... 43 6.2 In a special case of locally depolarized states, represented by points within the tetrahedron, the set of separable states can be characterized exactly as an octahedron. All states in the ball but not in the octahedron are entangled states which are not detectable by the entropic inequalities...... 44 6.3 An outline of our experimental set-up which allows to test for the violation of the entropic inequalities...... 45 6.4 Possible emissions leading to four-photons coincidences. The central diagram shows the desired emission of two independent entangled pairs – one by source

S1 and one by source S2. The top and the bottom diagrams show unwelcome emissions 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 Schr¨odingeras “...the characteristic trait of quantum 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 mechanics [1, 2, 3], quantum entanglement has been recently rediscovered as a physical resource in the context of quantum information science [4, 5, 6, 7]. This set of correlations, to which a classical counterpart does not exist, arises from the interaction between distinct quantum systems. Entanglement is instrumental in the improvements of classical computation and classical communication results, of which two particularly important examples are the exponential speedup of 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 published in 1935, EPR argued that quantum mechanics is not a complete theory of Nature, i.e. it does not include a full description of the physical reality, by presenting an example of an entangled quantum state to which it was not possible to ascribe definite elements of reality. EPR defined an 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 context of classical physics, but not in the context of quantum mechanics. The predictive power of quantum mechanics is limited to, given a quantum state and an observable, the probabilities of the different measurement outcomes. This feature led EPR to deem quantum mechanics as incomplete. The incompleteness of quantum mechanics, as understood by EPR, was to plague physicists for decades. On one hand the quantum mechanical formalism explained the behaviour of microscopical systems to a great degree of accuracy. On the other hand, it was conceptually unsatisfactory as a fundamental theory of Nature and the EPR argument seemed a valid one. It was not until John Bell published his seminal paper in 1964 [3], where he discussed the validity of the EPR assumptions, that light was shed into the matter. In his paper Bell does not make any assumption about quantum mechanics. It does, however, assume that our classical common sense view 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 diﬀerent types of measurements on their respective systems. The measurements performed in each pair are chosen at random and correspond to elements of reality. The expectation values of these observables depend of the probability associated with a given outcome and the actual value of the outcome. Bell then derived a set of inequalities that bound the expectation value of a linear combination of the observables. It turns out that certain entangled states theoretically violate these inequalities, which means that either quantum mechanics is an incomplete description of Nature or the EPR assumptions are incorrect. The only way to decide which is the case was by performing an experimental test of Bell’s inequalities. This test was realized with entangled pairs of photons in 1982 [10] and it shown the violation of the Bell’s inequalities, as predicted by quantum mechanics. This type of experimental test has subsequently been used to detect entanglement experimentally in physical systems [11].

1.2 Entanglement as a physical resource

Fundamental quantum effects, such as quantum tunnelling or stimulated emission, have yielded over the last century important technological breakthroughs, of which semiconductors or lasers are 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, the logical unit of information is the qubit, a two-level quantum system. The qubit differs from the bit in that is can be any superposition of ”0” and ”1”. In particular, a set of qubits can be in an entangled state. The possibility of exploiting these quantum correlations between qubits, for realizing computations faster than it would be possible classically, was first realized by Deutsch in 1985 [12]. The development of quantum algorithms that ensued culminated with a result by Shor for the efficient factoring the primes of a number [8]. The best classical algorithms for this task scale exponentially with the size of the number to be factored, which means that it is effectively impossible to factor large numbers. However, Shor’s algorithm can factor the primes in a time that scales polynomially with the number size, i.e. efficiently. This result is particularly relevant since the security of currently used cryptographic protocols is based on the difficulty of factoring large numbers. Therefore a quantum factoring machine would render these protocols useless. Ironically, entanglement turns out to be the key resource in one of the possible solutions to the security of cryptographic protocols. This solution, proposed by Ekert in 1991 [4], uses entangled states as the carrier of protected information. The security of the protocol comes from the fact that any attempt to gain access to the encrypted information, via a measurement on the state, will necessarily disturb the quantum correlations. As mentioned earlier, the amount of entanglement 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 also a physical resource of great practical importance. It is therefore no wonder that it has been extensively 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 both probing the limits of validity of quantum mechanics, as a physical theory, and for the monitoring of quantum information processes. Its success is intimately related to the successful development of theoretical tools that not only help us to further understand the properties of entanglement, CHAPTER 1. INTRODUCTION 3 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 to define entanglement as a mathematical property. It considers the density matrix representation of quantum states and attempts to derive conditions that the matrices must obey in order to represent an entangled state. This approach enabled the derivation of necessary and sufficient conditions for entanglement in systems of two or three qubits. These results were obtained by Peres [13] and the Horodeckis [14]. They pointed the way to a more general strategy of identifying the mathematical properties of entanglement, based on the theory of positive maps. I will return to this statement in more detail in the next chapter. However, a full characterization of the set of entangled states for high-dimensional bipartite systems is yet to be found. In particular the understanding of entanglement between more than two systems, multipartite entanglement, is at 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 with respect to some of the subsystems. A general framework for the classification of entanglement is yet to be developed and researchers have so far concentrated in studying specific classes of multipartite entangled states. I will present some examples of these classes in the next chapter that we believe illustrate simultaneously the complexity of multipartite entanglement and its great potential for quantum information processing. The second approach to entanglement research, the physical one, treats quantum states as properties of physical systems, that either exist in Nature or can be experimentally generated in the laboratory. This approach differs fundamentally from the mathematical one in that it focuses on the types of states actually generated in a given physical setting. The characterization or detection of entanglement in this case is accomplished by via tests that are tailored for the specific class of states considered. In the next chapter we will present the two most commonly used experimental entanglement tests. Rather than aiming at a full characterization of entanglement, this approach aims at developing techniques and methods for entanglement detection that are experimentally accessible. In particular, it tries to identify which properties of a given quantum system are relevant for entanglement detection. Providing a solution for this question will have important consequences on the realization of experiments in quantum information processing, since it will direct the experimentalists to a more efficient, and possibly easier, detection of entanglement in the laboratory. Despite all the effort devoted in recent years to the characterization of entanglement, the full understanding of entanglement’s properties still eludes researchers. My doctoral research aimed to contribute to our knowledge about entanglement by pursuing the physical approach. I have developed new methods for not only the detection of both bipartite and multipartite entanglement but also the characterization of certain properties of quantum states. These methods are experimentally 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 research results to include. I decided to include the results that were not only the most directly relevant to the subject of the dissertation, entanglement detection, but also the results that formed the most chronologically coherent set. It will become apparent that these results were obtained sequentially and that they are different instances of one research program. This program started from a rather abstract setting of quantum networks, specifically designed to measure state properties, and ended in the development of tailor-made experimental methods for the detection CHAPTER 1. INTRODUCTION 4 of entanglement in photons. However, I also pursued other research projects, such as the study of the computational complexity of quantum languages [15], the development of methods to generate classes of bound entangled states [16] and the investigation of methods to efficiently generate graph states [17].

1.5 Chapter outline

I will now present the outline of remainder chapters of the thesis. Chapter 2 introduces the basic concepts underlying the research results of the thesis. In particular it provides a mathematical description of entanglement and discusses in more detail the general methods to detect and characterize entanglement. Chapter 3 addresses the problem of estimating nonlinear functionals Trρk, k = 1, 2, ... of a general density operator ρ. The estimation method we proposed allows the direct estimation of these nonlinear functionals. Our method uses an interferometric network where a qubit undergoes single-particle interferometry and acts as a control on a swap operation between k copies of ρ. Chapter 4 extends the above result to a more general quantum information scenario, known as LOCC. In this scenario we consider two distant parties A and B that share several copies of a given bipartite quantum state ρAB and are only allowed to perform local operations and communicate classically. Chapter 5 investigates entanglement criteria based on nonlinear functionals of ρ that could be implemented in a simple, experimentally feasible way. Our method is based of particle statistics’ eﬀects and uses the fact that measuring the purity of ρ is tantamount to measuring the probability of projecting the state of two copies of ρ in its symmetric or antisymmetric subspaces. We extend of the nonlinear inequalities to the detection of multipartite entanglement. Chapter 6 investigates the experimental realization of the nonlinear entanglement test. We consider two copies of a polarization entangled pair of photons ρAB. We also analyze the realistic experimental scenario where the source of entangled photons is imperfect. Chapter 7 presents a conclusion to the thesis, with a summary of the main research results presented. CHAPTER 2

Basic concepts

2.1 State Vectors

Statistical predictions of quantum mechanics are based on two main concepts, quantum states and quantum observables. With every isolated physical system S, we associate a complex Hilbert space HS of a suitable dimension, so that quantum states are represented by time-dependent unit vectors | ψ(t)i ∈ 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 {|ψii} such that

A|ψii = ai|ψii , ai ∈ R. (2.1)

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

| ψ(t)i = U(t, t0) | ψ(t0)i , (2.2) † where U(t, t0) is a unitary operator, UU = 11. Given a quantum system described by a state vector | ψi and any observable A, represented by a Hermitian operator, we can calculate all statistical properties of A from the relation

† | ψ(t)i hψ(t) | = U | ψ(t0)i hψ(t0) | U , (2.4) and the average value of observable observable A will be written as

hAi = Tr ρA, (2.5)

5 CHAPTER 2. BASIC CONCEPTS 6 where ρ = |ψihψ| and the trace Tr ρA stands for the sum of the diagonal elements of ρA. The trace operation is linear, Tr (αA+βB) = αTr A+βTr B, and is basis-independent. The operator ρ is called density operator.

2.1.1 Subsystems Consider a quantum system S composed of two subsystems A and B. The Hilbert space associated 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 dim HS = dim HA · dim HB and any state |ψSi of the system S can be expressed as a linear superposition of elements of the type |ai⊗|bi, where |ai ∈ HA and |bi ∈ HB. Whenever convenient, we’ll also write |ai ⊗ |bi as |ai|bi or as |a, bi. If we introduce orthonormal bases, i.e. maximal sets of vectors {|aki} in HA and {|bmi} in HB, such that hak| ali = δkl, hbm | bni = δmn, then any vector in HS can be written as, X X 2 |ψSi = ckl|aki|bli , |ckl| = 1. (2.7) k,l kl

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

P 2 P 2 where k |αk| = l |βl| = 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 is basis-independent. Thus, if |ψSi is separable, we can associate state |ψAi with the subsystem A and state |ψBi with the subsystem B. Otherwise we need to resort to density operators in order to 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 type A ⊗ B, where A acts on HA and B acts on HB. We can choose operators bases, {Ak} acting on HA, {Bk} acting on HB, such that X S = SklAk ⊗ Bl. (2.11) k,l

X km S = Sln | aki hal | ⊗ | bmi hbn | . (2.12) k,l,m,n Any operator A pertaining only to sub-system A can be trivially extended to system S through 1

A → A ⊗ 11. (2.13) The average value of an observable S = A ⊗ B acting on S is given by

hψS | S | ψSi = hψS | (A ⊗ B) | ψSi (2.14) X ∗ = clnckm hal | hbn | (A ⊗ B) | aki | bmi k,l,m,n X ∗ = clnckm(hal | A | aki)(hbn | B | bmi). k,l,m,n

In the special case of an observable pertaining to one of the subsystems, i.e. if either A = 11 or B = 11, we obtain (we choose B = 11),

=TrρAA, (2.16)

Given ρAB, the density operator of a bipartite system, we obtain ρA, the reduced density operator of the subsystem A, by taking the partial trace over the subsystem B. Mathematically the partial trace operation

ρAB −→ ρA, (2.18) is deﬁned as Tr B(A ⊗ B) = ATr B. (2.19) Thus,

1The procedure for sub-system B is analogous. CHAPTER 2. BASIC CONCEPTS 8

= ρA.

2.2.1 Mathematical properties of density operators Density operators provide a description of quantum states. They can be deﬁned as such without any reference to state vectors. Let H be a ﬁnite-dimensional Hilbert space. A density operator ρ, on H, is a linear operator such that

• ρ is positive semi-deﬁnite, that is hφ|ρ|φi ≥ 0, for any |φi ∈ H.

• Trρ = 1.

Any linear positive semi-definite operator X on H is always Hermitian, with non-negative eigenvalues, and can be written as X = Y †Y for some Y [18]. Many inequalities regarding positive operators can be derived directly from hφ|X|φi ≥ 0 by special choices of |φi. In particular, if |φi has only two non-zero components, labelled by i and j, then the submatrix of X with the elements labelled by the indices i and j is also positive semi-definite. More generally, any submatrix of a positive semi-definite matrix, obtained by keeping only the rows and columns labelled by a subset of the original indices, is itself a positive semi-definite matrix and as such must 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 |Ψi ∈ H. The density operator of any pure state corresponds to a projection operator on that particular state, defined as

ρ = |ΨihΨ|, (2.21) which, like any projection operator, is idempotent:

ρ2 = ρ. (2.22) For example, the state of a qubit α|0i + β|1i is described by the density operator

ρ = (α|0i + β|1i)(h1|β∗ + h0|α∗) = |α|2 |0ih0| + αβ∗|0ih1| + α∗β|1ih0| + |β|2 |1ih1|, (2.23) or, in the matrix form, µ ¶ |α|2 αβ∗ ρ = . (2.24) α∗β |β|2

2 2 The diagonal elements ρ00 = |α| and ρ11 = |β| correspond, respectively, to the expectation values h0|ρ|0i and h1|ρ|1i, giving the probabilities of observing bit values 0 and 1 respectively. CHAPTER 2. BASIC CONCEPTS 9

2.2.2 Ensemble interpretation of density operators

Consider a quantum source which emits particles in states |Ψ1i, |Ψ2i... |Ψni with a priori probabilities p1,p2...pn. We will write it as an ensemble {pi,|Ψii} . In this case Ã ! Xn Xn Xn hSi = pihΨi|S|Ψii = piTrS|ΨiihΨi| = TrS pi|ΨiihΨi| = TrSρ. (2.25) i=1 i=1 i=1 The result depends on the observable S and on the quantum state, which appears in the expression above only as the combination Xn ρ = pi|ΨiihΨi|. (2.26) i=1

We call this operator the density operator that describes a mixture of pure states |Ψ1i, |Ψ2i... 2 |Ψni with weights p1,p2...pn. The operator ρ is not a projector any more, ρ 6= ρ, but it has all the properties we require for density operators (self-adjoint, semi-positive, unit-trace). If we refer to a single particle, we are uncertain as to which particular pure state |Ψii it is prepared in. However, it makes perfect sense to say that the particle is in the state ρ. Please note that many diﬀerent mixtures may lead to the same density operator: Xn Xn ρ = pi|ΨiihΨi| = qi|ΦiihΦi|. (2.27) i=1 i=1

Note the sets of pure states {|Ψii, |}, {|Φii, |} are not in general orthonormal. In fact, unless there 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:

µ 3 ¶ 4 0 ρ = 1 . (2.28) 0 4

2.3 Entanglement

We have previously introduced the concept of separable sates. However, there are states in HS which are not separable, i.e. they cannot be written as a simple tensor product of two states |ψAi and |ψBi (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 more quantum systems that does not have a classical counterpart. An example of an entangled state is the singlet state of two spin-half particles 1 |Ψ−i = √ (|↑i|↓i − |↓i|↑i) , (2.29) 2 where |↑i and |↓i denote respectively spin up and spin down with respect to a chosen quantization axis. CHAPTER 2. BASIC CONCEPTS 10

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

X ` ` ` ` ρ123...N = C`ρ1 ⊗ ρ2 ⊗ ρ3 ⊗ ... ⊗ ρN , (2.30) `

` P where ρj 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 state can be decomposed in any of all the possible convex sums of product states. We will discuss in a later section the most important results concerning the characterization and detection of entanglement. But ﬁrst, we will introduce the concept of superoperators, since they have proved particularly 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 obeys Eq.(2.4). Suppose now that S is composed of two sub-systems, A and B, and that we are interested in the time evolution of sub-system A only. We can, without loss of generality, choose the state of A to be ρA and the state of B to be the pure state |0i. The time evolution of the state of system S, ρA ⊗ |0ih0|, is given by

0 † ρ = UρA ⊗ |0ih0|U , (2.31) which is still a density operator describing system S. The time evolution of the state of subsystem A is then obtained by performing the partial trace, on sub-system B, of the state ρ0 of system S:

0 † ρA = Tr B(UρA ⊗ |0ih0|U ). (2.32) If we now consider an orthonormal basis |ii, i = 0, 1, ..., for sub-system B, Eq.(2.32) becomes X X 0 † † ρA = hi|U|0iρAh0|U |ii ≡ EiρAEi , (2.33) i i where Ei = hi|U|0i are operators, acting on sub-system A, and are trace-preserving: X X † † † Ei Ei = h0|U |iihi|U|0i = h0|U U|0i = 11. (2.34) i i 0 Eq.(2.33) deﬁnes a linear map L that takes linear operators ρA to linear operators ρA. Such a map, if the property in Eq.(2.34) is satisﬁed, is called a superoperator. The representation of the superoperator given in Eq.(2.33) is called the operator-sum representation.

2.4.1 Mathematical properties of superoperators A superoperator L : ρ → ρ0 that takes density operators to density operators has the following properties [18]:

• L is trace-preserving, that is Trρ0 = TrL(ρ) = 1. CHAPTER 2. BASIC CONCEPTS 11

• 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 ρ0 = L(ρ) is positive and the 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 third properties originate from the requirement that, assuming ρ to be a density operator, ρ0 will also be a density operator. The second property originates from our desire to reconcile the density operator time evolution and its ensemble interpretation. The first property of superoperators is quite straightforward to accept, since any density operator has, by definition, trace equal to one. The third property is perhaps less obvious. Clearly, L must be a positive map to assure that ρ0 will be a positive operator (necessary condition for ρ0 to be a density operator). But why must L be completely positive? The answer is: in order to assure that, if we decide to consider the action of the superoperator on an extended 00 system, 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 and superoperators. Consider the action of a superoperator Λ on half of the maximally entangled P state |ψ i = √1 N |ii|ii: J N i X 0 1 ⊗ Λ|ψJ ihψJ | → |iihj|Λ(|iihj|) = ρ . (2.35) ij The bipartite state ρ0 encodes all the properties of the superoperator Λ, as from it we can learn how each density matrix element is transformed by Λ

|iihj| → Λ(|iihj|) . (2.36) This establishes the equivalence between a completely positive map acting on density operators pertaining to a Hilbert space H of dimension d2 − 1 and a density operator pertaining to a Hilbert 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 distinguish between pure states of the form Eq.(2.7) and mixed states. Pure bipartite states are entangled iff the number number of terms of their Schmidt decomposition is greater than one. The Schmidt decomposition of |ψSi is defined as: X X 0 0 |ψSi = ckl|aki|bli = λi|aii|bii, (2.37) k,l i 0 0 where |aii and |bii are orthonormal bases for HA and HB, respectively, and λi are non-negative P 2 real coefficients such that i λi = 1. Any state of the form Eq.(2.7) admits a Schmidt decomposition [18]. Hence, given a pure bipartite state, the computation of the coefficients λi in the Schmidt decomposition is sufficient for entanglement detection. However, there are not any known efficient methods to determine experimentally the Schmidt coefficients of an unknown state Eq.(2.7). Therefore, other more accessible entanglement criteria were developed. An example is the entropic inequalities. Entropy measures uncertainty or our lack of information CHAPTER 2. BASIC CONCEPTS 12 about a particular physical property. Entropic inequalities, which quantify relations between the 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 reduced density operators pertaining to individual subsystems. They indicate that no matter which physical property is measured there is more uncertainty in the composite system than in any of its parts. Here S stands for several different types of entropies, including the regular von Neumann entropy S(%) = −Tr% log % and the Renýientropy S(%) = − log Tr%2 [21]. These inequalities depend on the spectrum of both the state of the composite system and the states of each individual subsystem, and provide necessary conditions for separability of bipartite pure states. We will introduce in a later chapter of the thesis an efficient method for the determination of 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 bipartite states such as Eq.(2.26), the Schmidt decomposition is no longer valid [18]. Therefore new methods to identify entangled states were developed. These methods are based on the theory of positive maps. Positive, but not completely positive maps are the most powerful tool in the detection of entanglement. These maps are not physical, that is, they cannot be directly implemented in the laboratory, but they provide the best mathematical criteria for the existence of entanglement in a given state. In fact, they provide a necessary and sufficient condition for the existence of entanglement [22]: a bipartite state ρAB ∈ HA ⊗ HB is entangled iff (11 ⊗ L)ρAB ≥ 0, for all 2 L ∈ HB . Unfortunately, very little is known about the structure of positive maps, even for small dimensional spaces like C⊗3. It is therefore very difficult to extract practical entanglement criteria from the above condition. Still, Peres [13] and the Horodeckis [14] have shown that the positive partial-transposition map provides a necessary and sufficient condition for systems of two or three qubits. This map preserves the eigenvalues of ρ, so it’s clearly positive and trace preserving. For example, let consider a generic density operator of a qubit. This is a 2 × 2 matrix of the form µ ¶ α γ∗ , (2.39) γ β where the coefficients α, β, γ are chosen such that Eq.(2.39) is a valid density operator. It is sometimes convenient to represent the density operators of qubits as −→ −→ P 1 + r · σ 1 + riσi ρ = = i=x,y,z , (2.40) 2 2 where 1 is the identity operator, −→r is a three dimensional vector of length smaller or equal to one and µ ¶ µ ¶ µ ¶ 0 1 0 −i 1 0 σ = , σ = , σ = , (2.41) x 1 0 y i 0 z 0 −1 are the Pauli operators. The action of the transposition map on the density operator of the qubit is

2 Or conversely, (L ⊗ 11)ρAB ≥ 0. CHAPTER 2. BASIC CONCEPTS 13

µ ¶ µ ¶ α γ∗ α γ →T . (2.42) γ β γ∗ β Suppose now that we consider a qubit, part of a larger system in the entangled state

1 |φ+i = √ (|0i|0i + |1i|1i) . (2.43) 2 If we now apply the transposition map to the second qubit, which corresponds to a situation in which we consider the extension of transposition to a larger system (11⊗T ), the density operator will suﬀer a partial transpose of its matrix elements:     1 0 0 1 1 0 0 0 1  0 0 0 0  1  0 0 1 0    →T   . (2.44) 2  0 0 0 0  2  0 1 0 0  1 0 0 1 0 0 0 1

1 1 1 1 The resulting density matrix has eigenvalues 2 , 2 , 2 and − 2 , so it’s not a valid density operator. The negativity under partial transposition is a signature of entanglement, even for more general cases. 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 be directly implemented in the laboratory. While this problem can be circumvented, by mathematically constructing completely positive maps out of the positive maps relevant for entanglement detection [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 positive maps, 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 quantum mechanics. While this is a completely different goal from the detection of entanglement, the fact that they were designed to capture the quantum essence of physical systems meant that they were also an entanglement test. In fact, they are the most widely used experimental entanglement 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 following scenario: two distant observers A and B share many identical pairs of particles; A and B can perform 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 means XAXB + YAXB + YAYB − XAYB = ±2. Hence, the expectation value of Q is CHAPTER 2. BASIC CONCEPTS 14

X E(Q) = p(xA, yA, xB, yB)(xAxB + yAxB + yAyB − xAyB) (2.46) x y x y A XA B B ≤ p(xA, yA, xB, yB) × 2 = 2, (2.47) xAyAxB yB 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

A XA = σz , (2.49) A YA = σx , (2.50) B B σz + σx XB = − √ , (2.51) 2 B B σz − σx YB = √ , (2.52) 2 on the singlet state |Ψ−i, we obtain that 1 hXAXBi|Ψ i = hYAXBi|Ψ i = hYAYBi|Ψ i = −hXAYBi|Ψ i = √ . (2.53) − − − − 2 √ Thus, hQi|Ψ−i = 2 2, which is in clear violation of Eq.(2.47) and implies that the state is entangled. The violation of this and other Bell’s inequalities has been extensively observed experimentally [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 the composite system, these inequalities fail to detect many entangled states currently produced in the laboratory. Hence, researchers have actively looked for other types of experimental entanglement tests.

2.6.2 Entanglement witnesses Entanglement witnesses W were recently introduced as a tool for experimental entanglement detection [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 operators acting on the composite Hilbert space HA ⊗ HB that obey the following properties:

• W is Hermitian, that is W † = W .

• Tr(W |a, biha, b|) ≥ 0, for all states |a, bi 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. CHAPTER 2. BASIC CONCEPTS 15

Thus, if we have Tr(W ρ) < 0 for some ρ, then ρ is entangled. In that case we say that W detects ρ. Every entanglement witness detects something [26], since it detects in particular the projector on the subspace corresponding to the negative eigenvalues of W. We will next give an example of an entanglement witness that detects bipartite entangled states. Consider an experimental setup that, due to the imperfections, produces the mixed rather than pure bipartite state of two qubits [27]

1 ρ = p|ψihψ| + (1 − p) , (2.54) 4 where |ψihψ| is the pure state generated under ideal experimental circumstances, 0 ≤ p ≤ 1 and 1/4 is the completely mixed state (white noise). The witness is constructed by ﬁrst computing the eigenvector corresponding to the negative eigenvalue of the partially transposed density operator ρTB . The witness is given by the partially transposed projector onto this eigenvector. If the Schmidt decomposition of |ψi is |ψi = a|01i + b|10i, with a, b ≥ 0, the spectrum of ρTB is given by

1 − p 1 − p 1 − p 1 − p { + pa2, + pb2, + pab, − pab}. (2.55) 4 4 4 4 Therefore ρ is entangled iff p > 1/(1 + 4ab). The eigenvector corresponding to the minimal eigenvalue λ− is given by 1 |φ−i = √ (|00i − |11i). (2.56) 2 Hence the witness W is given by   1 0 0 0 1  0 0 −1 0  W = |φ−ihφ−|TB =   . (2.57) 2  0 −1 0 0  0 0 0 1 Note that this witness does neither depend on p, nor on the Schmidt coefficients a, b. It detects ρ iff it is entangled, since we have that

− − TB − − TB Tr(|φ ihφ | ρ) = Tr(|φ ihφ |ρ ) = λ−. (2.58) Note also that in this particular case we just considered, if Tr(W ρ) ≥ 0, ρ is separable. This is not 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 bipartite entanglement. Hence, we know considerably less about its mathematical structure and experimental detection. Still, the general approach of the methods described in the previous section is equally suited to detect multipartite entanglement. In fact, Bell’s inequalities have been derived 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 approach to multipartite entanglement detection is similar to the bipartite case. Therefore we will use this section to try to capture the complexity of multipartite entanglement by presenting three examples of multipartite entangled states. These states were all introduced in the context of quantum information and have proved to be useful resources for quantum information tasks. CHAPTER 2. BASIC CONCEPTS 16

The classification of multipartite entanglement differs from the bipartite case in that it is difficult to compare the different types of multipartite entanglement that are possible in a given composite system. For example, multipartite states of N subsystems can be biseparable, i.e. admit the decomposition X i i ρ = ciρA ⊗ ρB, (2.59) i where A, B are two disjunct partitions of the composite system. How does one compare this type 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 of multipartite 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, we will equally deﬁne the maximally entangled state of N qubits:

1 |ψN i = √ (|0000....0iN + |1111....1iN ) , (2.60) 2 ⊗N where |iiii....iiN = |ii , 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

1 ρ = Tr (|ψ ihψ |) = (|00...0i h00...0| + |11...1i h11...1| ) . (2.61) m N−m N N 2 m m m m These states are particularly useful for multi-party quantum communication protocols, such as multiparty quantum coin ﬂipping [30].

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

1 |WN i = √ (|1000....0iN + |0100....0iN + |0010....0iN + ... + |0000....1iN ) . (2.62) N In this case all the qubits are again entangled with one another, but interestingly enough the state of any subset m of qubits is not separable. In fact, for the case of three qubits, the W state retains 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 multipartite over bipartite entanglement. This class of pure states is represented by a connected subset of a simple cubic lattice of qubits [32]. The cluster state is deﬁned as the set of states |φ{k}iC that obey the set of eigenvalue equations

(a) ka K |φ{k}iC = (−1) |φ{k}iC , (2.63) with the correlation operators CHAPTER 2. BASIC CONCEPTS 17

O (a) (a) (b) K = σx σz . (2.64) b∈nghb(a)

Therein, {ka ∈ {0, 1}|a ∈ C} is a set of binary parameters which specify the cluster state and nghb(a) is the set of all neighboring lattice sites of a. This class of states in cubic lattices with two or more dimensions is, together with single qubit measurements, suﬃcient for universal quantum computation [32]. It is remarkable how a multipartite entangled state is alone the computational 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 a series of operations, such that the ﬁnal result has logical meaning. Similarly to classical computations, quantum computations are described through quantum circuits or networks. These networks are a sequence of quantum gates, unitary operations that change the logical values of the qubits, acting on one or more qubits at a time. They are a very useful paradigm to describe the dynamical evolution of systems of qubits, where the emphasis is on the state of the system after the implementation of the quantum gate, rather than on the actual physical interaction that 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 arbitrary accuracy. 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 unitaries plus an entangling two-qubit gate [18]. Any single qubit unitary operator can be written in the form

−→ U = exp(iα)Rnˆ(θ) = exp(iα) exp(−iθnˆ · σ ), (2.65) where α, θ are real numbers and Rnˆ(θ) denotes a rotation by θ about then ˆ axis. However, the actual implementation of arbitrary rotations in a given physical qubit can be experimentally very challenging. Therefore, researchers have instead concentrated in ﬁnding a ﬁnite set of single qubit gates that can approximate an any unitary operation U to arbitrary accuracy δ, i.e.

²(U, V ) = max ||(U − V )|ψi|| ≤ δ, (2.66) |ψi where V is the unitary implemented instead of U, ²(U, V ) is the unitary error and the maximum is taken over all normalized states |ψi. A possible such set of gates is constituted by the Hadamard, π/4 and π/8 gates [18]: µ ¶ µ ¶ µ ¶ 1 1 1 1 0 1 0 H = √ , π/4 = , π/8 = . (2.67) 2 1 −1 0 i 0 eiπ/4 As for the two-qubit gate, it is a controlled operation, i.e. it is a quantum gate where the inputs have diﬀerent roles. One of the inputs is the control qubit while the other is the target qubit. The gate acts on the target qubit iﬀ the control qubit is in state |1i. A generic controlled- U gate is depicted in Fig. 2.1. The prototypical example of the entangling two-qubit gate is the controlled-NOT gate. It has the following matrix representation in the |control, targeti basis: CHAPTER 2. BASIC CONCEPTS 18

Truthtable C CT CT 0000 0101 U 101U(0) T 111U(1)

Figure 2.1: The controlled-U gate. The top line represents the control qubit and the bottom line represents the target qubit. U acts on the target qubit iﬀ the control qubit is in the logical state |1i.

  1 0 0 0  0 1 0 0  C − NOT =   . (2.68)  0 0 0 1  0 0 1 0 The C − NOT gate ﬂips the target qubit iﬀ the control qubit is in state |1i, otherwise it acts as the identity gate. Let us now understand why is it that this√ gate is an entangling gate. Consider the case where the control qubit is in state (|0i + |1i)/ 2 and the target qubit is in state |0i. The composite state of the two qubits is clearly separable. After the C − NOT ,

1 1 1 √ (|0i + |1i)|0i = √ (|00i + |10i) →CNOT √ (|00i + |11i), (2.69) 2 2 2 which is no longer separable and is in fact the maximally entangled state |φ+i mentioned earlier.

2.8.2 Interferometry As we mentioned earlier, the quantum network formalism provides us with a very useful set of tools to describe the dynamical evolution of physical systems of qubits. A particularly simple and relevant example that of quantum interferometry. Consider a single particle going through a Mach-Zender interferometer (Fig. 2.2). The incoming particle enters the interferometer from the lower left, in the path labelled |0i. It encounters a 50:50 beam-splitter that deﬂects the particle into arm |1i with probability p = 0.5. If the particle goes through arm |0i, it acquires a phase θ0, while if it goes through arm |1i 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 particle being detected in port |0i or |1i after the interferometer is given by CHAPTER 2. BASIC CONCEPTS 19

q- q P 2 æ1 0 ö 0 = cos ç ÷ è2 ø 50/50BeamSplitter q1 æq- q ö P sin 2 1 0 1 = ç ÷ è2 ø

q0 0

1 50/50BeamSplitter

Figure 2.2: The Mach-Zender interferometer.

µ ¶ θ − θ p = cos2 1 0 , (2.70) 0 2 µ ¶ θ − θ p = sin2 1 0 . (2.71) 1 2

This result can be easily understood if we translate the Mach-Zender interferometer into the language of quantum networks (Fig. 2.3). Let us encode our qubit in the two arms of the interferometer. The qubit is initially is state |0i. After the beam-splitter,√ which is nothing but a Hadamard gate, the state of the√ qubit becomes (|0i + |1i)/ 2. The qubit then acquires the iθ iθ phases θ0, θ1:(e 0 |0i+e 1 |1i)/ 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

|0i + |1i |0i + ei(θ1−θ0)|1i (1 + ei(θ1−θ0))|0i + (1 − ei(θ1−θ0))|1i |0i →BS √ →θ √ →BS , (2.72) 2 2 2 hence the probability of ﬁnding the qubit in state |0i or |1i after the interferometer is simply given 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 any mixed quantum state, which is due to the indistinguishability of state preparations. We have introduced superoperators, completely positive maps acting on quantum states, as the most general evolution of quantum systems. We have mentioned the Jamiolkowski isomorphism between superoperators and quantum states. Entanglement, and in particular its detection, is the main object of research of this thesis. We gave an overview of the main results concerning the CHAPTER 2. BASIC CONCEPTS 20

VISIBILITY

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 ϕ according to Eq.(2.70). mathematical characterization and experimental detection of entanglement. Finally, we introduced the circuit model of quantum computation and we shown its suitability to describe the dynamical 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 with A. K. Ekert, D. K. L. Oi, M. Horodecki, P. Horodecki, L. C. Kwek: A. K. Ekert, C. Moura Alves, 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 its spectrum, are of significant importance in quantum information science. They can be quantified in terms of linear or non-linear functionals of %. Linear functionals, such as average values of observables {A}, given by TrA%, are quite common as they correspond to directly measurable quantities. Non-linear functionals of state, such as the von Neumann entropy −Tr% ln %, eigenvalues, or a measure of purity Tr%2, are usually extracted from % by classical means i.e. % is first estimated and once a sufficiently precise classical description of % is available, classical evaluations of the required functionals can be made. However, if only a limited supply of physical objects in state % is available, then a direct estimation of a specific quantity may be both more efficient and more desirable [35]. For example, the estimation of purity of % does not require knowledge of all matrix elements of %, thus any prior state estimation procedure followed by classical calculations is, in this case, inefficient. However, in order to bypass tomography and to estimate non-linear functionals of % more directly, we need quantum networks [33, 36] performing quantum computations that supersede classical evaluations. In this chapter, we shall present and examine a simple quantum network that can be used as a basic building block for direct quantum estimations of both linear and non-linear functionals of any %. The network can be realized as multiparticle interferometry. While conventional quantum measurements, represented as quantum networks or otherwise, allow the estimation of TrA% for some Hermitian operator A, our network can also provide a direct estimation of the overlap 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 to modiﬁcations of visibility in interferometry. Consider a typical interferometric set-up for a single qubit: 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-U gate. The interference pattern is modiﬁed by the factor veiα = Tr [Uρ].

µ ¶ 1 0 ϕ = , (3.1) 0 eiϕ Hadamard gate, followed by a measurement in the computational basis. We modify the interferometer by inserting a controlled-U operation between the Hadamard gates, with its control on the qubit and with U acting on a quantum system described by some unknown density operator ρ, as shown in Fig. 3.1. The action of the quantum network is given by

1 |0i|ψi −→H √ (|0i + |1i) |ψi 2 1 −→c−U √ (|0i|ψi + |1iU(|ψi)) 2 φ 1 −→ √ (|0i|ψi + eiφ|1iU(|ψi)) 2 1 h ³ ´ ³ ´i −→H |0i |ψi + eiφU(|ψi) + |1i |ψi − eiφU(|ψi) . (3.2) 2 The action of the controlled-U on ρ modiﬁes the interference pattern:

1 1 P (φ) = (1 + veiαeiφ + ve−iαe−iφ + 1) = (1 + v cos (φ + α)) , (3.3) 0 4 2 by the factor Tr(|ψihψ|U) = TrρU = veiα [37], where v is the new visibility and α is the shift of the interference fringes, also known as the Pancharatnam phase [38]. Thus, the observed modification of the visibility gives an estimate of TrUρ, i.e. the average value of the unitary operator U in state ρ. Let us mention in passing that this property, among other things, allows the estimation of an unknown quantum state ρ as long as we can estimate TrUkρ for a set of unitary 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 |αiA|βiB = |βiA|αiB, for any pure states |αiA and |βiB. In this case, the modification of the interference pattern given by Eq. (3.3) can be written as, CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 23

· · ·a V a b ·b

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

v = TrV (% ⊗ % ) X Xa b = λrλshfj|hei| (|fsi|erihfs|her|) |eii|fji ij rs X X = λrλsδjsδirh ei | fs ih fj | er i ij rs X 2 = λiλj |h ei | fj i| ij

= 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 |0i at the output, P0, is related to the visibility by v = 2 P0 − 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 = |αihα| and %b = |βihβ| 2 the formula above gives Tr%a%b = |h α | β i| i.e. a direct measure of orthogonality of |αi and 2 |βi. If we put %a = %b = % then we obtain an estimation of Tr% . In the single qubit case, this measurement allows us to estimate the length of the Bloch vector, leaving its direction completely undetermined. For qubits Tr%2 gives the sum of squares of the two eigenvalues which allows 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 estimation of 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 deﬁne the shift V (k) as (k) V |φ1i|φ2i...|φki = |φki|φ1i...|φk−1i, (3.5) CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 24 for any pure states |φi. Such an operation can be easily constructed by cascading k − 1 swaps V . If we extend the network and prepare ρ = %⊗k at the input then the interference will be modiﬁed by the visibility factor,

Xk (k) ⊗k k k v = Tr V % = Tr % = λi . (3.6) i=1

Thus measuring the average values of V (k) for k = 2, 3...d allows us to evaluate the spectrum of % [35]. Although we have not eliminated classical evaluations, we have reduced them by a signiﬁcant amount. The average values of V (k) for k = 2, 3...d provide enough information to evaluate 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 collective measurements of several copies of %, have been proposed [40]. These methods essentially project the initial state ρ = %⊗n, which forms an operator on the n-fold tensor product space, onto orthogonal subspaces corresponding to irreducible representations of the permutation group of n 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 estimates of the ordered sequence of eigenvalues of %. The probability that the error in the spectrum estimation is greater than some ﬁxed ² 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 several interesting applications in which one of the inputs, say %a, is predetermined and the other is unknown. For example, projections on a prescribed vector |ψi, or on the subspace perpendicular to it, can be implemented by choosing %a = |ψihψ|. By changing the input state |ψi we eﬀectively “reprogram” the action of the network which then performs diﬀerent projections. This property can be used for quantum communication, in a scenario where one carrier of information, in state |ψi, determines the type of detection measurement performed on the second carrier. Note that as the state |ψi of a single carrier cannot be determined, the information about the type of the measurement 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 eigenvectors of %b without reconstructing the entire spectrum. In this case, the input states are of the form |ψihψ|⊗%b and we vary |ψi searching for the minimum and the maximum of v = hψ|%b|ψi. 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 |ψi and %b by, Ã ! X vψ = Tr |ψihψ| λi|ηiihηi| X i X 2 = λi |h ψ | ηi i| = λipi, (3.7) i i P where i pi = 1. This is a convex sum of the eigenvalues of %b and is minimized (maximized) 0 when |ψi = |ηmini (|ηmaxi). For any |ψi= 6 |ηmini (|ηmaxi), there exists a state, |ψ i, in the neighbourhood of |ψi such that vψ0 < vψ (vψ0 > vψ). Thus this global optimization problem can be solved using standard iterative methods, of which steepest decent [41] is an example. CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 25

Estimation of extremal eigenvalues plays a signiﬁcant role in the direct detection [35] and distillation [22] of quantum entanglement. As an example, consider two qubits described by the density operator %b, such that the reduced density operator of one of the qubits is maximally mixed. We can test for the separability of %b by checking whether the maximal eigenvalue of %b 1 does not exceed 2 [42].

3.2.4 State estimation

Finally, we may want to estimate an unknown state, say a d × d density operator, %b. Such an 2 operator is determined by d −1 real parameters. In order to estimate matrix elements hψ|%b|ψi, we run the network as many times as possible (limited by the number of copies of %b at our disposal) on the input |ψihψ| ⊗ %b, where |ψi is a pure state of our choice. For a ﬁxed |ψi, after several runs, we obtain an estimation of,

v = hψ|%b|ψi. (3.8)

In some chosen basis {|ni} the diagonal elements hn|%b|ni can be determined using the input states |nihn|⊗%b.√ The real part of the oﬀ-diagonal element hn|%b|ki can be estimated√ by choosing |ψi = (|ni + |ki)/ 2, and the imaginary part by choosing |ψi = (|ni + i|ki)/ 2. In particular, if we want to estimate√ the density operator of a√ qubit, we can choose the pure states, |0i (spin +z), (|0i + |1i) / 2 (spin +x) and (|0i + i|1i) / 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 is the use of a ﬁxed architecture network, controlled only by input data, to perform the estimation of properties of ρ.

3.2.5 Arbitrary observables We can extend the procedure above to cover estimations of expectation values of arbitrary observables A. This can be done with the network shown in Fig. 3.2, since estimations of mean values 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 A0 = γ1 + A is positive if −γ is the A0 minimum negative eigenvalue of A, we can construct the state % = % 0 = and apply our a A Tr(A0) interference scheme to the pair %A0 ⊗ %b. The visibility gives us the mean value of V, µ ¶ A0 v = hV i = Tr % , (3.9) %A0 ⊗%b Tr(A0) b which leads us to the desired value,

hAi%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 used to estimate certain properties of quantum channels. Recall that, from a mathematical point of view, a quantum channel is a superoperator, % → Λ(%), which maps density operators into density operators, and whose trivial extensions, 1k ⊗Λ do the same, i.e. Λ is a completely positive CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 26

P map. In a chosen basis the action of the channel on a density operator % = kl %kl|kihl| can be written as Ã ! X X Λ(%) = Λ %kl|kihl| = %klΛ(|kihl|) . (3.11) kl kl Thus the channel is completely characterized by operators Λ (|kihl|). In fact, with every channel Λ 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 operator 1 P P+ = d kl |kihl| ⊗ |kihl|, and we send only one particle through the channel, as shown in Fig. 3.3, we obtain P+ → [1 ⊗ Λ] P+ = %Λ, (3.12) where 1 X % = |kihl| ⊗ Λ(|kihl|) . (3.13) Λ d kl We may interpret this as mapping the |kihl|th-element of an input density matrix to the output matrix, Λ (|kihl|). Thus, knowledge of %Λ allows us to determine the action of Λ on an arbitrary state, % → Λ(%). If we perform a state tomography on %Λ we effectively perform a quantum channel tomography. If we choose to estimate directly some functions of %Λ then we gain some knowledge about specific properties of the channel without performing the full tomography of the channel. For example, consider a single qubit channel. Suppose we are interested in the maximal rate of a reliable transmission of qubits per use of the channel, which can be quantified as the channel 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 the capacities QC where C = ø, ←, →, ↔, stands for zero way, one way and two way classical communication. 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 sufficient condition for a one qubit channel to have non-zero two-way capacity. Namely, Q↔ > 0 iff %Λ CHAPTER 3. DIRECT ESTIMATION OF DENSITY OPERATORS 27

1 has maximal eigenvalue greater than 2 . 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, the 1 two qubit state, %Λ, is two-way distillable iﬀ the operator 2 ⊗ 1 − %Λ has a negative eigenvalue 1 (see [42] for details), or equivalently when %Λ has the maximal eigenvalue greater than 2 . This implies Q↔(Λ) > 0 because two-way distillable entanglement, which is non-zero iﬀ given state is 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 building block for direct quantum estimations of both linear and non-linear functionals of any density operator %. 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 functionals of any powers of density operators. The network has many potential applications ranging from purity tests and eigenvalue estimations to direct characterization of some properties of quantum channels. Finally let us also mention that the controlled-SWAP operation is a direct generalization of a Fredkin gate [45] and can be constructed out of simple quantum logic gates [36]. This means that experimental realizations of the proposed network are within the reach of quantum technology 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 with D. 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 estimate multi-copy observables, Tr[%k], of an unknown state % [47, 35, 23]. As mentioned before, these nonlinear functionals quantify important properties of %, such as the degree of entanglement or the spectrum. Therefore it would be very useful to be able to estimate them even when % is a bipartite state %AB shared by two distant parties, Alice and Bob, who can perform only local operations and communicate classically (LOCC). In this chapter we show that the estimation of non-linear functionals of quantum states admit LOCC implementation. We also show that Structural Physical Approximations [35, 23], an important tool for entanglement detection, can be constructed locally. This opens the possibility of the direct estimation of entanglement and some channel capacities using only LOCC. As a general remark, let us recall that a quantum operation Λ can be implemented using LOCC if it can be written as a convex sum X Λ = pk Ak ⊗ Bk, (4.1) k 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 local networks, one for Alice and one for Bob, in such a way that the global network is similar to the network with the controlled-shift. Unfortunately, the global shift operation V (k) cannot be 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 and d Bob share a number of copies of the state %AB ∈ H . They group them respectively into sets of k elements, and run the local interferometric network on their respective halves of the state ⊗k ρAB = %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 operators. Since Tr[V (k)%⊗k] is real, Alice and Bob can omit their respective phase shifters. CHAPTER 4. DIRECT ESTIMATION OF DENSITY OPERATORS USING LOCC 30

The individual interference patterns Alice and Bob record will depend only on their respective k reduced density operators. Alice will observe the visibility vA = Tr[%A] and Bob will observe k the visibility vB = Tr[%B]. However, if they compare their individual observations, they will be able to extract information about the global density operator %AB, e.g. about h ³ ´i k ⊗k (k) (k) Tr[%AB] = Tr %AB VA ⊗ VB . (4.2)

This is because Alice and Bob can estimate the probabilities Pij that in the measurement Alice’s interfering qubit is found in state |iiA and Bob’s in state |jiA for i, j = 0, 1. These probabilities can be conveniently expressed as 1 h ¡ ¢ ¡ ¢i P = Tr %⊗k 1 + (−1)iV (k) ⊗ 1 + (−1)jV (k) , (4.3) ij 4 AB A B hence the formula for the basic non-linear functional of %AB reads k Tr[%AB] = P00 − P01 − P10 + P11. (4.4)

In fact, the expression above is the expectation value hσz ⊗ σzi, measured on Alice’s and Bob’s qubits (the two qubits that undergo interference). Given that we are able to directly estimate k Tr[%AB] for any integer value of k, we can estimate the spectrum of %AB without resorting to a full 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 determining relevant parameters of density operators (see [23, 35] for more details). Basically the SPA of a mathematical operation Λ, denoted as Λ,˜ is a physical operation, a process that can be carried out in a laboratory, that emulates the character of Λ. More precisely, suppose Λ : Hd 7→ Hd is a trace preserving map which does not represent any physical process, for example, an anti-unitary operation 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 mixed state, represents a physical process, i.e. a completely positive map, as long as α is suﬃciently large. On top of this D, with its trivial structure, does not mask the structure of Λ. The Structural Physical Approximation to Λ is obtained by selecting, in the expression above, the 2 2 (d) (d) threshold value α = (d λ)/(d λ + 1), where −λ is the lowest eigenvalue of (1 ⊗ Λ)P+ and P+ is a maximally entangled state of a d × d system 1. Note that we impose the positivity condition on 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 formula (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 implementable but only in a probabilistic sense e.g. using experimental post-selection. There are many examples of mathematical operations which, though important in the quantum information contexts, do not represent a physical process. For example, mathematical criteria for entanglement involve positive but not completely positive maps [21] and as such they are not directly implementable in a laboratory — they tacitly assume that a precise description of a quantum state of a physical system is given and that such operations are mathematical transformations 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 can ˜ (d) be reduced to ΛP+ ≥ 0 CHAPTER 4. DIRECT ESTIMATION OF DENSITY OPERATORS USING LOCC 31

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 physical operation. But can it be implemented with LOCC? The positive answer is obtained by putting 1^⊗ Λ into the tensor product form (4.1). Let us start by writing it as

1^⊗ Λ = αD ⊗ D + (1 − α)1 ⊗ Λ µ ¶ 1 − α β = (1 − α + β)1 ⊗ Λ + D 1 − α + β 1 − α + β µ ¶ α −β + (α − β) D + 1 ⊗ D α − β α − β = (1 − α + β)1 ⊗ Λ˜ + (α − β)Θ˜ ⊗ D, (4.6) where 1 − α β Λ˜ = Λ + D, (4.7) 1 − α + β 1 − α + β α β Θ˜ = D + (−1). (4.8) α − β α − β Equation (4.6) does not represent a convex sum of physically implementable maps for any values of α and β but if we choose

β ≥ (1 − α)λd2 (4.9) α ≥ βd2, (4.10)

d ^ where −λ is the minimum eigenvalue of 1 ⊗ Λ(P+), 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 it can be implemented only with a certain probability of success. The minimal parameters α and β that satisfy inequalities Eqs. (4.9) and(4.10) are λd4 α = , (4.11) λd4 + 1 λd2 β = . (4.12) λd4 + 1

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. For example, 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 only LOCC. An important subclass of Bell diagonal states are the maximally correlated states, which rank two states equivalent (up to UA ⊗ UB transformations) to mixtures of two pure states, |ψ i = √1 (|0i|0i + |1i|1i) and |ψ i = √1 (|0i|0i − |1i|1i). The one-way distillable entanglement + 2 − 2 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 correlated states, we need to estimate only three parameters. CHAPTER 4. DIRECT ESTIMATION OF DENSITY OPERATORS USING LOCC 32

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

h i d2λ 1^⊗ Λ % ≥ , (4.13) AB d4λ + 1

d2 where −λ is the minimal eigenvalue of the state [(1 ⊗ 1) ⊗h(1 ⊗ Λ)]i (P+ ) [35]. Thus, by estimating 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 P TB to estimate the measure of entanglement [48], N (%AB) ≡ log ||%AB|| = log( i |λi|). Note that the computation of entanglement measures (see [49] for review) is known only for very particular cases. The measure introduced in [48] is valid for any shared bipartite state with a maximally mixed reduced density operator of at least one sub-system, and it is a function of the spectrum TB {λi} of the partially transposed matrix %AB ≡ 1 ⊗ T (%AB). It is worth mentioning that for the particular case of partial-transposition, a method that bypasses the implementation of a SPA was developed by Carteret [50]. This method simulates the action of the partial-transposition map 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 channel capacities. Let a completely positive map Λ : Hd 7→ Hd represent a quantum channel shared by Alice and Bob. Estimating the channel capacity can involve either channel tomography or direct estimation. In the case of tomography Alice prepares a maximally entangled pair of particles in d state P+ and sends one half of the pair to Bob. They now share the state

d %Λ = [1 ⊗ Λ] P+. (4.14)

From the JamioÃlkowski isomorphism [20], this bi-partite state encodes all properties of the channel Λ, so state tomography on %Λ is eﬀectively channel tomography on Λ. However, given a bi-partite state %Λ, Alice and Bob can also use the LOCC techniques to directly estimate its desired properties. For example, we have previously shown that a single qubit channel Λ has 1 non-zero channel capacity if and only if the maximal eigenvalue of %Λ is strictly greater than 2 (see [47] for details). This can be estimated directly via spectrum estimation, which in the case of two qubits requires three measurements of the type σz ⊗ σz as opposed to the 15 parameters required for the state estimation.

4.5 Summary

In summary, we have demonstrated that both direct spectrum estimations and the structural physical approximations can be implemented in the case of bi-partite states using only local operations and classical communication. This leads to more direct, LOCC type, detections and estimations of quantum entanglement and of some properties of quantum channels. Direct estimations of speciﬁc properties have the natural advantage over the state tomography because they avoid estimating superﬂuous parameters. CHAPTER 5

Entanglement Detection in Bosons

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

The implementation of almost any quantum information tasks requires precise knowledge on the entangled states being used. Hence, the development of “measurement tools” for the characterization 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 type inequalities [3], which are known to be quite ineﬃcient, in the sense that they leave many entangled states undetected [19]. Alternatively, one can perform a complete state tomography of the system [51], but this method requires the preparation of an exponentially large number of copies of the state and it is redundant, since not all parameters of the density operator are relevant for the entanglement detection. In this chapter we present a simple quantum network to detect multipartite entangled states through an entanglement test more powerful than the Bell-CHSH inequalities for all possible settings [52], albeit less powerful than full state tomography. The network is realized by coupling two 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 lattice or array of magnetic microtraps loaded with atoms in a Mott insulating state with ﬁlling factor one [53, 54]. Each of the atoms has two long lived internal states a and b which represent the qubit. The pairwise BS can be implemented by decreasing the horizontal barrier between the two rows of atoms.

5.1 Nonlinear entanglement inequalities

We start by introducing the set of inequalities used by our network for the detection of multipartite entanglement. The information-theoretic approach to separability of bipartite quantum systems leads to a set of entropic inequalities satisﬁed by all separable bipartite states [52]. We extend 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

X ` ` ` ` ρ123...N = C`ρ1 ⊗ ρ2 ⊗ ρ3 ⊗ ... ⊗ ρN , (5.1) `

33 CHAPTER 5. ENTANGLEMENT DETECTION IN BOSONS 34

rj x I ...... 1234jN-1N BSBSBSBSBSBSBS II ...... 1234jN-1N

z rj y

Figure 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 is ρ123...N . The total state of the system is ρ123...N ⊗ ρ123...N .

` P 2 where ρj is a state of subsystem j, and ` C` = 1. The purity Tr(ρ123...n) of ρ123...n, where n ∈ 1, 2, ..., N, is smaller or equal than the purity of any of its reduced density operators. For example,

2 2 2 2 2 Tr(ρ123...n) ≤ Tr(ρ123...n−1) ≤ Tr(ρ123...n−2) ... ≤ Tr(ρ12) ≤ Tr(ρ1) ≤ 1. (5.2) This set of nonlinear inequalities provides a set of necessary conditions for separability, i.e. if for any state % any of these inequalities is violated then % is entangled. For the case where ρ123...n 2 is separable and pure we have that Tr(ρ123...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 this functional 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 % ⊗ %. After implementing the network, we measuring the state of the control qubit. This measurement projects the state of the target qubits onto its symmetric subspace S% ⊗ %S†, if the control qubit is found in state |0i, or onto its antisymmetric subspace A% ⊗ %A†, if the control qubit is found in state |1i. Here, S = (1 + V )/2 and A = (1 − V )/2 are the symmetric and antisymmetric projectors, respectively, with V the swap operator previously introduced and 1 the identity operator. Hence the value of Tr(%2) is determined from the measurement of the expectation value, on state % ⊗ %, of the symmetric and antisymmetric projectors. We will next show how a BS transformation eﬀectively projects a pairs of bosons on its symmetric and antisymmetric subspaces.

5.2.1 Bipartite case

Let us ﬁrst consider the simple scenario of one pair of identical bosons, in state ρj ⊗ρj, impinging on the BS (as depicted in Fig. 5.1). In order to better grasp the action of the BS, it is convenient to consider the puriﬁcation of ρj ⊗ ρj. Writing ρj in its spectral decomposition CHAPTER 5. ENTANGLEMENT DETECTION IN BOSONS 35

ρj = λ0|ψ0ihψ0| + λ1|ψ1ihψ1|, (5.3) (j)† (j)† (j) (j) with λ0 +λ1 = 1, |ψii = αiak |vaci+βibk |vaci, where ak and bk , k = I,II, are the bosonic destruction operators for particles in site number j = 1 ··· N, internal state a and b, and row 2 2 I,II respectively. Setting |αi| + |βi| , i = 0, 1, the puriﬁcation of ρj ⊗ ρj is simply

X1 p p |ψi ⊗ |ψi = λi λj|ψii|ψji, (5.4) i,j=0

0 † After the BS, the resulting state ρj = UBSρj ⊗ ρjUBS, where

(j) (j) (j) aI,II − iaII,I UBS : a → √ (5.5) I,II 2 (j) (j) (j) bI,II − ibII,I UBS : b → √ (5.6) I,II 2 is the unitary time evolution operator of the BS, becomes

0 2 2 ρj = λ1|φ1ihφ1| + λ2|φ2ihφ2| + λ1λ2|φ3ihφ3| + λ1λ2|φ4ihφ4|, (5.7)

(j)† (j)† (j)† (j)† where |φii = (αI βI +αII βII )|vaci, α, β ∈ a, b, i = 1, 2, 3 are states with double occupancy (j)† (j)† spatial modes, i.e are states where the two bosons occupy the same site. As for |φ4i = (aI bII − (j)† (j)† aII bI )|vaci, 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 the single occupancy state originates from the antisymmetric component. Hence the BS eﬀectively projects ρj ⊗ ρj onto the symmetric (doubly occupied spatial modes) and the antisymmetric (singly occupied spatial modes) subspaces, with probabilities

1 1 1 P j = 1 − λ λ = Tr[(1 + V (2))ρ ⊗ ρ ] = + Tr(ρ2), (5.8) + 1 2 2 j j 2 2 j 1 1 1 P j = λ λ = Tr[(1 − V (2))ρ ⊗ ρ ] = − Tr(ρ2). (5.9) − 1 2 2 j j 2 2 j

5.2.2 Multipartite case We now extend the above two-boson scenario to the general situation (see Fig. 5.1) and consider two copies of a state of N bosons, undergoing pairwise BS. By correlating the probabilities of projecting 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 more concrete example, let us consider the probabilities for N = 3. We will label the subsystems 1, 2, 3, respectively:

1 Y3 P = Tr[ (1 ± V )ρ ⊗ ρ ] (5.10) ±1±2±3 23 i i 123 123 i=1 1£ = 1 ± Tr(ρ2) ± Tr(ρ2) ± Tr(ρ2) ± Tr(ρ2 ) ± Tr(ρ2 ) ± Tr(ρ2 ) 8 1 1 2 2 3 3 1,2 12 1,3 13 2,3 23 2 ¤ ±1,2,3Tr(ρ123) , CHAPTER 5. ENTANGLEMENT DETECTION IN BOSONS 36

0 0 where ±i,i0 = (±i)(±i), i, i = 1, 2, 3, and V1,2,3 stand for the swap operator acting on subsystem 1, 2, 3. The purities related to ρ123 are unequivocally determined by the eight probabilities

P±1±2±3 . For example,

2 Tr(ρ123) = 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 , 2 Tr(ρ12) = 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 , 2 Tr(ρ1) = 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 number antisymmetric projections (”-”) minus the probabilities of odd number in the subset, varied over all projections in the remaining bosons. The expression for the probabilities in Eq. (6.8) can be straightforwardly extended to states of N bosons, where we consider the expectation values of QN N the projector i=1(1 ±i Vi)/2, on ρ123...N ⊗ ρ123...N . In the N boson case, the 2 − 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 magnetic microtraps follows four steps: (i) Creation of two identical copies of the entangled state ρ123...N : Each of the two rows of bosons shown in Fig. 5.1 is realized by a 1D chain of entangled atoms. The entanglement can e.g. be created by spin selective movement and controlled interactions between atoms as described in [55, 56] or by entangling beam splitters as investigated in [57]. We assume that any hopping of atoms between the lattice sites is initially turned oﬀ and that the two chains consist of exactly one atom per lattice site [53, 54]. (ii) Implementation of the pairwise BS: This is achieved by decreasing the potential barrier between the two rows of atoms. In an optical lattice one can decrease the corresponding laser intensities [53] while in an array of magnetic microtraps electric/magnetic ﬁelds can be switched to change the barrier height [58]. The dynamics after lowering the potential barrier is described by the Hamiltonian H = HBS + Hint where (c.f. [53])

XN (j)† (j) (j)† (j) HBS = −J(aI aII + bI bII + h.c.) (5.14) j=1 X XN U U H = a a(j)†a(j)†a(j)a(j) + b b(j)†b(j)†b(j)b(j) + U b(j)†b(j)a(j)†a(j). (5.15) int 2 l l l l 2 l l l l ab l l l l l=I,II j=1

Here HBS describes vertical hopping of particles between the two rows with hopping matrix element J [59] and Hint gives the on-site interaction of two particles in a lattice site with internal state dependent interaction strengths Ua, Ub and Uab. For simplicity we assume that the interaction terms Hint can be neglected while the hopping is turned on, i.e. J À U, and we 1 assume Ua = Ub = Uab = U . 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 . CHAPTER 5. ENTANGLEMENT DETECTION IN BOSONS 37 implements the N pairwise BS. This can be seen by solving the Heisenberg equations of motion and calculating the time evolution of modes a and b:

(j) (j) (j) a(b)I (t) = cos(Jt)a(b)I − i sin(Jt)a(b)II (j) (j) (j) a(b)II (t) = cos(Jt)a(b)II − i sin(Jt)a(b)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 time τ. This introduces a phase θ = Uτ in each doubly occupied lattice site while it has no influence on 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 occupancy ones. (iv) Turning off the lattice and measuring the resulting interference pattern [60]: After time τ the particles are released from the trap such that their wave function dominantly spreads along the vertical direction x (see Fig. 5.1). The density profile resulting from the pair of atoms 0 2 2 j in state ρj will exhibit interference terms dependent on λ1, λ2, λ1λ2 and θ, so by varying the interaction phase θ we can determine λ1λ2 = P−j and subsequently P+j . For N pairs of atoms 2 2 the density profile will depend on λ1, λ2, 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 the

N pairs of atoms allows us to determine the diﬀerent joint probabilities P±1±2...±N , so by solving Eq. (6.8) we can detect multipartite entangled states which violate Eq. (5.2). We note that both the creation of the two copies of ρ123...N and the network can be implemented with current experimental 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 classes of states, such as maximally entangled states |ψi, Werner states ρW = p|ψihψ| + (1 − p)1 [19], and cluster states, the entanglement resource used in one-way computation [32]. In fact, our entanglement network does not presuppose any initial knowledge on the state, unlike entanglement witnesses or Bell inequalities, and its detection power is not affected by purity-preserving√ local unitaries. For example, our test detects entanglement in Werner states for p > 1/ 3, irrespective of the actual maximally entangled state |ψi defining the state. It also unequivocally identifies any maximally entangled multipartite state, since these state have the property of being pure while all related reduced density operators have purity 1/2. In the specific case of optical lattices, multipartite entanglement was recently generated experimentally [55, 56], via cold controlled collisions between nearest-neighboring atoms. The creation procedure worked as follows: (i) Start with a row of N atoms, one atom per lattice site, all in√ internal state |0i. (ii) Apply a π/2 pulse to the atoms, putting each atom in state (|0i+|1i)/ 2. (iii) Shift the lattice across one lattice site, let them interact for a variable length of time (the conditional phase acquired by the atoms depends on the interaction time), and shift the lattice back to its original position. This process generates a class of states |φii, where i = 1, ..., n is the total number of atoms in the row, CHAPTER 5. ENTANGLEMENT DETECTION IN BOSONS 38

V 0.5

0.25

0 f 0 p 2p (a) (b)

2 Figure 5.2: In Fig. 4.2(a), we plot the violation V of the inequalities Eq. (5.2), V1 = Tr(ρ123) − 2 2 2 2 2 Tr(ρ12) (dashed), V2 = Tr(ρ12) − Tr(ρ1) (grey) and V3 = Tr(ρ12) − Tr(ρ2) (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 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 purities are independent of N.

³ ´ 1 iφ |φ2i = |00i + |01i + e |10i + |11i , 2 µ ¶ µ ¶ φ φ |φ i = cos |000...0i + sin |Clusteri, (5.17) n 2 2 where |Clusteri is as defined in [32]. Since the entangled state of the N atoms, generated by the process described above, is be a pure state, it will violate the inequalities Eq. (5.2), for any 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) are always 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 generated by the controlled collisions, it is worth devoting some attention to the process itself. In this process, 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 means that the state of any equally numbered sets of adjacent atoms, located in different parts of the row, will be the same, as long as these sets do not include the two extremal atoms. If one of the sets includes and extremal atoms, then its state will be equal to the set that includes the other extremal 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 extremal atoms. We plotted the different purities associated with a cluster state of size N, as a function of φ, and we observed that indeed violation of Eq. (5.2) occurs for certain subsets of atoms. CHAPTER 5. ENTANGLEMENT DETECTION IN BOSONS 39

5.5 Degree of macroscopicity

Our network can also be used to study superpositions of distinct quantum macroscopic states which are of great importance for the better understanding of fundamental aspects of quantum theory [62, 63, 64]. There have been several proposals on how to create macroscopic superpositions 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 multipartite entangled states of the form

1 ⊗N ⊗N |ψi = √ (|φ1i + |φ2i ), (5.18) K

N N 2 where K = 2 + h φ1 | φ2 i + h φ2 | φ1 i and we define a parameter ² by the overlap ² = 2 1 − |h φ1 | φ2 i| . Recently, a measure based on ² for the effective size S of such superpositions of distinct macroscopic quantum states was introduced [69].√ It compares states of the form |ψi with generalized GHZ states of N atoms (|0i⊗N + |1i⊗N )/ 2, where ² = 1 for a generalized GHZ state. The effective size S of the state |ψi is given by S = N²2 [69]. We can determine S from the measurement of the purity of any reduced density operator of 2 Eq. (5.18). We derive an explicit formula for the purity ΠN−m = Tr(ρN−m), where ρN−m is the density operator ρN = |ψihψ| reduced by m subsystems. We find

1 + γm + γN + 4γN/2 + γN−m Π = , (5.19) N−m 2(1 + γN/2)2

2 2 with γ = 1 − ² = |h φ1 | φ2 i| .

5.5.1 Determination of ²

Suppose we create two identical BECs, each in state ρN , wait for a time tc to let their density operators be inelastically reduced via single particle loss processes to ρN−m ⊗ ρN−m0 , and then let the two BEC’s go through a BS like transformation. As an aside we note that the reduced density operators emerging from multi particle collisions not only depend on ², but also on |φ1i, |φ2i and thus could be used to gain further insight into the properties of the state. The BS can be implemented either through collisional interactions between the atoms in two arms of 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 latter method since it corresponds more directly to the situation of Fig. 5.1. The loss processes which 0 reduce the density operators ρN are stochastic so in general m 6= m which means that only N − n, where n = max{m, m0}, pairs of atoms will undergo pairwise BS in the lattice. Since only density profiles of pairs of vertical sites with two atoms contribute to the interference 2 pattern, measuring the collective density profile will determine Tr(ρN−n). Plots of different ΠN−n for an initial number of N = 300 atoms as a function of ² are presented in Fig. 5.2(b). The dependence of these curves on N is very weak but for constant ² the values of ΠN−n quickly tend towards 1/2 as n increases. Therefore from measuring the density profile the determination of ² 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 ² can be found by averaging over several runs of the experiment performed under identical initial conditions. We note that this measurement is considerably simpler than those in the previous entanglement detection schemes, since we do not require the ability to distinguish between individual pairs of bosons but only need to find the overall probability of projecting on the symmetric and antisymmetric subspaces. If the experimental setup allows to determine the number of pairwise CHAPTER 5. ENTANGLEMENT DETECTION IN BOSONS 40

Purity(N=300) 1

n=293

n=299 0.75 n=286

n=280

0.5 0 0.2 0.4 0.6 0.8 1 e

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 ², for N = 300 atoms. beam splitters N − n, the measurement can be performed in one run and inelastic processes are not necessary if one can measure the collective density proﬁle associated with a subset of the pairs of atoms.

5.6 Summary

In summary, we have presented and investigated a simple quantum network that detects multipartite entanglement, requiring only two identical copies of the quantum state and pairwise BS between the constituents of each copy. We have shown how the network can be implemented in optical lattices and magnetic microtraps, using current technology. As examples of its power we have applied the network to detect entanglement and imperfections in cluster states and shown that 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 quantum entanglement if of great practical importance. However, despite a remarkable progress in the field, entanglement still eludes both a rigorous mathematical classification and an efficient experimental detection. In particular, the most popular experimental methods of detecting entanglement in photons are based on inefficient tests, such as Bell’s inequalities, which leave many entangled states undetected. In this chapter we present a simple experimental technique that allows to test for entanglement of polarized photons. The test is more powerful than all the Bell-CHSH inequalities taken together [52]. The experimental implementation employs photon bunching and anti-bunching effects [71]. Consider a source which generates pairs of photons. The photons in each pair fly apart from each other to two distant locations A and B. Let us assume that the polarization of each pair is described by some density operator %, which is unknown to us. Our task is to determine whether % represents an entangled state or not. From a mathematical point of view we need to assert whether % can be written as a convex sum of product states [19], X % = pi |αiihαi| ⊗ |βiihβi|, (6.1) i P where |αii and |βii 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 mathematical tests that check for the existence of the decomposition Eq. (6.1) [72]. Of course, if we can measure polarizations of sufficiently many photons we can estimate the state %, but as long as our 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 as their corresponding mathematical tests [35], however, they rely on technology which is not yet available. Thus experimentalists are effectively left with the Bell-CHSH [24] inequalities or, more generally, with entanglement witnesses [73] as the method of choice. This is, to some extend, a heritage of the Einstein Podolsky Rosen programme [2], where the primary motivation was the refutation 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-CHSH inequalities.

41 CHAPTER 6. ENTROPIC INEQUALITIES 42

6.1 Entropic inequalities

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

2 2 Tr(%A) ≥ Tr(% ), 2 2 Tr(%B) ≥ Tr(% ), (6.2) where %A and %B are the reduced density operators pertaining to individual photons. The inequalities above are non-linear functions of density operators and are known to be stronger than all Bell-CHSH inequalities [52]. There are entangled states which are not detected by the Bell-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 by writing the density operator % in the basis of the Pauli operators σi, i = 1, 2, 3,   1 X % = 11 ⊗ 11 + a · σ ⊗ 11 + 11 ⊗ b · σ + T σ ⊗ σ  . (6.3) 4 ij i j i,j

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 correlations ti = Tr(% σi ⊗ σi), such that −1 ≤ ti ≤ 1. Please note that we require % to be a positive matrix thus only some of these nine parameters specify a density operator. In this parametrization the Bell-CHSH inequalities for all possible settings correspond to the set of the following three inequalities [76]:

2 2 t1 + t2 ≤ 1, 2 2 t1 + t3 ≤ 1, 2 2 t2 + t3 ≤ 1. (6.4)

In the parameter space spanned by t1, t2 and t3 the points satisfying all three inequalities form a solid common to three right circular cylinders of unit radii intersecting√ at right angles. The solid is also known as the Steinmetz solid [77], and has volume 8(2 − 2) ≈ 4.68629. The Steinmetz solid contains all separable states but also some entangled states. Using the same notation we can rewrite the two entropic inequalities Eq. (6.2) as

2 2 2 2 2 t1 + t2 + t3 ≤ 1 − |a − b |, (6.5) pwhere 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 values of parameters t1, t2 and t3 whenever the entropic inequalities Eq. (6.2) are satisﬁed, all of the Bell-CHSH inequalities Eq. (6.4) are also satisﬁed. The reverse does not hold. Thus the entropic inequalities are strictly stronger then all of the Bell-CHSH inequalities. This is illustrated in Fig. 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 CHAPTER 6. ENTROPIC INEQUALITIES 43

1 t 0.5 3 0 -0.5 -1 1

0.5

t2 0

-0.5

-1 -1 -0.5 0 t1 0.5

Figure 6.1: A graphical comparison of the Bell-CHSH inequalities with the entropic inequalities (6.2). All points inside the ball satisfy the entropic inequalities and all points within the Steinmetz solid satisfy all possible Bell-CHSH inequalities. NB not all the points in the outlining cube represent quantum states. CHAPTER 6. ENTROPIC INEQUALITIES 44

1 t 0.5 3 0 -0.5 -1 1

0.5

t2 0

-0.5

-1 -1 -0.5 0 t1 0.5

Figure 6.2: In a special case of locally depolarized states, represented by points within the tetrahedron, the set of separable states can be characterized exactly as an octahedron. All states in the ball but not in the octahedron are entangled states which are not detectable by the entropic inequalities. operators, also known as locally depolarized states (a = b = 0), we can provide a simple geometrical relationship between the set of separable states and those detected as entangled by the entropic inequalities. Following the same notation as above we ﬁrst represent the class of locally depolarized states (up to local rotations of the axes) by the tetrahedron spanned by the vertices [-1,-1,-1], [1,1,-1], [1,-1,1], [-1,1,1]. All locally depolarized and separable states form the octahedron deﬁned by the inequality

|t1| + |t2| + |t3| ≤ 1. (6.6)

The octahedron does not contain any entangled states. Thus in the case of locally depolarized states we have a clear classiﬁcation: octahedron corresponds to all separable states, all states in the ball but not in the octahedron are the entangled states which are not detectable by the entropic inequalities, all states in the Steinmetz solid but not in the ball are entangled states detectable 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 is illustrated in Fig. 6.2. It is quite remarkable that while the above reasoning has been performed for a speciﬁc basis, in which with the correlation matrix is diagonal, the experimental test of the entropic inequalities involves only a single setting and provides more information than all the settings of the Bell-CHSH inequalities. CHAPTER 6. ENTROPIC INEQUALITIES 45

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

6.2 Experimental proposal

Our experimental proposal for testing the entropic inequalities is based on the phenomenon of bunching and anti-bunching of photons. If two identical photons are incident on two different input ports of a beam-splitter they will bunch, i.e. they will emerge together in one of the two, randomly chosen, output ports. More precisely, all pairs of photons (in general all pairs of bosons) with a symmetric polarization state will bunch and all pairs of photons with an antisymmetric polarization state will anti-bunch i.e. photons will emerge separately in two different output ports of the beam-splitter. A beautiful experimental observation of this effect was reported by Hong, Ou, and Mandel over fifteen years ago [71], and more recently by Di Giuseppe et al [78]. Consider an experimental set up in outlined in Fig. 6.3. Sources S1 and S2 emit pairs of polarization-entangled photons. The entangled pairs are emitted into spatial modes 1 and 3, and 2 and 4. One photon from each pair is directed into location A and the other into location B. At the 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 the coherence time, effectively project on the symmetric and anti-symmetric subspace in the four dimensional Hilbert space associated with the polarization degrees of freedom. Let us consider four possible detections in this experiment: bunching at A and bunching at B, bunching at A and anti-bunching at B, anti-bunching at A and bunching at B, and finally, anti-bunching at A and anti-bunching at B. Anti-bunching at A (B) manifest itself in a coincidence detection in the two detectors at A (B). This is a preferable method of detection in all 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 with the four outcomes be, respectively, p00, p01, p10, and p11 (0 stands for bunching and 1 for anti- CHAPTER 6. ENTROPIC INEQUALITIES 46 bunching). They correspond to probabilities of projecting the state %⊗% of two pairs of photons on 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 shown in the previous chapter,

2 2 Tr(%A) = p00 + p01 − p10 − p11 ≥ p00 + p11 − p01 − p10 = Tr(% ), 2 2 Tr(%B) = p00 + p10 − p01 − p11 ≥ p00 + p11 − p01 − p10 = Tr(% ). (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 entangled photons 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 generates a 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 sources but 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-photon coincidences we can use ”phase marking” - for certain values of the phase difference between the two pumping beams we register only coincidences that were not corrupted by the spurious emissions. The description can be made more quantitative by analysing an effective Hamiltonian describing 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 = a1H a3V −a1V a3H and L = a2H a4V −a2V a4H are the linear combination of annihilation operators describing 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 state generated by this Hamiltonian can be written as

eiφ |Ψi = √ (a† a† − a† a† )(a† a† − a† a† )|vaci 10 1H 3V 1V 3H 2H 4V 2V 4H µ ¶ 1 1 1 + √ a†2 a†2 − a† a† a† a† + a†2 a†2 | vaci (6.10) 10 2 1H 3V 1H 1V 3V 3H 2 1V 3H µ ¶ e2iφ 1 1 + √ a†2 a†2 − a† a† a† a† + a†2 a†2 | vaci , 10 2 2H 4V 2H 2V 4V 4H 2 2V 4H where the ﬁrst term describes the desired two polarization-entangled pairs, each in the singlet state | Hi | V i − | V i | Hi, whereas the last two terms describe unwelcome four-photon states generated 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 CHAPTER 6. ENTROPIC INEQUALITIES 47

Figure 6.4: Possible emissions leading to four-photons coincidences. The central diagram shows the desired emission of two independent entangled pairs – one by source S1 and one by source S2. The top and the bottom diagrams show unwelcome emissions of four photons by one of the two sources. CHAPTER 6. ENTROPIC INEQUALITIES 48

3 1 3 p = p = (1 − cos 2φ), p = + cos 2φ. (6.11) ab ba 20 aa 4 20 In order to recover the coincidences associated with the desired singlet state we notice that for φ = 0 and φ = π/2 there are no spurious contributions to pab = pba and paa respectively. For these two phase settings the symmetric and antisymmetric superposition of the last two terms in Eq. (6.11)lead to additional symmetries at the input of the beam-splitters and cancels out the unwelcome outcomes. The use of symmetry properties of photonic states in post-selection of unwanted 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. Their power exceeds all linear tests such as the Bell-CHSH inequalities with all possible settings and entanglement witnesses. In fact, in a different context, our result can be viewed as the first experimental 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 our proposed setup was recently realized [80]. CHAPTER 7

Conclusion

In this thesis I have presented the main research results obtained during my doctorate. The main topic of my research was the detection of entanglement in physical systems. Chapter 1 and Chapter 2 were the introductory chapters, where I motivated and introduced the main concepts underlying my research. Chapter 3 addressed the problem of estimating nonlinear functionals Trρk, k = 1, 2, ... of a general density operator ρ. These functionals, of which the purity Trρ2 is an example, are known to be relevant quantities both in entanglement detection and in the characterization of ρ. In the standard estimation methods the functional is estimated from the classical description of the quantum state, i.e full state tomography is initially performed yielding the density operator matrix from which the functional is calculated. This procedure is both resource demanding and redundant, since a number of state parameters exponential in the dimension of ρ is measured in 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 the standard 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 interferometry and acts as a control on a swap operation between m copies of ρ, i.e the swap occurs conditional on the logical state of the qubit. By measuring the probability of ﬁnding the control qubit in either ”0” or ”1”, we directly estimate Trρm. From the knowledge of Trρm for m = 1, ..., d, where d is the dimension of the quantum state, we can determine the spectrum of ρ and other important quantities such as the von Neumann entropy. Chapter 4 extended the above result to a more general quantum information scenario, known as LOCC. In this scenario we consider two distant parties A and B that share several copies of a given bipartite quantum state ρAB and are only allowed to perform local operations and communicate classically, i.e each party can only act on its part/subsystem of the total state and send classical information, e.g. the outcome of a measurement. The LOCC setup is particularly relevant to information tasks where entangled states shared between the parties are used as a communication resource. The extension of our estimation method to the LOCC setup works as follows: each party implements the interferometric network earlier described on their respective set of halves ρA, ρB of the quantum state, measures the logical state of the respective control qubit and then correlates the results using classical communication. From the correlated results m m m A and B estimate not only TrρA , TrρB but also TrρAB, i.e from performing just local operations in 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 k k given bipartite state is separable, then TrρAB ≤ Trρα, α = A, B and k = 2, 3, ..., that could be implemented in a simple, experimentally feasible way. Even though our interferometric network directly estimates Trρk for any k, implementing it experimentally requires the ability to perform controlled swap operations on sets of physical copies of ρ, which unfortunately is not within practical reach of current technology. Nevertheless we introduced a significant experimental simplification, based of particle statistics’ effects, for the simpler inequality involving the purity Trρ2. 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 probability of projecting the state of two copies of ρ in its symmetric or antisymmetric subspaces. For bosons, the projection on these subspaces is simply accomplished by a beam-splitter transformation, after which the symmetric component of ρAB corresponds to two bosons in the same spatial mode 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 setting and we proposed an experimental realization with neutral atoms stored in optical lattices. In this case the experimental setup consists of two identical rows of N atoms each in a multipartite entangled state ρ123...N that undergo pairwise beam-splitter transformations. The beam-splitter is implemented by lowering the potential barrier between the pairs of atoms, allowing them to tunnel between the two sites. We also proposed a method to evaluate experimentally the macroscopicity ² of a given quantum superposition of states |ψi. Chapter 6 investigated the experimental realization of the nonlinear entanglement test in photonic 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 halves ρA impinge on beam-splitter A and the two halves ρB impinge on beam-splitter B, after which the number of photons in each of the four spatial modes is counted. From the probabilities of 2 2 2 detecting one or two photons at each mode we directly estimate TrρA, TrρB and TrρAB and check for the violation of the nonlinear inequalities. We analyzed the case where the source of entangled photons is imperfect, and we modified the experimental procedure to take the imperfection into account. This experiment was recently realized at Elsag SPA, Italy [80], and it successfully detected the singlet state. Bibliography

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