Detection of quantum entanglement in physical systems

Transcription

1 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

2 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. Entanglement 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.

3 Acknowledgements

4 Contents Abstract Acknowledgements i ii 1 Introduction Entanglement as a property of quantum systems Entanglement as a physical resource Detection and characterization of entanglement Outline of thesis Chapter outline Basic concepts State Vectors Subsystems Density Operators Mathematical properties of density operators Ensemble interpretation of density operators Entanglement Superoperators Mathematical properties of superoperators Jamiolkowski isomorphism Mathematical characterization of bipartite entanglement Mixed states Experimental detection of entanglement Bell s inequalities Entanglement witnesses Multipartite entanglement Maximally entangled state W State Cluster state Quantum networks Universal set of gates Interferometry Summary iii

5 CONTENTS iv 3 Direct estimation of density operators Modified interferometry Multiple target states Spectrum estimation Quantum communication Extremal eigenvalues State estimation Arbitrary observables Quantum channel estimation Summary Direct estimation of density operators using LOCC LOCC estimation of nonlinear functionals Structural Physical Approximations SPA using only LOCC Entanglement detection Channel capacities Summary Entanglement Detection in Bosons Nonlinear entanglement inequalities Estimation of the purities Bipartite case Multipartite case Realization of the entanglement detection network Detection of entanglement Degree of macroscopicity Determination of ɛ Summary Entropic inequalities Entropic inequalities Graphical comparison between Bell-CHSH and entropic inequalities Experimental proposal Realistic sources of entangled photons Conclusion 49 Bibliography 51

6 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 The Mach-Zender interferometer The quantum network corresponding to the Mach-Zender interferometer (ϕ = θ 1 θ 0 ). The visibility of the interference pattern associated with p 0 varies as a function of ϕ according to Eq.(2.70) A modified Mach-Zender interferometer with coupling to an ancilla by a controlled- U gate. The interference pattern is modified by the factor ve iα = Tr [Uρ] Quantum network for direct estimations of both linear and non-linear functions of a quantum state A quantum channel Λ acting on one of the subsystems of a bipartite maximally entangled state of the form ψ + = k k k / d. The output state ϱ Λ = 1 d kl k l Λ ( k l ), contains a complete information about the channel Network for remote estimation of non-linear functionals of bipartite density operators. Since Tr[V (k) ϱ k ] is real, Alice and Bob can omit their respective phase shifters 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 ρ N. The total state of the system is ρ N ρ N In Fig. 4.2(a), we plot the violation V of the inequalities Eq. (5.2), V 1 = Tr(ρ ) Tr(ρ 2 12 ) (dashed), V 2 = Tr(ρ 2 12 ) Tr(ρ2 1 ) (grey) and V 3 = Tr(ρ 2 12 ) Tr(ρ2 2 ) (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 v

7 LIST OF FIGURES vi 5.3 Plot of the purity Π N m for m = 1 (solid black), m = 7 (dashed black), m = 14 (solid grey) and m = 20 (dashed grey), as a function of ɛ, for N = 300 atoms 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 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 An outline of our experimental set-up which allows to test for the violation of the entropic inequalities Possible emissions leading to four-photons coincidences. The central diagram shows the desired emission of two independent entangled pairs one by source S 1 and one by source S 2. The top and the bottom diagrams show unwelcome emissions of four photons by one of the two sources

8 CHAPTER 1 Introduction The subject of this dissertation is the detection of quantum entanglement in physical systems. Quantum entanglement was singled out by Erwin Schrödinger as...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 conceptual 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

9 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 revolutionizing 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,

10 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

11 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 Trρ k, 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 ρ AB 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 ρ AB. 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.

12 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 H S of a suitable dimension, so that quantum states are represented by time-dependent unit vectors ψ(t) H S, and quantum observables by Hermitian operators acting in this space. Given a observable represented by the operator A, there is a set of vectors { ψ i } such that A ψ i = a i ψ i, a i R. (2.1) The vectors ψ i are called the eigenvectors of A, with respective eigenvalues a i. The set of values {a i } is called the spectrum of A. The time evolution of state vectors is unitary, i.e. ψ(t) = U(t, t 0 ) ψ(t 0 ), (2.2) where U(t, t 0 ) is a unitary operator, UU = 11. Given a quantum system described by a state vector ψ and any observable A, represented by a Hermitian operator, we can calculate all statistical properties of A from the relation A = ψ A ψ, (2.3) where A stands for the average value of A. In particular, when A is a projection operator, projecting on a one dimensional subspace spanned by vector ϕ, A = ϕ ϕ. In this case A = ψ ϕ 2 represents the probability, for a system in state ψ, to pass a test for being in the state ϕ. Quantum states can be equally well represented by projectors on the state vectors. Namely, if instead of states ψ we consider the corresponding projectors ψ ψ, then the time evolution of the state of the system will be given by ψ(t) ψ(t) = U ψ(t 0 ) ψ(t 0 ) U, (2.4) and the average value of observable observable A will be written as A = Tr ρa, (2.5) 5

13 CHAPTER 2. BASIC CONCEPTS 6 where ρ = ψ ψ 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 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 H S = H A H B. (2.6) The dimension of H S is dim H S = dim H A dim H B and any state ψ S of the system S can be expressed as a linear superposition of elements of the type a b, where a H A and b H B. Whenever convenient, we ll also write a b as a b or as a, b. If we introduce orthonormal bases, i.e. maximal sets of vectors { a k } in H A and { b m } in H B, such that a k a l = δ kl, b m b n = δ mn, then any vector in H S can be written as, ψ S = k,l c kl a k b l, c kl 2 = 1. (2.7) A particular subset of the states in H S can be written as a tensor product of state vectors of H A and H B, ψ S = ψ A ψ B = = kl kl ( ) ( ) α k a k β l b l k l (2.8) α l β l a k b l, (2.9) where k α k 2 = l β l 2 = 1. This requires (comparing Eq.(2.8) and Eq.(2.7)) that c kl = α k β l. (2.10) The states for which this holds are called separable states. Note that this decomposition is basis-independent. Thus, if ψ S is separable, we can associate state ψ A with the subsystem A and state ψ B 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 H S can be written as a superposition of operators of the type A B, where A acts on H A and B acts on H B. We can choose operators bases, {A k } acting on H A, {B k } acting on H B, such that S = k,l S kl A k B l. (2.11) The most common operator bases are formed from operators of the type ϕ i ϕ j. In our case we have a k a l, for operators acting on H A, and b m b n, for operators acting on H B (recall that a i and b j are, respectively, orthonormal bases in H A and H B ). This means that S can be expressed as

14 CHAPTER 2. BASIC CONCEPTS 7 S = k,l,m,n S km ln a k a l b m b n. (2.12) 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 ψ S S ψ S = ψ S (A B) ψ S (2.14) = c ln c km a l b n (A B) a k b m k,l,m,n = k,l,m,n c ln c km( a l A a k )( b n B b m ). 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), ψ S S ψ S = k,l,m,n = k,l,m,n c ln c km( a l A a k )( b n 11 b m ), c ln c km a l A a k δ nm, = k,l,m c lm c km a l A a k, =Tr c lm c km a k a l A, (2.15) k,l,m =Trρ A A, (2.16) where ρ A = k,l,m c lm c km a k a m is called the reduced density operator and is associated only with sub-system A. Recall that the density operator associated with the total system is ρ AB = ψ S ψ S = k,l,m,n c km c ln ( a k a l ) ( b m b n ). (2.17) Given ρ AB, the density operator of a bipartite system, we obtain ρ A, the reduced density operator of the subsystem A, by taking the partial trace over the subsystem B. Mathematically the partial trace operation ρ AB ρ A, (2.18) is defined as Thus, Tr B (A B) = ATr B. (2.19) 1 The procedure for sub-system B is analogous.

15 CHAPTER 2. BASIC CONCEPTS 8 Tr B (ρ AB ) = k,l,m,n = k,l,m,n c km c ln a k a l Tr b m b n (2.20) c km c ln a k a l δ mn = k,l,m c km c lm a k a l = ρ A 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 φ ρ φ 0, for any φ 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 φ X φ 0 by special choices of φ. In particular, if φ has only two non-zero components, labelled by i and j, then the submatrix of X 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 Ψ H. The density operator of any pure state corresponds to a projection operator on that particular state, defined as which, like any projection operator, is idempotent: ρ = Ψ Ψ, (2.21) ρ 2 = ρ. (2.22) For example, the state of a qubit α 0 + β 1 is described by the density operator ρ = (α 0 + β 1 ) ( 1 β + 0 α ) = α αβ α β β 2 1 1, (2.23) or, in the matrix form, ( α 2 αβ ρ = α β β 2 ). (2.24) The diagonal elements ρ 00 = α 2 and ρ 11 = β 2 correspond, respectively, to the expectation values 0 ρ 0 and 1 ρ 1, giving the probabilities of observing bit values 0 and 1 respectively.

16 CHAPTER 2. BASIC CONCEPTS Ensemble interpretation of density operators Consider a quantum source which emits particles in states Ψ 1, Ψ 2... Ψ n with a priori probabilities p 1, p 2...p n. We will write it as an ensemble {p i, Ψ i }. In this case ( n n n ) S = p i Ψ i S Ψ i = p i TrS Ψ i Ψ i = TrS p i Ψ i Ψ i = TrSρ. (2.25) 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 i=1 ρ = n p i Ψ i Ψ i. (2.26) i=1 We call this operator the density operator that describes a mixture of pure states Ψ 1, Ψ 2... Ψ n with weights p 1, p 2...p n. The operator ρ is not a projector any more, ρ 2 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 Ψ i 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: ρ = n p i Ψ i Ψ i = i=1 n q i Φ i Φ i. (2.27) i=1 Note the sets of pure states { Ψ i, }, { Φ i, } are not in general orthonormal. In fact, unless there is any degeneracy in the values p i, only one such set can be orthonormal. Now take, for example, this particular density operator of a qubit: ρ = ( ). (2.28) It can be viewed as the mixtures of 0 and 1 with the probabilities 3 4 and 1 4, or as a mixture of Ψ 1 = and Ψ 2 = with probabilities p 1 = 1 2 and p 2 = 1 2. Even though states 0 and 1 are clearly different from states Ψ 1 and Ψ 2, according to Eq.(2.25), these mixtures behave identically under any any physical investigation, i.e. we are not able to distinguish between different mixtures described by the same density operator. 2.3 Entanglement We have previously introduced the concept of separable sates. However, there are states in H S which are not separable, i.e. they cannot be written as a simple tensor product of two states ψ A and ψ B (states for which c kl α k β l ). 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 Ψ = 1 2 ( ), (2.29) where and denote respectively spin up and spin down with respect to a chosen quantization axis.

17 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] ρ N = l C l ρ l 1 ρ l 2 ρ l 3... ρ l N, (2.30) where ρ l j is a state of subsystem j, and l C l = 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 ρ A and the state of B to be the pure state 0. The time evolution of the state of system S, ρ A 0 0, is given by ρ = Uρ A 0 0 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 ρ of system S: ρ A = Tr B (Uρ A 0 0 U ). (2.32) If we now consider an orthonormal basis i, i = 0, 1,..., for sub-system B, Eq.(2.32) becomes ρ A = i i U 0 ρ A 0 U i i E i ρ A E i, (2.33) where E i = i U 0 are operators, acting on sub-system A, and are trace-preserving: E i E i = 0 U i i U 0 = 0 U U 0 = 11. (2.34) i i Eq.(2.33) defines a linear map L that takes linear operators ρ A to linear operators ρ A. 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 Mathematical properties of superoperators A superoperator L : ρ ρ that takes density operators to density operators has the following properties [18]: L is trace-preserving, that is Trρ = TrL(ρ) = 1.

18 CHAPTER 2. BASIC CONCEPTS 11 L is linear, that is L(α 1 ρ 1 + α 2 ρ 2 ) = α 1 L(ρ 1 ) + α 2 L(ρ 2 ), α 1 + α 2 = 1. L is a is completely positive map, that is, if ρ is positive, then ρ = 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, ρ 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 ρ will be a positive operator (necessary condition for ρ 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 system, ρ ext ρ, the resulting operator ρ = ρ ext L(ρ) will still be a density operator 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 state ψ J = 1 N N i i i : 1 Λ ψ J ψ J ij i j Λ ( i j ) = ρ. (2.35) The bipartite state ρ encodes all the properties of the superoperator Λ, as from it we can learn how each density matrix element is transformed by Λ i j Λ ( i j ). (2.36) This establishes the equivalence between a completely positive map acting on density operators pertaining to a Hilbert space H of dimension d 2 1 and a density operator pertaining to a Hilbert space H H of dimension 4d 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 ψ S is defined as: ψ S = k,l c kl a k b l = i λ i a i b i, (2.37) where a i and b i are orthonormal bases for H A and H B, respectively, and λ i are non-negative real coefficients such that i λ2 i = 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 λ i 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

19 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(ϱ A ) S(ϱ AB ), S(ϱ B ) S(ϱ AB ), (2.38) where ϱ AB is a density operator of a composite quantum system and ϱ A and ϱ 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(ϱ) = Trϱ log ϱ and the Renýi entropy S(ϱ) = log Trϱ 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 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 ρ AB H A H B is entangled iff (11 L)ρ AB 0, for all L H 2 B. Unfortunately, very little is known about the structure of positive maps, even for small dimensional spaces like C. 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 ( α γ γ β ), (2.39) where the coefficients α, β, γ are chosen such that Eq.(2.39) is a valid density operator. It is sometimes convenient to represent the density operators of qubits as ρ = 1 + r σ 2 = 1 + i=x,y,z r iσ i, (2.40) 2 where 1 is the identity operator, r is a three dimensional vector of length smaller or equal to one and σ x = ( ) ( 0 i, σ y = i 0 ) ( 1 0, σ z = 0 1 ), (2.41) are the Pauli operators. The action of the transposition map on the density operator of the qubit is 2 Or conversely, (L 11)ρ AB 0.

20 CHAPTER 2. BASIC CONCEPTS 13 ( α γ γ β ) ( T α γ γ β ). (2.42) Suppose now that we consider a qubit, part of a larger system in the entangled state φ + = 1 2 ( ). (2.43) 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: T (2.44) The resulting density matrix has eigenvalues 1 2, 1 2, 1 2 and 1 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 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, X A, Y A and X B, Y B, respectively; Each measurement is chosen randomly and has two possible outcomes: +1 and 1. Let us consider the quantity Q = X A X B + Y A X B + Y A Y B X A Y B. Note that X A X B + Y A X B + Y A Y B X A Y B = (X A + Y A )X B + (X A Y A )Y B. (2.45) Since X A, Y A = ±1, it follows that either X A + Y A = 0 or X A Y A = 0, which in turn means X A X B + Y A X B + Y A Y B X A Y B = ±2. Hence, the expectation value of Q is

21 CHAPTER 2. BASIC CONCEPTS 14 E(Q) = p(x A, y A, x B, y B )(x A x B + y A x B + y A y B x A y B ) (2.46) x A y A x B y B p(x A, y A, x B, y B ) 2 = 2, (2.47) x A y A x B y B where p(x A, y A, x B, y B ) is the probability that, before the measurements are performed, X A = x A, Y A = y A, X B = x B, Y B = y B. If we further notice that E(Q) = E(X A X B ) + E(Y A X B ) + E(Y A Y B ) E(X A Y B ), we obtain the Bell inequality E(X A X B ) + E(Y A X B ) + E(Y A Y B ) E(X A Y B ) 2. (2.48) However, if we now compute the expectation value of Q, with X A = σ A z, (2.49) Y A = σ A x, (2.50) X B = σb z + σx B, (2.51) 2 Y B = σb z σx B, (2.52) 2 on the singlet state Ψ, we obtain that X A X B Ψ = Y A X B Ψ = Y A Y B Ψ = X A Y B Ψ = 1 2. (2.53) Thus, Q Ψ = 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 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 H A H B that obey the following properties: W is Hermitian, that is W = W. Tr(W a, b a, b ) 0, for all states a, b in H A H B, 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.

22 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] ρ = p ψ ψ + (1 p) 1 4, (2.54) where ψ ψ 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 ρ T B. The witness is given by the partially transposed projector onto this eigenvector. If the Schmidt decomposition of ψ is ψ = a 01 + b 10, with a, b 0, the spectrum of ρ T B is given by { 1 p 4 + pa 2, 1 p 4 + pb 2, 1 p 4 + pab, 1 p 4 pab}. (2.55) Therefore ρ is entangled iff p > 1/(1 + 4ab). The eigenvector corresponding to the minimal eigenvalue λ is given by Hence the witness W is given by φ = 1 2 ( ). (2.56) W = φ φ T B = (2.57) 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 Tr( φ φ T B ρ) = Tr( φ φ ρ T B ) = 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]. However 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.

23 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 ρ = i c i ρ i A ρ i B, (2.59) 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 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: ψ N = 1 2 ( N N ), (2.60) where iiii...i N = i N, i = 0, 1. In this case all the qubits are entangled with one another, but the state of any subset m of qubits is separable ρ m = Tr N m ( ψ N ψ N ) = 1 2 ( m m m m ). (2.61) These states are particularly useful for multi-party quantum communication protocols, such as multiparty quantum coin flipping [30] 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 W N = 1 N ( N N N N ). (2.62) 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] 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} C that obey the set of eigenvalue equations with the correlation operators K (a) φ {k} C = ( 1) k a φ {k} C, (2.63)

24 CHAPTER 2. BASIC CONCEPTS 17 K (a) = σ (a) x b nghb(a) σ (b) z. (2.64) Therein, {k a {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] 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)rˆn (θ) = exp(iα) exp( iθˆn σ ), (2.65) where α, θ are real numbers and Rˆn (θ) denotes a rotation by θ about the ˆn 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. ɛ(u, V ) = max (U V ) ψ δ, (2.66) ψ where V is the unitary implemented instead of U, ɛ(u, V ) is the unitary error and the maximum is taken over all normalized states ψ. A possible such set of gates is constituted by the Hadamard, π/4 and π/8 gates [18]: H = 1 2 ( ) ( 1 0, π/4 = 0 i ) ( 1 0, π/8 = 0 e iπ/4 ). (2.67) 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 1. A generic controlled- U gate is depicted in Fig The prototypical example of the entangling two-qubit gate is the controlled-not gate. It has the following matrix representation in the control, target basis: