Entanglement Distribution in Many-Body Quantum Systems

Transcription

1 UNIVERSITÀ DEGLI STUDI DI BARI ALDO MORO Dipartimento di Matematica Dottorato di Ricerca in Matematica XXVII Ciclo A.A. 2014/2015 Settore Scientifico-Disciplinare: MAT/07 Fisica Matematica Tesi di Dottorato Entanglement Distribution in Many-Body Quantum Systems Candidato: Sara Di Martino Supervisore della tesi: Prof. Paolo FACCHI Coordinatore del Dottorato di Ricerca: Prof.ssa Addolorata SALVATORE

2 Contents Introduction iv 1 Quantum Information In a Nutshell Quantum Mechanics from its Postulates Density Operator formalism Quantum B(D)its 6 2 Entanglement: Passion at a distance Basic Definitions Schmidt Decomposition Measures of Entanglement Von Neumann Entropy Two important states: GHZ and W Mixed States The Peres-Horodecki Criterion Entanglement Witnesses and Positive Operators Completely Positive Maps 22 3 Entanglement for Indistinguishable Particles Spaces and Operators for Indistinguishable Particles Spatially Separated Identical Particles Peres Cluster Separability The n-particle Case Some Considerations Identical Particles: A Subalgebras Approach 32 4 The Problem of Maximally Entangled States Purity and Maximally Entangled States Perfect MMES and Frustration The Potential of Multipartite Entanglement Statistical Mechanics Approach 43 ii

3 Contents iii 4.4 Cactus and Other Diagrams Computation and Degeneracy of the graphs Computation of the Series 54 5 Is Entanglement Monogamous? The Concurrence Monogamy Inequality for Three-Qubit Systems Two old friends: GHZ and W Monogamy Inequality: A Step Beyond Our Conjecture: Notation and Toolkit An Interesting Hint The Four Qubit Case An Example Four-Qubit Strong Monogamy: Results 71 Conclusions and Outlook 75 Appendix A Pauli matrices 78 Appendix B Local Operations and Classical Communication 79 Appendix C Some results in classical coding theory 82 Subject Index 85 Bibliography 87 Acknowledgments 95

4 Introduction You ll stay with me? Until the very end J.K. Rowling The first 30 years of the 19th century were characterized by a profound revolution in the physics world. After the formulation of the theory of Relativity physicists had to question their perfect knowledge of natural world. In the same period a large number of experiments were highlighting the failure of classical physics in describing phenomena at the microscopic level. In 1922, for instance, Otto Stern and Walther Gerlach, in their famous experiment [45], showed that particles have an intrinsic angular momentum that can assume only quantized values. Furthermore, they discovered that a measurement of this angular momentum along an axis destroys information about its other components. The results of experiments like this one were impossible to explain with the physics theories known until that moment. The first results on the formulation of the new theory are due to the seminal works of physicist as Werner Heisenberg, Max Born, Hermann Weyl, Paul Dirac, et al. In this flourishing scenario the famous work of Erwin Schrödinger, Discussion of probability relations between separated systems [77] takes place. Here, for one of the very first time, the problem of interacting quantum system was considered. In particular, Schrödinger showed that after the interaction the two systems cannot be described in the same way as before even if they separate again. The systems have become entangled. This new feature of Quantum Theory troubled many physicists of that time, who started calling this new implementation spooky non-locality. One of them was Albert Einstein who in his famous paper in collaboration with Boris Podolsky and Nathan Rosen [33] sought to demonstrate that Quantum Mechanics is not complete and to reformulate the theory in a less troubling way, introducing a hidden variable model. It was only 29 years later, in 1964, after the work of John Bell [7], that the existence of entangled states could be tested experimentally, quitting the skepticism on the non-locality of Quantum Mechanics. iv

5 Introduction v The interest on entanglement however blew up after the development of Quantum Information Science [10, 66]. This completely quantum property, indeed, was considered the main resource for a quantum speed-up in Information Theory and Computation. Since then an emergent research activity has taken place in the field. At the foundations of Quantum Information Theory lies the possibility of storing information in a quantum state of a physical system. As Information Theory basic unit is the bit, it is possible to define the quantum analogous of a bit, called qubit that represents the fundamental unit in discrete variable Quantum Information. Moreover, we can extend the concept of qubit to physical systems with more than two levels. The basic unit in this case is called qudit. The first chapter of this thesis will be devoted to give a general mathematical description of the frame we are working on. This chapter is intended to give a very basic description of Quantum Mechanics from its postulates and to introduce the basic elements of Quantum Information Theory. For this aim we will distinguish two types of quantum states: pure and mixed, the first one describing the perfect preparation of a state, the latter a more realistic situation in which we have an uncertainty due to possible experimental errors. The helpfulness of entanglement in Quantum Information has been proven in many fields such as quantum cryptography [34], quantum dense coding [15] and quantum teleportation [14]. Unfortunately, entangled states have in general very complex structure so that the detection and measurement of entanglement are very involved problems. In the second chapter we will formalize the description of entanglement both for pure and mixed states. In the first class the presence of entanglement between two parties of the system can be revealed using the quantum version of the Shannon entropy, introduced by von Neumann [87]. Precisely, as the Shannon entropy measures the amount of uncertainty we have on a probability distribution, the von Neumann entropy measures how mixed is a quantum state. In this way we can consider the state of one of the two parties of the system and recover informations about the total amount of entanglement looking at its von Neumann entropy. Unlike pure states, mixed states have a more complex structures due to their definition as ensembles of pure states. Nowadays, a universal way to detect entanglement in mixed state is not known, however many studies have been done in this direction and in the second chapter of this thesis we will describe some of the most important results on this topic. The main aim of this thesis is to study how entanglement distributes among the parties of a many-body quantum system. The third chapter will be devoted to the study of entanglement in the very peculiar case of identical particle systems. In this instance, one of the postulates of Quantum Mechanics imposes the symmetry or antisymmetry of the states. It will be evident that this assumption is in general in contrast with the structure of separable state

6 Introduction vi given in the second chapter. However, we will show that in the traditional paradigm of physicists working in different laboratories (so that the identical particles are spatially separated) the symmetry or antisymmetry of the state is completely irrelevant. A new definition of entangled states is needed only in the case in which the particles are in the same spatial region and we will prove that the criteria to detect entanglement in the case of identical particles agree with the spatial separation results. In chapter 4 we will study in detail the structure of entangled pure states of multipartite systems. From the von Neumann entropy we can define a new functional, called purity, that can reveal the amount of bipartite entanglement of a given state. Taking the average of this functional over all the possible bipartitions of the system we can define a functional, called potential of multipartite entanglement, that reveals the amount of multipartite entanglement of the system. We will use this functional to define and study the class of multipartite maximally entangled states. In particular, we will analyze the case of qudit systems, extending results obtained for qubits in this more general frame. Finally, the fifth chapter will focus the attention on one of the properties that makes entanglement a completely quantum feature: the so-called monogamy. We will show that multipartite entanglement cannot be freely shared among all the parties of a quantum system, formulating the so-called monogamy inequality for pure and mixed states of 3-qubit systems. In particular, for mixed states we will introduce a new measure of entanglement, the concurrence. In the second part of the chapter we will extend the monogamy inequality to systems composed by of more than three qubits, and we will conjecture a class of inequalities for n-party quantum states, suppling numerical evidence of their validity.

7 Chapter 1 Quantum Information In a Nutshell For those who are not shocked when they first come across quantum theory cannot possibly have understood it. Niels Bohr The idea of a Quantum Computer was introduced by Feynman [44] in He had in mind the project of a computer whose basic operations were the ones allowed by quantum mechanics (unitary operators and measurements). The reason behind this idea was to move around the difficulty one can encounter trying to model quantum mechanical phenomena using a classical computer. Later, in 1994, Shor [80] proved that an hypothetical quantum computer could solve the factorization and the discrete logarithm problem in a polynomial time. Since then the theory of quantum information has become one of the most active fields in science. But all that glitters is not gold: we will see that the precepts of quantum mechanics make the theory difficult to be implemented. One of the problems to overcome is the collapse of the wavefunction, i.e. a measure on the quantum system destroys the quantum state one is analyzing. A simply solution of this problem can be to duplicate the state before measuring it. However, the no-cloning theorem [90, 30] states that it is not possible to produce copies of the state perfectly matching the original one. In this chapter we will introduce quantum mechanics starting from its postulates. Furthermore, we will provide some of the basic concepts in quantum information theory equipped with all the mathematical background needed. Before going further, we want to stress that this chapter is only a brief selfconsistent introduction to the theory and is far from being an exhaustive introduction to quantum mechanics or quantum information theory. If the reader is interested in a more complete description, we suggest to have a look at [76, 71]. 1

8 2 1.1 Quantum Mechanics from its Postulates In order to give a complete description of Quantum Mechanics it is useful to consider its axioms, their fundamental principles and the mathematical tools needed to formulate them. For this aim we will follow the presentation o references [71, 83, 86]. Postulate 1. The configuration space of a quantum system is a complex separable Hilbert space H and the possible states of this system are represented by the vectors of H which have norm one. The normalization condition of the vectors representing the states reflects the interpretation as probabilities of the square moduli of their components. The notation we will use to indicate the state vectors will be the one introduced by Dirac in [31]. An element of the Hilbert space will be denoted by the symbol ψ, called ket, while its adjoint will be denoted by ψ, called bra. Following this notation the scalar product of two elements will be denoted by and their external product by ψ φ, (1.1) ψ φ. (1.2) From now on we will focus on finite dimensional Hilbert spaces, i.e. H = C n for some positive integer n. The reason of this choice will be clarified in the next sections. For a more complete discussion on Quantum Mechanics on general Hilbert spaces see [76, 83]. Since our aim is to describe a physical theory, the next step is to formalize the concepts of measurement of observables such as the spin. Postulate 2. Each observable a corresponds to a linear self-adjoint operator A (A = A ) acting on H. Postulate 3. The expectation value for a measurement of a, when the system is in the state ψ H, is given by ψ Aψ = Aψ ψ, (1.3) which is real for all ψ H. Since the operators we are considering are self-adjoint we can consider their spectral decompositions: A = λ k P k, (1.4) k

9 3 where λ k R are the eigenvalues of A and P k are the orthogonal projections on the eigenspaces of A satisfying: P k P l = δ kl P k, (1.5) P k = P k, (1.6) where δ kl is the Kronecker delta defined as: { 1 if l = m δ lm = 0 if l m. (1.7) In quantum mechanics the possible outcomes of a measurement of an observable corresponds to the eigenvalues of the operator representing the observable. If the state of the system is ψ, then ψ P k ψ, (1.8) is the probability to measure the outcome λ k. Moreover, after the measurement, if this probability is not zero, the state of the system becomes P k ψ ψ P k ψ. (1.9) This phenomenon, known as collapse of the wave function, rules the behaviour of two measurements associated with non commuting operators. Let a and b be two observables. If the corresponding operators, A and B do not commute then it is not possible to find a common basis of eigenvectors. This means that the two observables cannot simultaneously have definite values. Thus, performing a measurement of A will influence the outcome of a subsequent measurement of B. The last postulate we are going to state describes the time evolution of the system. Even if we are not going to use it during the thesis, we want to recall it for completeness. Postulate 4. The time evolution is given by a strongly continuous one-parameter unitary group U(t). The unitarity of the group is due to the conservation of probability. Indeed if we assume that the system is in the state ψ(0), at time t = 0, then its evolution can be written as ψ(t) = U(t) ψ(0). (1.10) The conservation of probability then reads ψ(0) ψ(0) = ψ(t) ψ(t) (1.11)

10 4 and the only consistent way to realize this task is with an unitary operator U(t). The evolution of the system is governed by the Schrödinger equation: i d ψ = H ψ, (1.12) dt where H is the generator of the one-parameter unitary group that governs the evolution. Moreover, unitarity of the evolution is equivalent to self-adjointness of its generator, as stated in Stone s theorem [61]: Theorem (Stone). Let U(t) be a strongly continuous one-parameter group of unitaries on a Hilbert space H. Then there is a unique self-adjoint operator H on H, called the generator of the group, such that U(t) = e iht/ for all t. The four axioms we describe here form the very first basis for a correct mathematical formulation of quantum mechanics. One can construct the whole theory keeping in mind these very basic principles that link the physics to the mathematics behind it. 1.2 Density Operator formalism In the previous section we have used the Dirac notation to express the physical states of the system. In this section we want to introduce a formalism that will give us a more general view of the theory. This necessity comes out from the awareness that not all the preparations of a state are perfect. Thus, sometimes the only information we have about the system is that it is in one of the states in the set { ψ k } k, and has probability p k to be found in the state ψ k. Since we are still describing physical states we have to take into account the normalization condition prescribed by the first axiom, so that the first requirement is p k = 1. (1.13) k If we know the exact state of the system, i.e. we are dealing only with one element of the Hilbert space ψ, we say that the state is pure, otherwise we call the state mixed. It can be easily understood that the formalism used until now is no longer convenient to describe the new features of the theory, so it is better to introduce a new notation. Each state describing the physical system can be associated to an operator ρ : H H, (1.14)

11 5 called density operator, in the following way. Consider the ensemble {p k, ψ k }, where the states ψ k are not necessarily orthogonal (notice that even a pure state can be seen as an ensemble in which for one k, p k = 1, while for j k p j = 0), then ρ = k p k ψ k ψ k. (1.15) Theorem A density operator has the following properties: 1. ρ is normalized, i.e. trρ = 1; 2. ρ is self-adjoint; 3. ρ is non-negative. Proof. The proof of the normalization goes as follows: tr(ρ) = tr( k p k ψ k ψ k ) = k p k ψ k ψ k = k p k = 1, (1.16) where we used the normalization condition of the states in the ensemble. Self-adjointness comes naturally if we notice that the p k s are probabilities and, therefore, real numbers. To prove the positivity of ρ we have to prove that ψ H, ψ ρ ψ 0. ψ ρ ψ = k p k ψ ψ k ψ k ψ = k p k ψ ψ k 2 0, (1.17) where we used the non negativity of the p k. All the operators satisfying the three properties in theorem are well-defined density operator. As a remark, we notice that a density operator corresponds to a projection, i.e. ρ 2 = ρ, if and only if the corresponding state is pure. It is worth noticing that different ensembles can give rise to the same density operator, as showed by the following example. Example Consider the finite dimensional Hilbert space H = C 2. The canonical (computational) basis for this space can be expressed as: ( ) ( ) :=, 1 :=. (1.18) 0 1 The two ensembles {( ) ( )} 3 1 4, 0, 4, 1, (1.19)

12 6 and {( ) ( )} , , 2, , (1.20) are both associated to the density operator ρ = (1.21) 4 Let us suppose to have a bipartite physical system S, i.e. a system composed by two subsystems A, B 1 such that S = A B with A B =. In this case the Hilbert space that describes the system can be written as the tensor product of two Hilbert spaces, H = H A H B, describing the two subsystems. Moreover, given an orthonormal basis for H A, { ψ (k) A }, and an orthonormal basis for H B, { ψ (k) B }, we can construct an orthonormal basis for H { ψ (kj) AB = ψ(k) A ψ(j) B }. (1.22) We can ask if, given the density operator of the whole system, is possible to obtain the density matrices of the two subsystems. The answer to this question is given by the partial trace. If the density operator of the total system is ρ = k p k φ (k) AB φ(k) AB, (1.23) then the partial trace over subsystem B acts as ρ A = tr B (ρ) = k ψ (k) B ( j ) p j φ (j) AB φ(j) AB ψ (k) B. (1.24) It is easy to prove that the operator ρ A is still a density operator and is called reduced density operator of subsystem A. From now on we will use the word state referring to both vectors of the Hilbert space and density operators. The meaning will be evident from the context. 1.3 Quantum B(D)its At the beginning of this chapter we have stressed that in this thesis we will focus on finite dimensional Hilbert spaces. The reason of this interest lies in a direct transposition from classical to quantum information theory. In the first one the fundamental 1 Notice that we have used the same notation to indicate the operators (in the previous section) and the parties of the system, however the meaning of the symbols will be always clear from the context.

13 7 unit is the binary digit (bit), i.e. a quantity that can assume only two values, generally indicated with 0 and 1. We can implement a bit with every physical system that can exist in either of two possible distinct states. The equivalent of the bit in quantum information theory is the quantum bit (qubit) [71]. The very first difference we can notice between bits and qubits is that the former, being a quantum object, needs to be identified as an element of a Hilbert space. In particular, the one we consider is the two-dimensional space H = C 2 where the values of the bit are translated into two orthormal vectors that constitute a basis generally addressed as computational basis, eq. (1.18), that we recall here: 0 := ( 1 0 ), 1 := ( 0 1 ). (1.25) Mathematically the state of a qubit is an element of H, i.e. a complex superposition of the two vectors of the basis: ψ = α 0 + β 1, (1.26) with α, β C and where the normalization condition is expressed as α 2 + β 2 = 1. (1.27) One of the possible physical implementations of a qubit is a system composed by a spin- 1 particle, like for instance an electron. We can encode two orthogonal 2 directions of one of the components of the spin in { 0, 1}, obtaining as states all the possible orientation of the spin in the three-dimensional space. A density operator of one qubit is essentially a 2 2 Hermitian matrix. It can be expressed using a suitable basis for the space, that is {I, σ x, σ y, σ z }, where I is the identity matrix and the σ k s, for k = x, y, z, are the so-called Pauli matrices: ( ) ( ) ( ) i 1 0 σ x :=, σ 1 0 y :=, σ i 0 z :=. (1.28) 0 1 (for a more detailed description of the Pauli matrices see appendix A). Thus every density operator can be written as ρ( p ) = 1 (I + p σ), (1.29) 2 where p is the vector of the components of ρ, chosen in such a way that ρ( p ) satisfies all the requirements for a density operator, i.e. 0 p 1. Moreover, since the Pauli matrices are traceless, here we need the coefficient 1/2 for normalization reasons. Since the connection of vectors p with density operators is a one-to-one correspondence we can associate each one-qubit state with a point in a unit 3-ball called

14 8 0i i = 1 2 0ih ih1 1i Figure 1.1: The Bloch sphere has all the possible pure states on the boundary while in the centre lies the maximally mixed state. Bloch sphere, see fig In particular, the points on the boundary corresponds to density matrices with vanishing determinant and therefore to pure states. On the other side, the centre of the sphere is the maximally mixed state: ρ = (1.30) 2 Until now our discussion was focussed on two-level systems (qubits). Even considering finite dimensional spaces we can ask what happens if we move from the space C 2 to C d, with d a positive integer. In analogy with the two-dimensional case, we call this kind of systems qudits, if d is the dimension of the corresponding Hilbert space. As we did for qubits we can define the canonical (computational) basis: 0 := , 1 := ,..., d 1 := (1.31) The transition from qubits to qudits can seem a mathematically nuisance; nevertheless this enlarged space describes physical systems with more than two levels. We can indeed consider a particle with a generic spin s and in this case a measure of one of its component can give only 2s + 1 values: m s with m s = s, s 1,..., s.

15 Chapter 2 Entanglement: Passion at a distance 1 I would not call that one but rather the characteristic trait of quantum mechanics E. Schödinger Entanglement is one of the most intriguing features of quantum mechanics 2. It was first introduced in 1935 by Schrödinger [77]: When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or ψ-functions) have become entangled. Since then it played a crucial role for applications in quantum information such as quantum teleportation [14] and quantum key distribution [11, 13, 34], but even in a variety of other areas ranging from quantum field theory to condensed matter, statistical physics, thermodynamics, and biology [4, 59]. Despite its central importance, however, the physical understanding and mathematical description of its essential 1 This expression was coined by R. D. Gill in [48] 2 This is one of the most used incipit in papers regarding entanglement. It can seem an overstatement to convince ourselves that we are studying something cool, but we hope to convince the reader that it is not only our belief. 9

16 10 characteristics remain highly nontrivial tasks, particularly when many-particle systems are analyzed. In this chapter we will provide the basic definitions that characterize this quantum property and we will discuss some of its main aspects. Moreover, we will analyze the progress that has been done in its mathematical description and comprehension. We want to stress once again that the systems we consider will be finite dimensional unless differently specified. For a review on continuos variable systems see [1, 43]. 2.1 Basic Definitions We start this section, and so the treatment of entanglement, from its very basic definition. Definition 2.1. A pure state ψ H = H A H B is called separable if there exist two pure states ψ A H A and ψ B H B, such that ψ = ψ A ψ B. Otherwise the state is called entangled. The relevant difference between entangled and separable states lies in the impossibility to create the first ones locally. The transition from a separable to an entangled state cannot involve only local operations. Example The very first example one can give of entangled pure states are the states of the Bell basis[66]. They are states of two qubits (and this is the minimum we can ask when we want to talk about entanglement). The physical scenario in which two two-level systems are interacting is described by the Hilbert space H = C 4 = C 2 C 2. The four states, expressed in the canonical basis, φ ± = 1 2 ( 00 ± 11 ); (2.1) ψ ± = 1 2 ( 01 ± 10 ), (2.2) are called Bell States. It is easy to prove that these states are not separable. Indeed let us take, for instance, the state ψ + (we can do the same reasoning for the other three states). If we want to express it as a tensor product of two states, ψ A = j a j j and ψ B = j b j j, we have to solve the system { ak b j = 1 2 a k b k = 0 for k j (2.3) and it is easy to see that it has no non-trivial solutions.

17 11 An important property of the Bell states is that they form an orthonormal basis of entangled states for the Hilbert space H. Essentially, the main difference between classical correlations and entanglement is that in the latter case the state provides more information about the global system than about its two parties, whereas this can never happen for a classical state. In the case explained in the previous example, for instance, each of the states considered is pure, but if we trace over one of the two subsystem we remain with a state that is completely mixed. In other words, the state of the global system is deterministic, while the best we can say about the subsystems is that they can be found in two different states with equal probability. Following the definition of entanglement for pure states we can extend it to mixed states. Definition 2.2. A mixed state ρ : H H is said separable if it can be written as where ρ (k) X entangled. ρ = k p k ρ (k) A ρ(k) B, (2.4) are density matrices on H X, with X = A, B. Otherwise the state is said The question that here arises naturally is how to distinguish entangled states from separable ones. The interesting thing is that, despite the simplicity of the definition it is in general quite hard, at least for mixed states, to check whether a state is separable or not. Indeed, one can think that the entanglement of a quantum state can be easily found by looking at the structure of the state, i.e. at its eigenvectors. Actually, this is not true as the next example shows. Example The separable state ρ = 1 4 ( )( ) + 1 ( )( ), (2.5) 4 has as eigenvectors: φ 1 = 1 2 ( ) (2.6) φ 2 = 1 6 ( ), (2.7) (notice that these are not all the eigenvectors of ρ but only the ones relative to nonzero eigenvalues). In particular the first one is one of the Bell states that we proved to be entangled. A more difficult problem arises when one asks about the amount of entanglement. As we will see there are answers to this question for pure states while it becomes far more involved in the case of mixed states.

18 Schmidt Decomposition Let us consider the following theorem for pure states of bipartite systems. Theorem (Schmidt Decomposition). Given a pure state ψ of a bipartite system there exist orthonormal states ψ (k) A of subsystem A, and orthonormal states φ(k) B of subsystem B such that ψ = λ k ψ (k) A φ(k) B, (2.8) k = 1, known as Schmidt coef- where λ k are non-negative real numbers such that k λ2 k ficients. Proof. Let { ψ (k) A } be an eigenbasis of the reduced density operator of subsystem A. Then ρ A = λ 2 k ψ (k) A ψ(k) A, (2.9) k for some real λ k, with k λ2 k = 1. At the same time we can take an orthonormal basis { φ (j) B } for H B. The state ψ H A H B can be written as ψ = k,j α k,j ψ (k) A φ(j) B k ψ (k) A (k) φ, (2.10) B (k) where φ B j α kj φ (j) (k) B. Notice that here the vectors { φ B } are no more normalized. If we compute once again the reduced density operator ρ A we find: ( ) ρ A = tr B ( ψ ψ ) = tr B ψ (k) A ψ(j) (k) (j) A φ B φ B. (2.11) k,j Then, since ( ) (k) (j) tr B φ B φ B = i φ (i) B (k) (j) φ B φ B φ(i) B = φ (j) B (k) φ, (2.12) B we can rewrite eq. (2.11) as ρ A = k,j (j) (k) φ B φ B ( ψ(k) A ψ(j) A ). (2.13) Comparing eqs. (2.9) and (2.13) we have: (j) (k) φ B φ B = λ2 kδ kj, (2.14)

19 13 where δ kj is the Kronecker delta. This means that the vectors { and can be normalized: φ (j) B B = λ j In conclusion we have that is exactly eq. (2.8). ψ = k (j) φ B } are orthogonal (j) φ B. (2.15) λ k ψ (k) A φ(k) B, (2.16) Definition 2.3. The number of k s such that λ k 0 is the Schmidt number of ψ. Let us take a closer look at the Schmidt decomposition of ψ. If we take the partial trace over one of the two subsystem we have ρ A = tr B ( ψ ψ ) = k ρ B = tr A ( ψ ψ ) = k λ 2 k ψ (k) A ψ(k) A, λ 2 k φ (k) B φ(k) B. (2.17) Suppose that the two subspaces H A and H B have different dimensions, say m and n respectively, with m > n. Since the decomposition (2.17) holds for both ρ A and ρ B then, ρ A has 0 as an eigenvalue, possibly degenerate with at least m n degree of degeneracy. In particular, they are equally mixed. The Schmidt number gives a suitable way to recognize if a pure state is entangled or not. Indeed separable pure states give rise to reduced density operators that are also pure. Vice versa, if the state is entangled then the reduced state has to be necessarily mixed. We will see that this consideration is at the very basis of the study of entanglement in pure states. 2.3 Measures of Entanglement We can introduce a class of functions suitable to measure the amount of entanglement that a state presents [70]. Definition 2.4. A function E : B(H) R is a measure of entanglement if it satisfies: 1. 0 E(ρ) 1, for all density operators ρ B(H), and E(ρ) = 0 if and only if ρ is separable; 2. E is convex; 3. E is non increasing under Local Operations and Classical Communication (LOCC).

20 14 Sometimes this may be changed according to the different aspects of entanglement one wants to catch. There can be found, for instance, functions that are considered measures of entanglement that are not convex. On the other side there are more requirements that can be written in the list, such as the additivity E(ρ σ) = E(ρ) + E(σ), (2.18) or the continuity of the function. Nevertheless, from now on we will focus on the three points of the definition and we will analyze them. The first condition requires that all the possible measures agree on separable states. Moreover a normalization condition is added, for simplicity. The convexity E( p k ρ k ) p k E(ρ k ), (2.19) k k is a property that is very convenient mathematically and physically. It captures the loss of information we have moving from a selection of identifiable states to a mixture of these states. To explain the third property we have to introduce first the concept of LOCC [21]. In order to formalize the possible operations that can be performed on a quantum system we have to take into account that quantum parties are possibly distributed in distantly separated laboratories. The idea of local operations (LO) is that they can be performed in well-controlled environments without the disturbance induced by communication over long distances. Moreover we can avoid the problem of making such operations totally independent simply allowing people in different laboratories to classically communicate (CC). In this way they can coordinate the quantum operations they are performing, see fig This is, in words, what we intend for local operations and classical communication. For a mathematical description of LOCC see appendix B. Alice Bob Figure 2.1: Alice and Bob performing LOCC protocols in different laboratories As can be easily understood LOCC can only create classical correlations among the parties, so that via LOCC we can only create separable states. This explains the reason of the third requirement in the definition. Nevertheless, we can make this

21 15 request weaker asking that E does not increase on average under LOCC. In this case E is called entanglement monotone. An interesting feature that comes out from the definition is the invariance of the measure under local unitary operations (notice that this kind of operations are local and invertible). The importance of this consideration lies in the fact that a unitary operation does not change the state from a mathematical point of view so it is reasonable to ask that its level of entanglement remains the same Von Neumann Entropy Regarding pure states we can always measure the entanglement between two parties of a given bipartition looking at the mixedness of the reduced density matrices. For this aim it is useful to introduce a measure for the entropy of the state of a system. In classical information theory, given a classical probability distribution X, the Shannon entropy [79] measures the amount of information we gain (on average) learning the value of X. In other words it measures the amount of uncertainty we have on X before learning its value. Essentially, given the probability distribution X = {p 1,..., p n } (with k p k = 1), its Shannon entropy is H(X) H(p 1,..., p n ) k p k log p k, (2.20) where the logarithm is taken with base two and 0 log 0 is defined to be 0. It is easy to see that if there is only one k such that p k = 1 than H(X) = 0, meaning that we are not gaining information learning the value of the variable (we are sure of that before learning it). Example The Shannon entropy associated to the toss of a coin is ( 1 H(X) = 2 log log 1 ) = 1, (2.21) 2 meaning that in this case the uncertainty is the biggest possible. We can generalize the definition of Shannon entropy for quantum states, replacing classical probability distributions with density operators. If {λ k } is the set of the eigenvalues of the state ρ then we can define the von Neumann entropy as S(ρ) = λ k log λ k, (2.22) k where we impose the same conditions as above. Another way to express the von Neumann entropy, that does not depend directly on the eigenvalues of ρ is: S(ρ) = tr(ρ log ρ). (2.23)

22 16 Example The von Neumann entropy of the state is ρ = 1 ( ) (2.24) 2 S(ρ) = 0. (2.25) If we take a closer look on the state in the previous example we can notice that it is pure ρ = 1 ( )( ). (2.26) 2 In fact we can understand the von Neumann entropy as a measure of the mixedness of the qubit state: it is 0 for pure states, 1 for maximally mixed states. As we said in section 2.2 we can easily identify bipartite entanglement in pure states looking at the Schmidt number of the reduced density operator. Therefore, the von Neumann entropy of the reduced density operator is a good tool to discriminate separable from entangled states. Moreover, for bipartite pure states this one is the only function that satisfies the conditions in definition 2.4 [58]. From a computational point of view it can be more convenient to introduce a new functional called purity π(ρ) = tr(ρ 2 ), (2.27) that is linked to the first order expansion of the von Neumann entropy: S(ρ) = tr(ρ log ρ) tr(ρ(ρ I)) = 1 π(ρ). (2.28) As for the von Neumann entropy, it is possibile to use the purity to detect and, in a certain sense, measure 3 the entanglement between the parties of the bipartition. Indeed, we can define the local purity: π AB π tr A π tr B, (2.29) as the purity of the reduced density matrices. From the purity it is possible to define many functions that detect entanglement, see for instance [29]. In the next chapters we will discuss deeply this function and we will examine its properties. 3 Notice that the purity is not a proper measure of entanglement according to the definition we gave. Indeed, it does not satisfy the first condition in definition 2.4.

23 Two important states: GHZ and W Many examples of entangled states can be given, however there are states that are considered characteristic for their properties. As a very first example in section 2.1 we saw the Bell states. For three qubits we have two pure states that are relevant to describe entanglement, i.e. the W-state [32]: W = 1 3 ( ), (2.30) and the GHZ-state [52]: GHZ = 1 2 ( ). (2.31) From now on we will refer to these states considering not only the states themselves but even their equivalence classes under local unitary operations. Let us begin with the GHZ-state. Since this state is symmetric under permutations of the three qubits, we can focus on one of them and then extend the results obtained to the other two. If we trace over one of the three qubits we obtain the reduced density operator: Therefore, the purity of the reduced state, that is: ρ (GHZ) AB = 1 ( ). (2.32) 2 π(ρ (GHZ) AB ) = tr(ρ (GHZ) 2 AB ) = 1 2. (2.33) Moreover, the reduced density operator represents a separable state. This means that even if the GHZ-state presents the minimum of the purity, i.e. a maximal level of entanglement, its reduced states are all separable. In this case the entanglement is described as fragile [32]. The case of the W-state is slightly different. Indeed, if we trace over one of the three qubits (notice that even this state is invariant under permutations of the qubits), we obtain: ρ W AB = 1 ( ). (2.34) 3 The corresponding purity π(ρ W AB) = tr(ρ (W ) 2 AB ) = 5 9 > 1 2, (2.35) is not minimal, i.e. the entanglement of this state is lower than the one of the GHZstate. However, the entanglement in the W-state survives tracing over any of the three qubits. In this case the entanglement is called robust [32].

24 Mixed States The problem of detecting and measuring entanglement in mixed states is much more complex than in pure states. In the previous section we have solved the problem of identifying entangled pure states looking at the mixedness of the states of the subsystems. On the contrary, the complete characterization of the convex set of separable mixed states reveals itself to be extremely involved [93]. However, despite all the difficulties, many efforts have been done in this direction. In particular, as we will see in the next sections, nowadays we are able to detect and measure entanglement in systems with a low number of particle and with low dimensions in general. It is easy to imagine from where all the difficulties come from. In fact, the characterization of entanglement in mixed states cannot in general be done looking only at the structure of the state. We have shown, in example 2.1.2, that even the eigenvector decomposition of the state is not a privileged one. Therefore, to identify entanglement in a quantum state one has to look at all the possible pure-state decompositions of the state. In principle, this kind of decomposition can be made up of an arbitrary number of states. Luckily, the Carathéodory theorem [20, 82] bounds the dimension of the ensemble we are searching for. Theorem (Carathéodory). Let S R n. If x R n lies in the convex hull of S, i.e. is a convex combination of elements of S, then x lies in the convex hull of a subset S 1 S, with S 1 n + 1. The Carathéodory s theorem helps us to reduce the dimension of the ensemble in the decomposition. Indeed, we can always assume that the number of states in this set exceeds by 1 the dimension of the system. Of course this does not solve the problem, even if it makes things less difficult. Another interesting point is that, at least at the theoretical level, there is a way to extend a measure of entanglement valid for pure states, to mixed states. This tool is called convex roof and acts as follows. If E is a suitable measure of entanglement for pure state we can consider E(ρ) = min p k E( ψ k ), (2.36) {p k, ψ k } where the minimum is taken over all the possible pure states decompositions of ρ, i.e. ρ = k p k ψ k ψ k. Notice that the convexity requirement for the function E is crucial to define the convex roof, indeed we have k E(ρ) = E( k p k ψ k ψ k ) k p k E( ψ k ). (2.37)

25 19 The ensemble that minimize E is called optimal decomposition of ρ. It is not hard to see that finding this optimal decomposition is not so simple in general and the minimization in the definition is not easy to tackle. 2.5 The Peres-Horodecki Criterion The very first step toward the concrete detection of entanglement was done in 1996 simultaneously by Peres [69] and the Horodecki s [57]. Their criterion, known as Positive Partial Transposition Criterion (PPT), is based on the partial transposition operation that, roughly speaking, consists in transposing only the part of the operator related to given subsystem. To formalize this concept mathematically let us use latin letters to refer to subsystem A and greek letters to refer to B. In particular, if { m } is an orthonormal basis for H A, and { µ } is an orthonormal basis for H B, then we can write the element of ρ in this basis as: ρ mµ,nν = m, ν ρ n, ν. Using this new notation the partial transpose of ρ with respect to subsystem A is the matrix ρ P T mµ,nν ρ nµ,mν. (2.38) Theorem (Peres-Horodecki criterion). If a quantum state ρ is separable with respect to a given bipartition, the operator obtained partially transposing the state with respect to that bipartition is positive. Proof. Let us suppose the system S can be divided as S = A B. From the definition 2.2 of separability, ρ can be written as: ρ = k p k ρ (k) A ρ(k) B. (2.39) If we partially transpose ρ with respect to subsystem A we obtain the operator ρ P T = k p k (ρ (k) A )T ρ (k) B. (2.40) Since the initial state was separable, the ρ (k) A themselves are density operators k. Therefore, (ρ (k) A )T are also positive operators so that the operator ρ P T is still positive. The necessary condition expressed in the previous theorem is valid for Hilbert spaces of arbitrary dimension. Moreover, it was proven in [57] that it becomes sufficient for low dimensional system as stated in the next theorem. Theorem A quantum state ρ acting on the space C 2 C 2 or C 2 C 3 is separable if and only if its partial transpose is a positive operator.

26 20 Starting from these two theorems we can define the functional N (ρ) = 2 i σ i (ρ P T ), (2.41) where {σ i (ρ P T )} is the set of the negative eigenvalues of ρ P T. This functional is called Negativity and gives us a simple operational way to identify the entanglement of a given state. 2.6 Entanglement Witnesses and Positive Operators In this section we will introduce another criterion to detect entanglement. Differently to the one in the previous paragraph this one will be non-operational, in the sense that it is state dependent. Just before introducing this criterion it can be helpful to recall some mathematical concepts [73]. Definition 2.5. Two sets A and B in a locally convex space are separated by an hyperplane if there is a continuous real-valued functional f and an a R such that f(x) a, x A, and f(x) a, x B. Moreover, if the inequalities are strict then A and B are strictly separated. The central point will be the next corollary of the Hahn-Banach theorem [73]. Theorem Let A and B be disjoint convex sets in a Banach space. Then if A is compact and B is closed, they can be strictly separated by a hyperplane. With the help of this theorem, applied to the set of separable states and the set composed by an entangled state ρ, we can prove the following [19]. Theorem For every entangled state ρ acting on H A H B there exists a Hermitian operator W such that tr(w ρ) < 0, (2.42) and for every separable state σ. tr(w σ) > 0, (2.43) Proof. It is easy to prove that the set of separable states is convex and compact. Moreover, the set composed by only one point is compact. Since we are in the hypothesis of theorem we can find an hyperplane that strictly separate ρ from the set of separable states, i.e. a real functional f such that f(ρ) < β < f(σ), (2.44)

27 21 for every separable state σ. Moreover, for the Riesz representation theorem [73], there exists a unique Hermitian operator W such that for all the states ω. Eventually, we can take obtaining and for all separable states σ. f(ω) = tr(ω W ), (2.45) W = W βi, (2.46) tr(w ρ) = tr( W ρ βiρ) = f(ρ) β < 0, (2.47) tr(w σ) = tr( W σ βiσ) = f(σ) β > 0, (2.48) A graphical representation of this theorem can be found in fig Moreover, ρ W Separable states Figure 2.2: Graphic representation of an entanglement witness. from this theorem it comes out the following definition. Definition 2.6. A Hermitian operator W is called an entanglement witness if and only if there exists an entangled state ρ such that tr(w ρ) < 0 and tr(w σ) > 0 for every separable state σ. It is easy to note the theoretical power of the entanglement witnesses. Unfortunately, theorem does not give us an insight on how to construct them given a state. Nevertheless, there are many results on how to construct witnesses for some classes of state. For a review on entanglement witnesses see [23, 24].

28 Completely Positive Maps The concept of entanglement witness can be embedded in the more general frame of positive maps. We need to consider operators acting on the space of bounded operators on an Hilbert space (B(H)), the so-called supeoperators or maps, see appendix B. Definition 2.7. A linear map Λ : B(H) B(H) is called positive if it maps positive operators into positive operators, i.e. for all ρ 0. Λ(ρ) 0, (2.49) This definition is motivated by the need of mapping quantum states into quantum states. For the same reason, generally speaking the positiveness is associated with the request for the map to preserve the trace. Nevertheless, sometimes it is not sufficient to deal only with positive maps, for we need a stricter requirement. Definition 2.8. A positive linear map Λ : B(H) B(H) is called completely positive if the map Λ I A is positive for every finite tensor product extension, with I A the identity map on the space B(H A ). An example of a completely positive map is the unitary evolution a quantum system. In this situation the complete positivity describes a system evolving without interacting with the environment. On the other hand the partial transposition is an example of a map positive but not completely positive. We will see that positive but not completely positive maps are important in entanglement theory since they give an insight about the entanglement of the system. Indeed, following the work of Woronowicz [91, 92] on positive maps, a link can be found between Hermitian operators and completely positive maps given by the Choi Jamiołkowski isomorphism [22, 60]. It can be described as follows. To every linear map Λ, given an orthonormal basis { e j } j for the Hilbert space H, we can associate the state ρ Λ = j,k e j e k Λ( e j e k ). (2.50) This association is an isomorphism that leads us to the following [57]. Theorem A Hermitian operator W is an entanglement witness if and only if the corresponding map via the Choi-Jamiołkowski isomorphism is a positive but not completely positive map.

29 23 Consequently we have the link between positive maps and entangled quantum states. Theorem A quantum state ρ is entangled if and only if there exists a positive map Λ such that (Λ I A )ρ 0. To conclude this discussion about entanglement witnesses and positive operators we want to show how considering low dimensional systems simplifies the whole thing even in this case. On a two-dimensional Hilbert space any positive map Λ can be decomposed as Λ = Λ 1 + Λ 2 T B, (2.51) where Λ 1 and Λ 2 are completely positive maps and T B is the partial transposition with respect to subsystem B. One of the important consequences of this decomposition is that it makes the Peres-Horodecki criterion a necessary and sufficient condition. In fact, the lack of complete positivity in a map is due only to the presence of partial transposition, that is a positive operator if restricted to separable states.

30 Chapter 3 Entanglement for Indistinguishable Particles Ci sono soltanto due possibili conclusioni: se il risultato conferma le ipotesi, allora hai appena fatto una misura; se il risultato è contrario alle ipotesi, allora hai fatto una scoperta. E. Fermi A composite quantum system can include subsystems that have all the same physical properties. In this case any measurement that can be done on the system treats them exactly in the same way and cannot be affected by permuting their order. This kind of subsystems are called identical, or indistinguishable and their state undergoes the symmetrization postulate [65]: Postulate 5. The states of a system containing N identical particles are necessarily either all symmetric or all antisymmetric with respect to any permutation of the particles. Since the early days of quantum mechanics it has been realized that indistinguishability of quantum particles is a fundamental feature of the theory and has important consequences for the interpretation of physical phenomena [65]. In particular the study of correlations between identical particles has attracted a lot of attention in recent years. The indistinguishability of the particles seems in contrast with the Hilbert space tensor product structure on which the concept of entanglement is based. To overcome this problem different definitions of entanglement have been considered, based on different ideas. In [46, 47], for instance, the properties of the Schmidt eigenvalues and the von Neumann entropy of the single-party reduced density operators are studied. A different approach to the problem has been showed by Benatti 24

31 25 et al. [8] who rephrased the concept of separability in terms of commuting algebras of observables. Moreover, the use of representation theory of the symmetric group can lead to distinguish the entanglement of pure states using a proper generalization of the notion of Schmidt rank [51]. Finally, an approach through the GNS construction [17] has been recently proposed in [6] based on the use of the general idea of the restriction of states to subalgebras. Even if in the last years entanglement for identical particles has been studied a lot, sometimes the symmetry and antisymmetry of the states create problems at a conceptual level, especially when one refers to the standard paradigm of Alice and Bob making experiments in different labs. Is this paradigm valid? Is correct to consider the definition of entanglement we gave in the previous section? This chapter, based on [27], will be devoted to give an answer to these questions. In particular, we will elaborate on an idea introduced by Peres [68] showing that if we impose that the identical particles are spatially separated the indistinguishability does not affect at all the structure of the reduced density operator. Moreover, the condition of spatial separability has been already recognized as the natural request in order to recover the distinguishability of identical particles in experiments with fermions and bosons [54, 53]. 3.1 Spaces and Operators for Indistinguishable Particles Let us consider a quantum system composed of n particles. Here any particle has two relevant sets of degrees of freedom related to its spin and its position in space. Therefore, the Hilbert space H k describing the state of the k-th particle is itself the product of a spin space, that we will denote by h k, and a spatial space l k : H k = l k h k. (3.1) The spin space of each particle is finite dimensional and its dimension depends on the spin of the particle itself. Namely, If the k-th particle has spin s k then h k = C 2s k+1 is a 2s k + 1 dimensional complex vector space. On the other side, the spatial space is the space of square integrable functions l k = L 2 (R 3 ). Therefore, every pure state of the system is described by a normalized wave function ψ(x 1, σ 1,..., x n, σ n ), with x k R 3 and σ k = s k, s k + 1,..., s k 1. 1 The notation that we are using here to define the component of the finite dimensional space is slightly different from the one that we have used until now. This mismatch is due to the fact that here we want to underline the physical properties of the spin and not only the mathematics that lies behind it.

32 26 According to the previous setting, if the system is composed of n identical particles (same mass m and spin s), the space we have to consider is the tensor product of n times the one-particle Hilbert spaces H total = H n = (l h) n, (3.2) The indistinguishability implies that any property of the composite system has to be invariant under a permutation of the particles and, as we said, this leads to a strong limitation on the possible states of the system. This condition can be translated in the fact that the admissible vectors have to belong to a proper subspace of H total. The symmetric group S n acts on H total has, W π ψ(x 1, σ 1,..., x n, σ n ) = ψ(x π(1), σ π(1),..., x π(n), σ π(n) ), π S n. (3.3) Such action provides a unitary representation of S n in H n in the sense that: W π W τ = W πτ and W π = W 1 π = W π 1, π, τ S n. (3.4) Then, the indistinguishability of the n particles can be recast in the requirement that π S n the W π leave the state ψ unchanged apart from a constant W π Ψ = λψ. (3.5) The only two values of λ that are consistent with the linearity of quantum mechanics [65] are λ = ±1. So we arrive at the symmetrization postulate for which the vector states describing n particles have to belong to the symmetric subspace H n = {ψ H n : W π ψ = ψ, π S n }, (3.6) in which case the particles we are describing are bosons, or to the antisymmetric subspace H n = {ψ H n : W π ψ = sgn(π)ψ, π S n }, (3.7) in which case the particles are fermions. Formally, we can use the orthogonal projections [18] Π + = 1 W π, and Π = 1 sgn(π)w π (3.8) n! n! π S n π S n to map the Hilbert space H total onto the above subspaces: H n = Π + H total, and H n = Π H total. (3.9) To what concern the algebra of operator, we have to impose that the outcomes of a measurement of an n-particle state ρ have to be invariant with respect to any

33 27 permutation on the state W π ρw π. Even in this case the indistinguishability condition imposes that not all operators in B(H n ) are appropriate observables of the system. In particular, the admissible observables X are the ones that lie on the subalgebra of the exchangeable operators: X = W π XW π. (3.10) Every one-particle operator A B(H) can be lifted to an observable of n indistinguishable particles using the map, called second-quantization functor dγ : B(H) B(H n ), that symmetrizes the operator in the n particles space [18] dγ(a) = A I I + I A I + + I I A, (3.11) where I is the identity operator on the space of a single particle. It is evident from the definition that the symmetric and antisymmetric subspaces are left invariant by the operator dγ(a). We can generalize this definition to any k-particle operator, with k n. For n-particle operators the explicit expression of the second quantization functor is dγ(a 1 A 2 A n ) = π S n A π(1) A π(2) A π(n) (3.12) As a particular case, consider ( ) Π spatial = dγ (P 1 I) (P n I) (3.13) where the P i s act on the spatial space l, while the identity operators act on the spin space h. It is easy to verify by inspection that Π spatial is a projection operator provided that the P i s are orthogonal projections P i P j = δ ij P i. 3.2 Spatially Separated Identical Particles The aim of this section is to show mathematically the validity of what can be consider a simple intuition. The idea is that it is possible to obtain a separable state from a global state describing a system of identical particles under the hypotheses that the particles are spatially separated. In practical terms we will consider projections for the spatial degrees of freedom, under the assumption that their domains are disjoint. After performing a partial trace over the spatial degrees of freedom we will obtain a separable quantum state in the spin degrees of freedom. The Hilbert space describing two indistinguishable particle is H 2 or H 2, depending on the nature of the two particles, where H = l h = L 2 (R 3 ) C 2s+1. (3.14)

34 28 In the case of identical particles, the reduced state ρ spin (of the spin degrees of freedom), obtained via the partial trace over the spatial degrees of freedom, is defined as: { ( )} tr(ρ spin A B) = tr ρ dγ (P A) (Q B), (3.15) A, B B(h), where ρ is a generic state in H 2 (or H 2 ), A, B are two observables acting on spins and P, Q are projections onto the spatial regions Ω 1 and Ω 2 of R 3, where the two spin measurements are respectively performed. Notice that the reduced state will in general depend on the choice of the projections P and Q. If we substitute the definition of the second quantization functor into this expression, we obtain an explicit expression for the right hand side of Eq. (3.15): { ( )} tr ρ dγ (P A) (Q B) = tr {ρ (P A) (Q B)} + tr {ρ (Q B) (P A)}. (3.16) Moreover, each one of the two terms in the sum can be rewritten in the form tr {ρ (P A) (Q B)} = tr {(P I) (Q I)ρ (P A) (Q B)}, (3.17) and tr {ρ (Q B) (P A)} = tr {(Q I) (P I)ρ (Q B) (P A)}. (3.18) Now let us suppose that ρ is the density operator of a pure state ρ = ψ ψ with ψ = 1 ] [(f ξ) (g η) ± (g η) (f ξ), (3.19) 2 where ξ, η represent one-particle spin states and f, g are one-particle spatial wavefunctions such that P f = f and Q g = g. The spatial separation requirement can be rephrased in terms of the projection operators, imposing that the regions of the space in which they act are disjoint, i.e. Ω 1 Ω 2 =. Then formally we have P Q = QP = 0. Notice that the requirement that the operators (measurements) acts on different regions of the space is in perfect agreement with the standard setting of quantum communication, where Alice and Bob are spatially separated, whence the wavefunction f vanishes outside Ω 1 and, similarly, g is zero outside Ω 2. Plugging this state into eq. (3.16) we obtain { ( )} tr ρ dγ (P A) (Q B) = 1 2 tr {( ξ ξ η η )(A B)} tr {( η η ξ ξ )(B A)} = tr {( ξ ξ η η )(A B)}. (3.20)

35 29 Therefore, from definition (3.15) we finally get of the reduced state of the spins ρ spin = ξ ξ η η, (3.21) that is nothing but a pure separable state of the two spins. In the same way, if we start from a generic symmetric or antisymmetric state ψ = 1 2N N i=1 [ ] (f i ξ i ) (g i η i ) ± (g i η i ) (f i ξ i ), (3.22) here we do not care of the normalization of the state, we obtain the state ρ spin = 1 ( ) f i f j g i g j ξ i ξ j η i η j, (3.23) N i,j which is, in general, mixed and entangled. We want to stress, once again, that the structure of ρ spin is not constrained to satisfy any particular symmetry so it can be a generic state of two distinguishable spins in C 2s+1 C 2s+1. The reduced state has no memory of the antisymmetric or symmetric structure of the initial state (where also the spatial degrees of freedom are considered) as long as the spin measurements are spatially separated [54]. Under the spatial separability assumption and considering the spin degrees of freedom, the particles become distinguishable from the entanglement point of view. Of course, if the particles are localized in the same region, P Q 0, these results are no longer true Peres Cluster Separability The results previously obtained fit perfectly in Peres framework on cluster separability [68]. Definition 3.1. Given two states u and v in H, v is said remote with respect to u if for any operator A with support in a spatial neighborhood of u, Av is vanishingly small. Let us consider a state of two identical particles ψ = 1 2 ( u v ± v u ), (3.24) where the orthogonal states u and v describe two particles that are far apart. If the one-particle operator A B(H) is non-vanishing in a neighborhood of u, then Av is vanishingly small and its lifting dγ(a) = A I + I A yields ψ dγ(a)ψ = u Au. (3.25)

36 30 because the terms involving v will vanish. We can notice that even if a direct consequence of the symmetrization and antisymmetrization is that the two particles are always entangled, this entanglement has no effect if we focus on localized observables. Indeed, there is no quantum prediction on one particle that is affected from the presence of another particle, identical to the first one in a remote part of the universe. In particular, consider the state (3.19) of two localized identical particles (bosons or fermions) with supp f Ω 1, supp g Ω 2, and Ω 1 Ω 2 =. In terms of the above notations, the projection operators P and Q onto Ω 1 and Ω 2, respectively, are mutually orthogonal P Q = 0. Therefore the state g η is remote (in the sense of definition 3.1) with respect to state f ξ and then, for any operator A B(C 2s+1 ) we can reapply the results obtained in eq. (3.25) ψ dγ(p A Q I)ψ = f ξ (I A)(f ξ). (3.26) The n-particle Case The result obtained until now can be generalized to a system of n identical particles. Let us consider a state ψ of n particles. For bosons the state belongs to H n (the symmetric space), while for fermion it belongs to H n (the antisymmetric space). In order to trace over the spatial degrees of freedom we have to consider n projections P 1,..., P n acting on disjoint spatial regions Ω 1,..., Ω n, each of which corresponds to n spatially separated spin experiments. As we did for two particles, we can define the reduced density matrix of the spins as ( tr ρ spin n i=1 If ρ is a pure state, ρ = ψ ψ with ) { A i = tr ρ dγ ( n ) } P i A i. (3.27) i=1 ψ = 1 n! π S n ɛ π n ( ) fπ(i) ξ π(i), (3.28) i=1 where ɛ π = 1 for bosons and ɛ π = sgn(π) for fermions, the partial trace leads to { tr ρ dγ = 1 n! ( n i=1 π S n tr ) } P i A i = { n } tr (P π(i) I) ψ ψ (P π(i) A π(i) ) π S n { n i=1 i=1 ( f π(i) ξ π(i) f π(i) ξ π(i) A π(i) ) }. (3.29)

37 31 Thus the reduce states is ρ spin = n ξ i ξ i, (3.30) and we can make the same conclusion as for two indistinguishable particles. 3.3 Some Considerations i=1 The aim of this section is to better clarify the interplay between indistinguishability of the global wavefunction and entanglement of local states. Recall that the oneparticle Hilbert space is a product H = l h, where h is the spin space and l the position space. In the previous section we have seen that, whenever some localization precludes the identical particles to share the same spatial state, there is no obstruction on the structure of the reduced state of the internal (spin) degrees of freedom. The spatial separation makes the reduced states insensitive to the quantum statistics of the global state. To be more clear, let us see what happens if the particles share the same spatial state. Let us focus on the fermionic setting H n. It will be useful to introduce an orthonormal basis { µ } µ 1 of l. A state ψ H n whose n factors share the same spatial state belongs to H = µ 1( µ h) n = l (h n ) (3.31) which can be considered as a proper subspace of H n. The partial trace of a state ρ D(H ) acting on such a space provides a reduced state with fermionic character: tr l D(H ) = D(h n ). (3.32) Note that Eq. (3.32) holds also if we consider the subspace, describing a symmetric spatial wavefunction, H = ( l n) (h n ), tr l nd(h ) = D(h n ). (3.33) On the other hand, if the spatial part of the state is antisymmetric, the proper subspace H = (l n ) ( h n) (3.34) is mapped by the partial trace onto the n-fold symmetric product of h: tr l nd(h ) = D(h n ). (3.35) The above three examples should be enough to understand that the statistics of the global state of n identical particles can be hidden, preserved or even changed if we focus on the local states of some very particular subspaces.

38 Identical Particles: A Subalgebras Approach We want to conclude this chapter analyzing briefly the case in which the particles are located in the same region. In this case the usual definition of separability, the one that we gave in the previous chapter, is too restrictive and a new approach is needed. In this section we will focus on the works of Benatti et al. [8, 9] who have given a definition of separability related to subalgebras of observables. Definition 3.2. Consider B(H), the algebra of all bounded operators acting on H. A pair (A 1, A 2 ) of commuting unital subalgebras of B(H) is defined to be an algebraic bipartition of B(H). Notice that this definition of bipartition does not require that the two parties reproduce the whole space. Definition 3.3. A state ρ on B(H) is defined to be separable with respect to the bipartition (A 1, A 2 ) if for any operator of the form A 1 A 2, with A 1 belonging to A 1 and A 2 belonging to A 2, we have tr(ρa 1 A 2 ) = k λ k tr(σ k A 1 )tr(ω k A 2 ), (3.36) with λ k 0, k λ k = 1 and where σ k and ω k are states on B(H). Notice that if the particles are distinguishable, this definition leads to the standard notion of separability. Indeed, in this case natural tensor product decomposition of the Hilbert space H is translated in a tensor product decomposition of the algebra of its operator B(H) and so from eq. (3.36) we recover the standard definition of separability. In particular, in [9] it is proved the following characterization for the separability of the state. Theorem A mixed state ρ of n identical particles is separable with respect to bipartion (A 1, A 2 ) if and only if it is the convex combination of projections on pure states separable with respect to (A 1, A 2 ). Otherwise, the state is entangled. We can easily show that this correspondence comes out naturally when we assume the spatial separation. For the sake of clarity, let us consider two fermions. Then the relevant algebra is the algebra of all bounded operators on H 2, with H = l h. The subalgebra of operators concerning the spin degrees of freedom is A = {dγ(i A) A B(h)}. (3.37) As in Section 3.2, we take two projections P, Q acting on the spatial space l.

39 33 The subalgebras of B(H 2 ) corresponding to spin measurements by Alice and Bob, respectively, are defined by: { ( ) } A 1 = dγ (P A 1 ) (Q I) A 1 B(h), (3.38) { ( ) } A 2 = dγ (P I) (Q A 2 ) A 2 B(h). (3.39) The commutator between two element of this algebra, is [ ( ) ( )] dγ (P A 1 ) (Q I), dγ (P I) (Q A 2 ) = [P A 1 Q I + Q I P A 1, P I Q A 2 ] +[P A 1 Q I + Q I P A 1, Q A 2 P I] = QP I P Q A 1 A 2 P Q I QP A 2 A 1 +P Q A 1 A 2 QP I QP A 2 A 1 P Q I, (3.40) for A 1, A 2 B(h). As we can see, this commutator identically vanishes if and only if P Q = 0, that is if and only if the particles are spatially separated. Therefore we can conclude that the subalgebras A 1 and A 2 commutes if and only if the domains of the projections P and Q do not overlap, namely [A 1, A 2 ] = 0 iff P Q = QP = 0. (3.41) This result is consistent with the discussion in Section 3.2: if two systems are spatially separated the statistics has no effect at the level of the internal degrees of freedom.

40 Chapter 4 The Problem of Maximally Entangled States If one has really technically penetrated a subject, things that previously seemed in complete contrast, might be purely mathematical transformations of each other. J. von Neumann In [49] Gisin et al. characterize pure and symmetric maximally entangled states of n qubits as the states such that all their partial traces are maximally mixed. This criterion can be generalized for all types of pure states and a valuable measure of entanglement that can be used is the purity, introduced in section In the general context of multipartite systems we can consider the average purity over all possible bipartitions of the system. If we require that this quantity reaches its minimum we obtain the so-called Multipartite Maximally Entangled States (MMES) [41]. The characterization of this kind of states is not simple, indeed it passes trough the minimization of a functional on the set of pure states of fixed dimension. For qubits this problem has been widely studied [38, 50, 78]. The main purpose of this chapter will be to analyze the general case of qudits. At first we will analyze the purity functional and we will introduce the problem of MMES and their structure. In particular, we will generalize previous results on qubit-mmes to the case of qudits. Then we will tackle the minimization problem with a statistical mechanics approach [28]. We will examine the high-temperature expansion of the distribution function of the average bipartite purity proving that the series characterizing the expansion converges. Moreover, we will introduce a diagrammatic technique to analyze 34

41 35 the behavior of each term of the series. 4.1 Purity and Maximally Entangled States In this section we will deeply analyze the purity of a generic system of n particles. The finite dimensional Hilbert space that describes this kind of systems is H = (C d ) n. The orthonormal computational basis for this product space can be denoted with { k } k, where k Z n d and Z d = Z/d Z is the cyclic group with d elements. Indeed, we can create a correspondence between the basis of the space and the strings of length n on d symbols 1. Example To fix the ideas on the notation that will be used, consider the orthonormal computational basis of two qutrits: { 00, 01, 02, 10, 11, 12, 20, 21, 22 }. (4.1) Since we have not done it yet, it is time to give a formal definition of bipartition. Definition 4.1. A bipartition of a system S = {1, 2,..., n} of n parties is a pair (A, Ā) such that A Ā = S and A Ā = φ. Furthermore if A = n A and Ā = n Ā = n n A are the dimensions of the two subsystems then the bipartition is called balanced if [ n ] [ ] n + 1 n A = and n 2 Ā =, (4.2) 2 where [x] denotes the integer part of x. From now on, without loss of generality, we will assume n A n Ā, indeed, the bipartitions (A, Ā) and (Ā, A) play the same role in our consideration. Generally speaking a system composed by n > 2 particles has a number of partition that scales with the binomial coefficient as: ( ) n, (4.3) n A [ n 2 ] see fig Every pure state ψ H admits a Fourier expansion in terms of the basis { k } k ψ = k Z n d 1 Notice that this is what we have done by tacit agreement until now. n A z k k, (4.4)

42 36 Figure 4.1: A system with three components has 3 = ( 3 1) different bipartitions. where z k = k ψ C. Using this expansion, we can rewrite the purity as: π A (ψ) = tr ( tr Ā ( ψ ψ ) 2) ( = tr ( ) z k z l δ ka l A k A l A ) 2 k,l = tr z k z k z l z l δ kā l δ Ā k k A l Ā l Ā A k A l A = k,k,l,l Z n d z k z k z l z l δ kā l Ā δ k Ā l Ā δ k A l A δ k A l A, (4.5) k,k,l,l Z n d where we used the notation k A to indicate the substring of k belonging to A, and where δ lm is the Kronecker delta, defined in eq. (1.7). Proposition Given a state ψ H of n qudits and a bipartition (A, Ā), the following holds 1. π A (ψ) = π Ā (ψ), 2. 1/N A π A (ψ) 1, where N A = d n A is the dimension of H A. Moreover the upper bound is reached only by separable states. Proof. The reduced density matrix of a given state when we trace over subsystem Ā is of the form ρ A = z kl k l, (4.6) k,l Z n A d where z kk R and z kl = z lk, because of the self-adjointness of the density operators. We are searching for the extremal values of the function π A (ψ) = zkk 2 (4.7) k Z n A d

43 37 in the simplex N A = {(z ll ) n l Z A d l z ll = 1, 0 z ll 1}. (4.8) The minimum is reached at the interior point for which dπ A = 0, i.e. z kk = 1 N A, (4.9) for every k Z n A d, while the maximum is reached at the frontier, i.e. z kk = δ lk, for some l Z n A d. The states that saturate the lower bound of the purity for some bipartition are called maximally entangled with respect to that bipartition. The Bell states are examples of maximally entangled states (here there is no need to specify the bipartition since this is unique in systems with only two components). Definition 4.2. A state ψ such that π A (ψ) = 1/N A with respect to every bipartition (A, Ā), i.e. maximally entangled with respect to every bipartition of the system, is called perfect multipartite maximally entangled state (perfect MMES). To determine if a given state ψ is a perfect MMES it is sufficient to check if it satisfies the minimization condition for all the balanced bipartitions. Indeed, if a state as a reduced density matrix of the form with respect to subsystem A, then ρ A = I N A, (4.10) ρ B = I N B, (4.11) for every smaller party B A. Therefore, the problem of finding perfect MMESs can be tackled studing the average purity over all the balanced bipartitions. Definition 4.3. The average purity over all possible balanced bipartitions is called potential of multipartite entanglement and is given by the following π ME (ψ) = ( ) 1 n [ n ] 2 A =[n/2] π A (ψ). (4.12) As for the purity we can define a bound for the potential of multipartite entanglement.

44 38 Proposition The potential of multipartite entanglement satisfies the following bound: 1/N A π ME (ψ) 1, (4.13) with N A = d [n/2], for every state ψ H. We will see that the lower bound 1/N A of the potential of multipartite entanglement is not always attained. This justifies the following. Definition 4.4. A state ϕ that minimize π ME π ME (ϕ) = min{π ME (ψ) : ψ H, ψ ψ = 1}. (4.14) is a multipartite maximally entangled state (MMES). Let us underline, once again, that the difference between a MMES and a perfect MMES lies in the saturation of the lower bound of the potential of multipartite entanglement. Example The GHZ-state is a perfect MMES, indeed we have seen in section that the purities with respect to all the possible bipartitions are Perfect MMES and Frustration One of the questions that arise naturally from the previous discussion is about the general structure of a perfect MMES for given values of d (the dimension of each subsystem) and n (the number of subsystems). With an abuse of notation we can say that the Bell states are perfect MMES for systems of two qubits 2 while for n = 3 qubits the only perfect MMES, up to local and unitary transformations, is the GHZ-state. The problem of characterizing a perfect MMES has not always such an easy solution. In [50] Gour et al. proved that for n = 4 qubits a perfect MMES cannot be found and that the minimum of the average purity is π (4) ME = 1 3 > 1 4 = 1 N A. (4.15) When the lower bound of the potential of multipartite entanglement cannot be saturated, the system is said to be frustrated. In this case the requirement that the purity is minimal for all the bipartitions generates conflicts among them. For system of n = 5, 6 qubits there are examples of perfect MMES, see [38], while for n 8 qubits a perfect MMES does not exist as proved by Scott in [78], using the classical error correction theory. 2 There is no multipartite entanglement here.

45 39 The case of n = 7 qubits is still an open problem and it has been conjectured that a perfect MMES cannot exist [16]. Up to now only guesses about the minimum of the potential of multipartite entanglement have been done. Recently we have found a 7-qubit state with the lowest π ME found until now, to the best of our knowledge. Its potential of multipartite entanglement is π ME = , (4.16) and the distribution of its purities for the balanced bipartitions can be found in figure 4.2. An interesting feature of this state is that all bipartitions are frustrated, i.e. none nπa n7, stato Sara, Μ , Σ ΠA Figure 4.2: Purity distribution for our 7-qubit state. of them reaches the minimum for the corresponding purity. Frustration appears when one or more bipartitions cannot reach their minima. Enlarging the dimensions d of each subsystem, at fixed n, tends to eliminate this problem. To prove this statement some results of classical coding theory are needed. For a brief review of this results see appendix C, while for a general introduction to classical coding theory see [84]. Let us consider a code C = {c j }, with N A codewords of length n and alphabet Z d. Using the codewords of C we can construct the n-qudit state: ψ = 1 N A c j. (4.17) NA If the minimal Hamming distance of C is greater than [ n 2 ], after the partial trace over a balanced bipartition all the off-diagonal terms, tr Ā ( c j c k ), vanish. Indeed tr Ā ( c j c k ) = j=1 l Z n Ā d l c j c k l 0, (4.18)

46 40 if and only if c j and c k have at least n Ā symbols in common. Therefore, for this kind of states, π ME reaches its minimum and the state in eq. (4.17) is a perfect MMES. It remains to prove the existence of such a code. In particular we want to address the existence of a MDS (see appendix C) code of the form { (n, N A, n A + 1) d if n is even (n, N A, n A + 2) d if n is odd. (4.19) Indeed, these are the biggest values of the parameter permitted by the Singleton bound (see appendix C). The presence of d [ n 2 ] terms in the sum come from the necessity to make the diagonal terms of ρ A exactly 1 N A for each bipartition (A, Ā). Then, we can surely find a MMES if we can write such code with exactly d [ n 2 ] word. From the bounds given in appendix C we arrive at the following. Theorem If a perfect MMES of n qudits exists 3. d n 1 (4.20) Remark. Notice that in appendix C we give this bound for linear codes. Indeed, we can always take d as the smallest power of a prime number greater or equal than n 1. Notice that the bound in the previous theorem prove that enlarging the dimension of the single subsystems the frustration tends to disappear. However it gives us only a hint on the presence of frustration. For instance in the case of 5 qubits a perfect MMES exists, while for the bound in the theorem we need d 3. Nevertheless it works well for systems of 4 particles. In [55] a perfect MMES of 4 qutrits is shown. We want to remark that if a MDS code of the form previously given exists then we can find a perfect MMES, the converse does not hold. It can be possible indeed to construct MMES that cannot be linked to classical codes. 4.2 The Potential of Multipartite Entanglement Before introducing the main results of this chapter let us analyze more in detail the potential of multipartite entanglement. In particular we will show an explicit formula for the π ME for states of qudits, following the idea presented in [38] in the case of qubits. 3 This theorem was first stated by A. V. Thapliyal during a talk done in Bristol.

47 41 Let us recall that, using the Fourier expansion of the state, we can write the purity of a pure state, with respect to a given bipartition as (4.5): π A (ψ) = z k z k z l z l δ kā l δ Ā k δ Ā l k Ā A l δ A k A l A, (4.21) k,k,l,l Z n d If we average the purity over all the possible bipartitions with fixed dimension, we can define the coupling function, [38]: (k, k ; l, l ; n A ) = 1 2 (k, k ; l, l ; n A ) (k, k; l, l ; n Ā ), (4.22) where ( ) 1 n (k, k ; l, l ; n A ) = δ k n A l A δ ka l δ A k lāδ Ā k. (4.23) Ā l Ā A A =n A Definition 4.5. The Hamming weight of a string k, that we indicate with k, is the number of symbols that are different from the zero-symbol of the alphabet used. If the alphabet is binary, i.e. it is Z 2 = {0, 1}, then the Hamming distance is nothing but the number of 1 in the strings. We can use the definition of Hamming weight to rewrite the coupling function in a more convenient form. Theorem The coupling function has the following expression: (k, k ; l, l ; n A ) = δ k+k,l+l δ 0,(k l) (k l) f(k l, k l, n A ), (4.24) where and f(k, l, n A ) = 1 ( ) 1 [( ) ( )] n n k l n k l +, (4.25) 2 n A k n A l n A k ± l = (k j ± l j ) j and k l = (min{k j, l j }) j. (4.26) Proof. The coupling function ( ) 1 n (k, k ; l, l ) = n A A =n A 1 2 (δ k A l A δ k A l A δ k Ā lāδ k Ā l Ā + δ k A l A δ k A l A δ k Āl Ā δ k Ā l Ā ), (4.27) is non zero if and only if for some subset A of S = {1, 2,..., n}, with A = n A, we have k A = l A, k A = l A, k Ā = l Ā, k Ā = l Ā. (4.28)

48 42 This imposes that if j A, i Ā or equivalently that for j S and k j = l j, k j = l j, k i = l i, k i = l i, (4.29) k j l j = 0 or k j l j = 0, (4.30) k j l j = 0 or k j l j = 0. (4.31) Putting these conditions all together we have that 0 if and only if k + k = l + l and (k l) (k l) = 0. (4.32) It remains to count the number of bipartitions (A, Ā) that contribute to the sum in eq. (4.27). For this aim let us call S 0 = {i S k i = l i = k i = l i}, (4.33) S 1 = {i S k i l i or k i l i} (4.34) S 2 = {i S k i l i or k i l i }. (4.35) From the previous discussion it is easy to see that S 1 S 2 = φ and that S = S 0 + S 1 + S 2. In this new notation a bipartition (A, Ā) for which the contribution of the first term in the sum does not vanish, i.e. δ k A l A δ ka l A δ k lāδ Ā k 0, (4.36) Ā l Ā is characterized by the two inclusions A S 1 + S 0 and Ā S 2 + S 0. Furthermore, since A Ā = φ and A Ā = S, then A = S 1 + A S 0 and Ā = S 2 + Ā S 0. Finally, we can conclude that their number is equal to the binomial coefficient ( S0 A S 1 ) ( ) ( n S1 S 2 n k l k l = = n A S 1 n A k l The same result can be obtained for the second term in the sum: swapping the role of A and Ā. δ k A l A δ k A l A δ k Āl Ā δ k Ā l Ā ). (4.37) (4.38) Remark. The binomial coefficients in Eq. (4.25) is intended to be zero if one of its arguments is negative. Since we want to focus on balanced bipartitions, from now on we will omit the parameter n A both in and f, understanding that n A = [ ] n 2. In this way, using the coupling function, the potential of multipartite entanglement can be written as π ME = (k, k ; l, l )z k z k z l z l. (4.39) k,k,l,l Z n d

49 Statistical Mechanics Approach The minimization problem of the potential of multipartite entanglement can be handled following a statistical mechanics approach [39]. Roughly speaking we will consider the free energy of a suitable classical system at a fictitious temperature and we will recover the original problem in the zero temperature limit. We define the Hamiltonian H(z) = π ME (ψ), (4.40) where z = (z k ) is the vector of the Fourier coefficient in the expansion of the state: ψ = k z k k. (4.41) Let us consider M vectors and the ensemble {m j }, whit m j the number of vectors with fixed potential of multipartite entanglement, H = ɛ j. We want to find the distribution that maximizes the quantity Ω = M! Π j m j!, (4.42) under the constraints j m j = M and j m jɛ j = ME, where E is the average π ME. In particular, if we let M we recover the canonical ensemble with partition function Z(β) = dµ(z)e βh(z), (4.43) where ( (N 1)! dµ(z) = δ 1 ) z π N k 2 dz k d z k (4.44) k k is the unitarily invariant measure over pure states induced by the Haar measure over U(H) through the mapping ψ = U ψ 0, for a given state ψ 0, [94]. Here β plays the role of the inverse of the temperature, so that for β + only the configurations that minimize the Hamiltonian survive. In other words, we recover the MMES in the limit β +. On the other side, if β 0, eq. (4.43) yields to the distribution of typical states. Using the partition function, the average energy can be written as H β = 1 Z(β) dµ(z)he βh(z) = ln Z(β). (4.45) β The high-temperature expansion (β 0) of this energy distribution function is: ( β) m 1 H β = (m 1)! κ(m) β=0 [H], (4.46) m=1

50 44 where κ (m) β=0 [H] = (H H 0) m m 1 m 1 0 = ( 1) β H β. (4.47) m 1 The first useful result about the series in eq. (4.46) is that it converges. Indeed, notice that the Hamiltonian that we are considering is bounded (H H 0 ) m 0 c m, (4.48) whit c a positive constant. This bound lets give a estimation of the series: m=1 ( β) m 1 (m 1)! κ(m) β=0 [H] m=1 ( β) m 1 (m 1)! cm = c e βc, (4.49) that is convergent with convergence radius +. This statistical mechanic approach has been studied for qubits in [42], in the bipartite case, and in [39, 40] in the multipartite case. Here we want to analyze the multi-qudit case. 4.4 Cactus and Other Diagrams In this section we will use the diagrammatic technique introduced in [40] to control each term of the series (4.46). First of all let us consider the quantity: H m 0 = m (x, x ; y, y )z x z x z y z y 0. (4.50) x,x,y,y Z n d An explicit form of this quantity requires the product of m functions. To simplify the notation we introduce the vector k = (k 1,..., k 2m ), with k j Z n d. Therefore, H m 0 = k,l Z 2mn d m j=1 Theorem The following equality holds: H m 0 = k Z 2mn d 2m (k 2j 1, k 2j ; l 2j 1, l 2j ) z kj z lj 0. (4.51) m p S 2n j=1 j=1 (k 2j 1, k 2j ; k p(2j 1), k p(2j) ) N(N + 1)... (N + 2m 1). (4.52)

51 45 Proof. Since e i arg z 0 = 0, the only non-vanishing terms in Eq. (4.50) are those such that {z kj } = {z lj }. This means that a permutation p S 2n exists such that z lj = z p(kj ). In other words: H m 0 = k Z 2mn d m p S 2n j=1 Furthermore it can be proved, see [40], that: k j=1 In the case we are discussing 2m j=1 2m (k 2j 1, k 2j ; k p(2j 1), k p(2j) ) z kj 2 0. (4.53) j=1 z qj 2m j 0 = (N 1)! k j=1 m j! (N 1 + k j=1 m j)!. (4.54) z kj 2 0 = that proves the statement of the theorem. 1 N(N + 1)... (N + 2m 1), (4.55) In order to introduce the diagrammatic technique, let us define the square brackets: k Z 2mn d [p(1) p(1 ),..., p(m) p(m )] := (k 1, k 1 ; k p(1), k p(1 ))... (k m, k m ; k p(m), k p(m )), (4.56) with p S 2m a permutation acting on 2m elements. Here we rewrite (k 2j 1, k 2j ; k p(2j 1), k p(2j) ) (4.57) as (k j, k j ; k p(j), k p(j )); this does not change at all the expressions, but simplifies the notation used. With the use of the new square brackets eq. (4.53) becomes: H m 0 = 1 N(N + 1)... (N + 2m 1) p S 2m [p(1) p(1 ),..., p(m) p(m )], (4.58) and we can give a graphic representation of it. Each pair (k j, k j ) represents a vertex of a graph from which two edges go out and two go in. The first two edges are represented by k p(j) and k p(j ), and the latter are k j and k j, see fig. 4.3.

52 46 k p(j0 ) k p(j) k j k j 0 Figure 4.3: We can associate to each couple (k i, k i ) a vertex with 4 edges, two going in and two going out. (k 1,k 1 0) (k 2,k 2 0) Figure 4.4: Graph with two points representing [1 2, 1 2 ]. Example The square brackets [1 2, 1 2 ] yields the graph in fig As we will see this new framework makes the study of eq. (4.47) simpler. Furthermore, we can rephrase some previous results in terms of these diagrams. Indeed, eq. (4.32) can be interpreted as a conservation law, i.e. the flow going into a vertex has to be the same of the flow that goes out, see fig Moreover, the symmetries of the function are translated in the degeneracy of the graphs. For instance, the square brackets in the example above are the same as and so on. [1 2, 1 2 ] = [2 1, 2 1 ] = [1 2, 1 2] = [2 1, 1 2] = [1 2, 2 1 ] (4.59) Example In the graphs notation we have H 0 = 1 N(N + 1) ([1 1 ] + [1 1]) = 1 N(N + 1) ( + ) (4.60) Definition 4.6. A graph such that any two vertices are linked by no more than two edges, and such that it has no cycles of the form in fig. 4.5, is called cactus, otherwise the graph is called non-cactus.

53 47 Example The graph in fig.4.4 is a cactus. Figure 4.5: A graph with a cycle is a non-cactus Computation and Degeneracy of the graphs In this section we will study in details the graphs introduced in the previous section. In particular, we will compute the contribute that each graph gives to the series and its degeneracy. For this aim we will divide each graph in its component and we will compute the degeneracy and the contribution to the series of each single component. Definition 4.7. We call leaf the component of a graph represented by [..., j p(j ),... ] = k j k j 0 k p(j0 ). (4.61) Lemma A leaf at the vertex (k j, k j ) gives the contribution: N A + N δ Ā kj,k p(j ). 2 (4.62) Proof. In a graph the contribute of a leaf is: [..., j p(j ),... ] = (k j, k j ; k j, k p(j )). (4.63) k j k i :i j k i We can notice that k j appears only in the last part of the sum so that, isolating it from the rest, we have: k j (k j, k j ; k j, k p(j )) = δ kj,k p(j ) k j f(k j k j, 0) N A + N = δ Ā kj,k p(j ). (4.64) 2

54 48 Example As an example consider the graph in fig If we compute the contribution of the upper leaf as in the previous lemma we find = k1 δ k1,k 1 N A + N Ā 2 = N N A + N Ā. (4.65) 2 Figure 4.6: The only graph with a single vertex. Definition 4.8. A loop, see fig. 4.7, is the part of a graph represented by [..., p(i) j,..., p(j) i,... ] = k p(i) k j 0 k p(j). (4.66) k i k i 0 k j Theorem Each graph representing a cactus gives a contribution ( ) v NA + N N Ā, (4.67) 2 where v is the number of vertices in the graph. Proof. We can compute the contribution of a graph decomposing it in its elementary parts. From definition 4.6, we can gather that a cactus has at least one leaf. Moreover, notice that after computing the contribution of the leaf, the remaining terms in the square brackets constitute a graph with v 1 vertices. This new graph is essentially the previous one with v vertices but without a leaf. Moreover, taking away the vertex of the leaf transforms a loop into a leaf. Besides, removing a leaf leaves invariant the characteristic of the graph to be a cactus. We can iterate the computation obtaining a contribute of (N A + N Ā ) 2 (4.68)

55 49 for each vertex. At the end of this computation the remaining term of the square brackets will be 1 = N, (4.69) and this concludes the proof. k k p(i) k j 0 k p(j) k i k i 0 k j Figure 4.7: A loop in a graph. Computing the contribution of non-cactus graphs is not as simple as in the cactus case. Nevertheless, we can give an upper bound for it. Theorem A loop gives a contribution that is always lower or equal than d [ n+1 2 ]. Proof. We can isolate the contribution of each single loop obtaining = k i,k j [..., p(i) j,..., p(j) i,... ] (k i, k i ; k p(i), k j ) (k j, k j ; k p(j), k i ). (4.70) Using the expression of the coupling function in theorem we have (k i, k i ; k p(i), k j ) (k j, k j ; k p(j), k i ) = k i,k j k i,k j δ ki +k i,k p(i) +k j δ kj +k j,k p(j) +k i δ 0,(ki k p(i) ) (k i k p(i) ) δ 0,(kj k p(j) ) (k j k p(j) )f(k i k p(i), k i k p(i) )f(k j k p(j), k j k p(j) ). (4.71) If we define l = k i k p(i) and we conveniently rearrange the variables in the δ- function we have (k i, k i ; k p(i), k j ) (k j, k j ; k p(j), k i ) k i,k j = δ ki +k j,k p(i) +k p(j) δ 0,(ki k p(i) ) l f(k i k p(i), l) f(k i k p(i), l + k p(i) k j ) l kj δ kj +k j,k p(j) +l+k p(i) δ 0,(kj k p(j) ) (k j k p(j) ). (4.72)

56 50 k j k j k n+1 k (n+1) 0 k m k m Figure 4.8: Pinching operation Finally, the last sum leads to k i,k j (k i, k i ; k p(i), k j ) (k j, k j ; k p(j), k i ) = δ ki +k j,k p(i) +k p(j) f(k i k p(i), l) f(k i k p(i), l + k p(i) k j ) l δ 0,(ki k p(i) ) l δ 0,(ki k p(i) ) (l+k p(i) k j ). (4.73) It is straightforward to prove that f(k, l) 1. Moreover, from the assumption that the binomial coefficient is zero if one of its argument is negative, we have [ ] n + 1 l max{n A, n Ā } =, (4.74) 2 and so the last expression in (4.73) can be bounded by d [ n+1 2 ]. We will use he last part of this section to give an explicit way to compute the degeneracy of graphs. In particular, we will show a recursive way to compute the degeneracy of a generic (n + 1)-vertex graph knowing only the degeneracy of the n-vertex graph from which it is generated. For this aim we will call the latter mother graph and the ones generated from it daughters. Definition 4.9. We define pinching the operation that connects two edges adding a vertex to the graph (see fig. 4.8). The following proposition illustrate the degeneracy of this operation. Proposition Adding a vertex through pinching increases the degeneracy by: 1. 4 if the four vertices are non degenerate or if they degenerate into two but the edges have different directions; 2. 2 if the four vertices degenerate into one;

57 if the four vertices degenerate into two and the directions of the two edges are the same. Remark. From now on we will suppose to start from a n-vertex graph and to add the vertex labeled (n + 1, (n + 1) ). Proof. In the first case (fig. 4.8) the mother graph has the form then the daughter graph can be represented by: [..., j p(i ),..., m p(l ),... ], (4.75) [..., n p(i ),..., (n + 1) p(l ),..., j m]. (4.76) Swapping n and n + 1 or j and m leaves the graph unchanged, therefore the degeneracy of the new one has an extra factor 4 compared with the degeneracy of the mother. In the second case, see fig. 4.9(a), the n-vertex graph can be represented by and the pinching leads to the representation that is equivalent to [..., j j,... ], (4.77) [..., (n + 1) (n + 1),..., j j ], (4.78) [..., (n + 1) (n + 1),..., j j ]. (4.79) Since there are no other possibilities, compared to the degeneracy of the mother, the degeneracy of the daughter has an extra factor 2. In the last case, fig. 4.9(b), we start from the representation and arrive to [..., i j,... ], (4.80) [..., (n + 1) (n + 1),..., i i ], (4.81) where again we have an extra factor 2 of degeneracy. The only case that remains unexplored in the previous proposition is the one in which the four vertices degenerate into two and the two edges degenerate into one. However, this operation is the same as making a leaf germinate, fig In this case, before the creation of the leaf the graph is associated to [..., i p(j ),... ], (4.82)

58 52 k j kj k i k j k n+1 k i k j 0 k n+1 k (n+1) 0 k i 0 k j 0 k (n+1) 0 k j 0 k i 0 (a) (b) Figure 4.9: (a) Four vertices degenerating into one. (b) Four vertices degenerating into two. while after the pinching the representation becomes [..., (n + 1) p(j ),..., i (n + 1) ]. (4.83) Even in this situation we have 3 other square brackets that leads to the same graph. Indeed we can swap n + 1 and (n + 1), i and p(j ) or the elements in the last couple. Remarkably, the pinching operation let us construct all (n + 1)-vertex graphs starting from the n-vertex graphs. In addition it gives a convenient way to compute the degeneracy. However, it is not true in general that the degeneracy of the daughter graph is the degeneracy of the mother multiplied by the degeneracy of the pinching. In fact, the addition of a point can break the symmetry of the graph, so that, when this happens we have a factor lower or equal than n + 1 = (n + 1)!, (4.84) n! meaning that the more the graph is symmetric the lower is its degeneracy. Furthermore, sometimes the action of pinching different couples of edges gives rise to the same graph. Proposition The degeneracy of a graph is degeneracy of the mother degeneracy of the pinching different ways in which pinching the edges gives rise to the same graph the symmetrization factor (in case). Making the pinching between edges with the same direction of the flux gives rise to graphs that are different from the ones obtained by pinching edges with the same direction. The difference between the two classes is shown in fig. 4.12(b) and fig. 4.12(c) where the two graphs differ only for the flux of the arrows. We are now ready to give a bound for the degeneracy of the graphs. Theorem The degeneracy m of a n-point graph satisfies m 2 2n n! (4.85)

59 53 k 0 n+1 k n+1 k i Figure 4.10: Germination of a leaf. Proof. To compute the degeneracy of a graph in the worst case we have to compute all the configurations [p(1) p(1 ),..., p(n) p(n )] that lead to the same graph. In the worst case the pinching operation gives a factor 4 for each point and every permutation of the points leaves the configuration unchanged, so that the degeneracy, in the worst case is 4 n n!. Example The only graph with one vertex is the one in fig It has degeneracy 2. From this graph we can generate the two connected graphs with two vertices. The one in fig. 4.3 is obtained by a non degenerate pinching, then its degeneracy is = 16. (4.86) The one in fig is generated by a degenerate pinching then its degree of degeneracy is 2 2 = 4. (4.87) Figure 4.11: A non-cactus with two vertices. Example The graph in fig. 4.12(a) is generated from the one in fig by the germination of a leaf. Its degeneracy is where 3 is the symmetrization factor = 192, (4.88)

60 54 (a) (b) (c) Figure 4.12: All connected non-cactus with three vertices Example The graphs in fig. 4.12(c) generates from a pinching in graph in fig Its degree of degeneracy then is 16 4 = 64, (4.89) while the graph in fig. 4.12(b) is generated by pinching from the one in fig. 4.11, therefore its degeneracy is = 16. (4.90) 4.5 Computation of the Series The graph representation introduced in the previous section gives us important information about the series (4.46). Indeed, using this diagrammatic technique we can prove the following. Theorem For all m 1 H m 0 2 2m m! d nm 2 (N + 1)... (N + 2m 1) [ C 1 (m) + C ] 2(m), (4.91) d where C 1 and C 2 are positive functions of the parameter m only, that do not depend on d, representing respectively the number of cactus and non-cactus at fixed number of vertices.

61 55 Proof. From eq. (4.58) we know that we can write H m 0 in terms of the square brackets: H m 0 = 1 N(N + 1)... (N + 2m 1) p S 2m [p(1)p(1 ),..., p(m)p(m )]. (4.92) If we divide the set the permutation in S 2m in the ones that generate a cactus, P 1, and the ones that generate a non-catus, P 2, then H m 2 2m m! 0 N(N + 1)... (N + 2m 1) [ ( ) m NA + N ( C 1 (m) N Ā + C 2 (m) N d [ n+1 2 ] ) ] m 1, (4.93) 2 where C 1 (m) and C 2 (m) are respectively the non equivalent cactus and non-catus with m vertices. Since for n sufficiently large so that n A n Ā, recalling that N A = d [n/2], we arrive at the conclusion: [ H m 2 2m m! d nm 2 0 C 1 (m) + C ] 2(m). (4.94) (N + 1)... (N + 2m 1) d Finally, we can arrange this results all together in the series (4.46). = + m=1 + ( β) m 1 H β = (m 1)! (H H 0) m 0 (4.95) m=1 ( β) m 1 m ( ) m ( 1) m t H t 0 H m t 0. (4.96) (m 1)! t t=0 Summing up we proved the convergences of this series in section 4.3. Moreover, from theorem it is clear that since C 1 and C 2 do not depend on d, in the limit d the contribution to the sum due to the presence of the non-cactus graphs goes to zero and only the presence of cactus becomes relevant. Heuristically, we can attribute the presence of frustration in the system to the relevance of the non-cactus graphs in the series.

62 Chapter 5 Is Entanglement Monogamous? It is impossible for any number which is a power greater than the second to be written as a sum of two like powers. I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain. P. de Fermat Until now we have presented problems related to entanglement in pure states. In this chapter we want to analyze the distribution of bipartite and multipartite entanglement both in pure and mixed states. We have stressed that entanglement is one of the features that signs the difference between quantum and classical mechanics. One of the key properties that differentiate entanglement from classical correlations is its monogamy: entanglement cannot be freely shared among multiple parties. This is a consequence of the no-cloning theorem [90] that, as we said, states that is impossible to create a perfect copy of an unknown quantum state. In this chapter we will analyze the monogamy of entanglement for qubits systems. In particular we will show how it can be formalized in the case of tripartite systems and we will examine the generalization of this formulation to multiqubit systems. In the next section we will define one of the basic tool to investigate entanglement in qubit systems: Wootters concurrence. This will be essentially our first approach with a measure of entanglement for mixed states. We will see the basic motivations that leads to the definition of this function and how to compute it. In section 5.2 we will introduce the concept of monogamy of entanglement for systems of three qubits. In particular we will follow the work of Coffman, Kundu e Wootters [25] to 56

63 57 formulate the mathematical description of the problem. The sections that will came after will be devoted to the generalization of the previous results. Grounding on [74] we will conjecture a formulation for the monogamy of entanglement in systems with more than 3 qubits, explaining the motivations behind it. Moreover, we will provide some analytical results and numerical evidence that strengthen the validity of our reasoning. 5.1 The Concurrence In section we have introduced the von Neumann entropy as a suitable measure of entanglement for bipartite pure states. When Bennett et al. [12] introduced this entropy they proved that it measures the amount of singlet states, i.e. states of the form 1 ( ), (5.1) 2 that can be extracted from n copies of the state we want to measure. More precisely, suppose that Alice and Bob share n copies of a given state ψ. If the entanglement of the state is E(ψ) = tr(ρ log 2 ρ), where ρ is the partial trace of ψ ψ over one of the two subsystems, then the n states can be converted using LOCC into m singlet states, with m E(ψ). (5.2) n An interesting point is that we can address the m singlet states either as the number of singlets we can extract, or the number of singlets we need to create the n copies of the state. However, this is true for pure states, but it is no longer valid for mixed states. The Entanglement of Formation is, essentially, the number of singlets we need to create the n copies of the state. In the following we will see that we can write a function to measure it and that this function has a simple analytic expression for mixed states. The entanglement of formation for mixed states is defined as the convex roof of the von Neumann entropy, i.e. E(ρ) = min k p k E(ψ k ), (5.3) where we recall that the minimum is taken over all the possible ensembles of pure states {p k, ψ k } such that ρ = k p k ψ k ψ k. From now on we will refer to the entanglement of formation as the entanglement of the system.

64 58 In 1997 Hill et al. [56] found an exact formula for the entanglement of formation for all density matrices of two qubits having at most two non-zero eigenvalues. Later, in 1998, Wootters [88] generalized this result for arbitrary states of two-qubit systems. To explain this exact formula, let us define for pure states of a single qubit, the spin-flip operator, that acts as: ψ = σ y ψ, (5.4) where ψ is the complex conjugate of ψ, i.e. ψ = k z k k if ψ = k z k k, and σ y is the Pauli matrix ( ) 0 i σ y =. (5.5) i 0 For more than one qubit we can perform the spin flip simply applying the above transformation to all qubits, while for mixed states of two qubits we can define it as: ρ = (σ y σ y )ρ (σ y σ y ), (5.6) where again ρ is the complex conjugate of ρ (not the transposed complex conjugate) and we used the Hermiticity of σ y. We can use the spin-flip to define the matrix R = ρ ρ. Notice that even if R is non- Hermitian in general, both ρ and ρ are positive operators, therefore the eigenvalues of R are non-negative real numbers. If {λ 1, λ 2, λ 3, λ 4 } is the set of the square roots of the eigenvalues of R in nondecreasing order, then we can define the Concurrence as C(ρ) = min{0, λ 1 λ 2 λ 3 λ 4 }. (5.7) Using the concurrence the Entanglement of Formation can be written as where E(C) = σ=± E(ρ) = E(C(ρ)), (5.8) 1 + σ 1 C σ 1 C log 2 2. (5.9) 2 Moreover, it can be proven that the concurrence itself is a measure of entanglement [89]. In particular its square, that is still a monotonic function, can be also used as a measure of entanglement and in literature is known as 2-tangle, or simply tangle [25]. It is worth noticing that in the special case of pure states, R has only one non-zero eigenvalue and the concurrence reduces to: where ρ A = tr B ( ψ ψ ). C( ψ ψ ) = 2 det(ρ A ), (5.10)

65 5.2 Monogamy Inequality for Three-Qubit Systems Entanglement presents some interesting features that are completely quantum in the sense that they are not valid for classical correlations. One of them is the so-called monogamy of entanglement (MoE) that, roughly speaking, bounds the amount of entanglement the parties of the system can share. To better understand what we are going to do, let us consider the following problem, see fig Given a tripartite system S = (A, B, C), if party A is maximally entangled with B, can A be, at the same time, maximally entangled with C? The answer to this question is negative. Indeed, if we want A maximally entangled with B we have to impose that their state is one of the Bell states. This is a constraint that forces the state of A and B to be pure and so to be separable with respect to the third party C. Of course, an analogue reasoning can be done the other way round considering A maximally entangled with C.? 59 Bob Alice Charlie Figure 5.1: The monogamy of entanglement described using a common situation in real life. This problem can be generalized asking something less: the amount of entanglement party A shares with party B, bounds in some way the entanglement A can share with C? The answer here is positive and is given by the so-called monogamy of entanglement. The first formalization of MoE was done in 2000 by Coffman, Kundu, and Wootters [25] for three-qubit systems. In particular, they gave a quantitative constraint, known as monogamy inequality, of the entanglement of the three-parties in terms of the 2-tangle. Theorem (Monogamy inequality (pure states)). Given a pure state ρ of a threequbit system, the following inequality holds: where ρ XY = tr Z (ρ), for X, Y, Z = A, B, C. C 2 (ρ AB ) + C 2 (ρ AC ) 4 det ρ A, (5.11)

66 60 Proof. If ρ = ψ ψ is a pure state of three qubits, then the reduced density operators ρ AB and ρ AC have at most two non-zero eigenvalues. This is a consequence of the Schmidt decomposition, theorem This means that we can write the 2-tangle as C 2 (ρ AB ) = (λ 1 λ 2 ) 2 = λ λ 2 2 2λ 1 λ 2 tr(ρ AB ρ AB ), (5.12) and the same holds for C 2 (ρ AC ). Moreover it can be proved by direct computation that tr(ρ AB ρ AB ) = 2(det(ρ A ) + det(ρ C ) det(ρ B )), (5.13) using the fact that the reduced density operators ρ X have all unitary trace. Eq. (5.12) and (5.13), together, lead to the result. The right hand side of eq. (5.11) can be interpreted as a 2-tangle even if it is not computed for a state of two qubits. Indeed it has the same form of a concurrence for pure states. We will address it as C 2 A(BC)(ρ) = 4 det ρ A, (5.14) and we will understand it as a measure of the entanglement between A and the rest of the system. We can extend the new 2-tangle to mixed states using the convex roof: C 2 A(BC)(ρ) = min {p k, ψ k } k p k CA(BC)(ρ 2 (k) A ), (5.15) where the minimum is taken over all the possible pure-state decompositions of ρ and ρ (k) A = tr BC( ψ k ψ k ). (5.16) Therefore we can enlarge the monogamy inequality (5.11) to mixed states. Theorem (Monogamy inequality (generic states)). Given a state ρ of a threequbit system, the following inequality holds: C 2 (ρ AB ) + C 2 (ρ AC ) C 2 A(BC)(ρ). (5.17) Proof. Once we have proved the inequality for pure states, the proof for mixed states comes naturally. Indeed, if {p k, ψ k } is the optimal decomposition of ρ for CA(BC) 2 (ρ), then we can use it to compute all the other quantities, obtaining: k p k C 2 (ρ (k) AB ) + C2 (ρ (k) AC ) k p k CA(BC)(ρ 2 (k) A ), (5.18) where agin we used the notation ρ (k) XY = tr Z( ψ k ψ k ). Since k p kc 2 (ρ (k) XY ) C2 (ρ XY ), we eventually obtain the result.

67 61 + Figure 5.2: The monogamy inequality gives a bound for the entanglement three qubits can share. Eq. (5.17) is the answer to the problem we formulated at the beginning of this section. It says that the entanglement between A and the other two qubits, taken as a whole, cannot be less than the sum of the individual entanglements between A and each of the two remaining qubits, see fig From this point of view the difference between left and right hand side of eq. (5.17) can be interpreted as a quantifier of the entanglement genuinely shared among the three qubits. It can be seen as the entanglement that the three parties share when we leave out the bipartite entanglement of the system. In fact, one can define the so-called residual 3-tangle or, in short, 3-tangle, of a pure state ρ = ψ ψ as τ (3) A B C (ρ) := (C2 A (BC) C 2 (ρ AB ) C 2 (ρ AC ))(ρ). (5.19) Interestingly, this quantity does not depend on the focus qubit, in this case A, that we privilege in the decomposition. The three-tangle is a full-fledged measure of the genuine tripartite entanglement of any three-qubit pure state ψ [32, 59] Two old friends: GHZ and W To give an example of how the monogamy inequality works let us compute it for the two states that we introduced in section Let us recall that the GHZ-state is defined as, see eq. (2.31): GHZ = 1 2 ( ). (5.20) We have showed that if we trace over one of the three qubits this becomes separable. In other words the two concurrences of the reduced density matrices are zero: C 2 (ρ (GHZ) AB ) = C 2 (ρ (GHZ) AC ) = 0. (5.21) Essentially for the GHZ-state the monogamy inequality (5.11) reduces to 4 det ρ (GHZ) A 0, (5.22)

68 62 that is always verified. For the W-state, see eq. (2.30), the situation is different. Indeed, let us recall that it is defined as W = 1 ( ), (5.23) 3 and that tracing over one of the three qubits we obtain ρ W = 1 ( ), (5.24) 3 (here we have not specified the bipartition because the W-state is symmetric for permutations of the qubits). Therefore, the concurrence of the reduced state is C 2 (ρ W AB) = C 2 (ρ W AC) = 4 9. (5.25) Moreover, if we trace over two of the three qubits we obtain the state ρ W A = , (5.26) 3 and so 4 det ρ W A = 8 9. (5.27) Thus in this case the monogamy inequality reads: = 8 9. (5.28) Notice that in this case the monogamy inequality (5.11) is saturated, meaning that all the entanglement that the three particles shares is pairwise, i.e. there is no tripartite entanglement. 5.3 Monogamy Inequality: A Step Beyond In the last part of their paper [25], Coffman, Kundu, and Wootters conjecture the possibility of a generalization of the monogamy inequality for systems of more than three-qubits. Their guess was answered by Osborne and Verstraete [67] several years later, in To show their result we indicate with q j, for j = 1,... n, the n qubits of the system. If ρ is a general state of n qubits, the following holds C 2 q 1 (q 2...q n)(ρ) C 2 (ρ q1 q 2 ) + C 2 (ρ q1 q 3 ) + + C 2 (ρ q1 q n ). (5.29)

69 It is worth underlining that it sufficient to prove this type of inequalities for pure states. Indeed, we can always consider the optimal decomposition of ρ with respect to the left hand side and follow what we did in the proof of theorem Inequality (5.29) has an interpretation similar to the n = 3 case, see fig The entanglement between q 1 and the rest bounds the sum of the individual pairwise entanglements involving q 1 and each of the other n 1 qubits q j (j = 2,..., n). However, unlike the case n = 3, this time the difference between left and right hand Figure 5.3: The generalized monogamy inequality for n-qubit system proved by Osborne et al. in [67]. side in (5.29) just gives a rough indicator of all the leftover entanglement not distributed in pairwise form, and does not give the genuinely n-partite entanglement that the parties share. Until now, attempts to construct generalized monogamy inequalities in n-qubit systems have been considered trying to use, for instance, different measures of entanglement [37, 50, 26, 35], but these have not led to clear recipes to isolate the genuine n-partite entanglement, nor have resulted in a general sharpening of (5.29) for arbitrary states. In [74] we have proposed a set of monogamy constraints, more general than the ones studied before. The main idea is that the residual in (5.29) contains m-partite contributions which involve m = 3, 4,..., n 1 qubits, in all possible combinations encompassing the focus qubit q 1. Therefore, we moved from the hypothesis that it is amenable to a further decomposition. The relevant aspect here is that heuristically one can expect that all the multipartite contributions of this type are independent and that the global bipartite entanglement between q 1 and the rest of the system can be obtained adding up all these quantities. We postulate a hierarchy of strong monogamy inequalities that limit the distribution of entanglement, bi- and multipartite, in n-qubit systems. Their general form is

70 64 the following τ (1) q 1 (q 2...q n) (ρ) + n k>j=2 [τ (3) n τ (2) (ρ q1 q 2 ) j=2 } {{ } 2-partite q 1 q j q k ( ψ )] µ } {{ } 3-partite n l=2 q 1 q 2 q l 1 q l+1 q n ( ψ )] µ n 1, [τ (n 1) } {{ } (n 1)-partite (5.30) where we indicate with τ (j), for j = 2,..., n 1, a suitable function that measures the j 1-partite entanglement and with τ (1) a measure for the entanglement between one qubit and the rest of the system. Notice that here we included a sequence of rational exponents {µ m } n 1 m=2, with µ 2 1, whit the aim of regulating the weight assigned to the different m-partite contributions. If we use the notation j m = (j1 m,..., jm 1) m to indicate the vector which spans all the ordered subsets of the index set {2,..., n} with (m 1) distinct elements, we can write our strong monogamy inequality in the short-hand notation: τ (1) q 1 (q 2...q n 1 n) m=2 j m [τ (m) q 1 q j m 1 q j m m 1 ( ψ )] µm. (5.31) Our conjecture is that, provided a suitable choice of the entanglement measures and their weight, {τ (m), µ m }, the inequality (5.31) holds for arbitrary states of n qubits, together with its variants for different choices of the privileged qubit. A constraint like this holds for the distribution of entanglement in permutationallyinvariant continuous variable Gaussian states [2, 3], and this gives a strong hint that a similar sharing structure should hold for entanglement in finite-dimensional systems too, although no supporting evidence was obtained prior to this work. Before going further in the discussion on our conjecture we want to underline that the powers µ m parametrize a whole class of monogamy constraints, for any given choice of the measures of entanglement. Any choice of this sequence, with the constraint µ 2 1, gives a new monogamy inequality, generalizing the one in [25]. Moreover, the verification of (5.31) given a set {µ m} implies its validity for all {µ m } {µ m}. This means that the aim should be to prove the inequalities fixing each µ m to be as small as possible. The best choice as possible, of course, is µ m = 1 m.

71 Our Conjecture: Notation and Toolkit In this section we want to precise the prescriptions we adopt and to give all the tools we need in order to show the validity of our conjecture. First of all we have to specify the measures τ (k) we want to use in practice. As the 3-tangle, we define the pure-state residual k-tangle τ (k) as the difference between left and right hand side in (5.30), k 1 τ (k) q 1 q 2 q k ( ψ ) := τ (1) q 1 (q 2 q k ) ( ψ ) [τ (m) q 1 q j m q 1 j ( ψ )] µm. (5.32) m m 1 m=2 j m In this way, we can reshape the conjectured strong monogamy inequality (5.30) into the non-negativity of the residual τ (k) q 1 q 2 q k ( ψ ) 0, (5.33) where the ordering of the subscripts in (5.32) reflects the choice of the focus qubit, which occupies the first slot (we do not expect permutation invariance for n > 3). The next step is the extension of the residual k-tangle τ (k) for mixed states. Given a state ρ of n qubits we define its residual tangle using a non-conventional convex roof procedure, [ τ (k) q 1 q 2 q k (ρ) := inf {p j, ψ j } j p j τ (k) q 1 q 2 q k (ψ j )] 2, (5.34) where the minimization is taken over all possible pure-state decompositions of the state ρ = r p j ψ j ψ j. The reason of this choice lies in the fact that for n = 3, the definition (5.34) reduces to the mixed-state extension of the three-tangle τ (3) as defined in [85], which has been proven to be an entanglement monotone [25, 32, 59]. = Cq 2 i q j, with the square of the concurrence. Using Eqs. (5.32) (5.34) in a recursive way, we can define every m-partite term τ (m) (for m 2) appearing in the n-qubit strong monogamy inequality (5.30), in terms of the corresponding residual m-tangle rescaled by a suitable exponent µ m. Moreover, for n = 2, we recover the standard pairwise tangle, τ (2) q i q j An Interesting Hint The simplest (non-trivial) examples of states for which the strong monogamy holds are the generalized GHZ- and W-states to n 4 qubits, defined as W n = 1 n ( 0 n n 1 ) (5.35) GHZ n = 1 2 ( 0 n + 1 n ), (5.36)

72 66 where x n denotes the string with n equal symbols x. We begin by investigating the strong monogamy constraint (5.30) in its sharpest form (µ m = 1 m), on states defined as the superpositions of this two states. In particular we consider states of the form Φ n α,β,γ := α 0 n + β W n + γ 1 n, (5.37) with α, β, γ C, α 2 + β 2 + γ 2 = 1. Notice that states of this type are permutationally invariant, so that all the calculations are simplified. Rewriting the state Φ n α,β,γ as ( Φ n α,β,γ = 0 n m α 0 m + + ) m n β W m (5.38) n m n β W n m 0 m + γ 1 n m 1 m, (5.39) for 1 m n 1, where W 1 1, we can observe that the the particular choice γ = 0 makes all the residual multipartite terms vanish τ (1) ( Φ n α,β,0 ) = 0 = τ (2) ( Φ n α,β,0 ), (5.40) being the state separable. This consideration leads us to the following inductive result. Assume the strong monogamy inequality (5.30) holds for arbitrary pure states of m < n qubits, then for the n-qubit states Φ n α,β,γ one has, by direct calculations: τ (1) = 4 n 2 [ n 2 α 2 γ 2 + (n 1) β 2 ( β 2 + n γ 2 ) ], (5.41) τ (2) 4 β 4 n 2, (5.42) τ (n 1) 4 n β 2 γ 2, (5.43) τ (m) = 0 (5.44) for 2 < m < n 1. Substituting these into eq. (5.32), one finds τ (n) q 1... q n ( Φ n α,β,γ ) 4 α 2 γ 2 0, (5.45) which proves the validity of the strong monogamy inequality (5.30) for the n-qubit states (5.37).

73 The Four Qubit Case It is not hard to convince ourselves that proving the strong monogamy conjecture for n qubits is in general a formidable challenge. Indeed, at variance with the 3-qubit case, we do not have a closed formula for all the tangles we defined for general mixed states and computing all the minima in (5.30) is a non-trivial task. In the following we focus on the 4- qubit case, verifying the conjecture analytically on relevant multi-qubit states, and providing a comprehensive collection of analytical and numerical evidence in support of the strong monogamy hypothesis. Once again we want to stress that the extension to n-qubit mixed states ρ comes automatically once they are established for arbitrary pure states ψ. For this reason, it will be enough to keep the subsequent analysis focused on globally pure states ψ, for which at least the left hand side of (5.30) is computable via eq. (5.14). Unfortunately, a preliminary numerical exploration of the 4-qubit strong monogamy reveals that the choice µ m = 1 is too strong to hold, as it leads to negative residual four-tangles on a small subset of states. We proceed testing the strong monogamy inequality for successive level of the hierarchy and we set µ m := m/2 (m 2). Then eq. (5.30) specializes to (see fig. 5.4 for a graphical representation): τ (1) q 1 (q 2 q 3 q 4 ) τ (2) q 1 q 2 + τ (2) q 1 q 3 + τ (2) q 1 q 4 + [τ (3) q 1 q 2 q 3 ] [τ (3) q 1 q 3 q 4 ] [τ (3) q 1 q 2 q 4 ] 3 2, (5.46) where we focus on qubit q 1 and we omitted the state ( ψ ) for brevity.... Figure 5.4: The strong monogamy inequality for 4-qubit systems. Notice that all the quantities in (5.46) are well defined. In particular the bipartite terms τ (1) and τ (2) are computable using the concurrence as described in section 5.1, while the tripartite terms τ (3) are to be evaluated on the reduced mixed state ρ ijk of qubits q i, q j, and q k, that have rank 2 for the Schmidt decomposition theorem, via the prescription in Eq. (5.34). It is worth noting that the 3-tangle of three-qubit pure states admits the following closed expression [25]. If the Fourier expansion of the pure state in the computational

74 68 basis is ψ = 1 r,s,t=0 z rst rst, then τ (3) q 1 q 2 q 3 ( ψ ) = 4 z z z001 2 z z010 2 z z100 2 z (z 000 z 111 z 001 z z 000 z 111 z 010 z 101 +z 000 z 111 z 100 z z 001 z 110 z 010 z 101 (5.47) +z 001 z 110 z 011 z z 100 z 011 z 010 z 101 ) +4(z 000 z 011 z 101 z z 111 z 100 z 010 z 001 ). However, the problem to face is always the same. To date, there is no closed formula for the three-tangle of three-qubit mixed states. Nevertheless, there are some special cases in which the problem is, at least partially, solved [36] and a method to determine when the 3-tangle vanishes for rank-2 states can be found in [62]. Our way to tackle the problem is to looking for easy-to-treat upper bounds for the 3-tangle, τ (3)up q i q j q k τ (3) q i q j q k, in such a way that a lower bound to the residual four-tangle of eq. (5.32) is τ (4)low q 1 q 2 q 3 q 4 := τ (1) q 1 (q 2 q 3 q 4 ) 4 j=2 τ (2) q 1 q j 4 k>j=2 and the strong monogamy inequality is verified proving that [ τ (3)up q 1 q j q k ] 3 2, (5.48) τ (4)low q 1 q 2 q 3 q 4 0. (5.49) The particular bound we exploit is the one introduced by Rodriques, Datta, and Love (RDL) in [75]. Theorem (Rodriques-Datta-Love bound). Let ρ, π B(H) be two density operators satisfying supp(π) supp(ρ). Then there exists a positive constant k > 0 such that the operator defined as σ ρ := ρ + k (ρ π), (5.50) D(ρ, π) where D(ρ, π) = ρ π 1 is the trace distance, is a state, and such that rank σ ρ < rank ρ. Moreover, if E : B(H) R is a non-negative convex function bounded from above, then D(ρ, π) E(ρ) E(π) D(σ ρ, π) (E(σ ρ) E(π)). (5.51)

75 69 Notice that every measure of entanglement and entanglement monotone satisfies the hypotheses of the theorem to be non-negative, convex and bounded from above, so that we can use this result to bound, in particular, the 3-tangle. Furthermore, from theorem 5.4.1, we can construct an algorithm to practically evaluate τ (3)up q 1 q j q k for every given rank-2 mixed state ρ [75]. Using its spectral decomposition, we can write ρ as ρ = λ (1 λ) 2 2, (5.52) k k, for k = 1, 2, are the normalized eigenvectors of ρ. We can construct a simplex S 0 containing mixtures of (up to) four pure sates Z l (l = 1,..., 4) of the form Z l = zl 2 ( 1 + z l 2 ), (5.53) with vanishing 3-tangle. Therefore, z l C are the complex roots of the fourth-order equation τ (3) ( 1 + z 2 ) = 0, (5.54) defined via eq. (5.47). If the rank-2 state ρ belongs to the simplex S 0, then τ (3) (ρ) = 0. On the contrary, we can define the uniform mixture π = Z l Z l, (5.55) l=1 and use theorem to find the constant κ > 0 such that σ ρ := ρ + κ ρ π 1 (ρ π) (5.56) describes a three-qubit state. Notice that for the theorem rank σ ρ < rank ρ = 2, so that σ ρ is a pure state. One has then τ (3) (ρ) τ (3)up (ρ) := ρ π 1 φ φ π 1 τ (3) ( φ ), (5.57) where τ (3) ( φ ) can be computed from eq. (5.47) An Example Let us consider superposition states of the form ψ p = p W p W 4, (5.58)

76 70 Normal-form states G x (unnormalized) 1 G 1 abcd = a+d 2 + b+c 2 ( ) + a d 2 ( ) + b c 2 ( ) ( ) 2 G 2 abc = a+b a b ( ) + ( ) 2 2 +c( ) G 3 ab = a( ) + b( ) G 4 ab a+b = a( ) + ( ) 2 i ( ) + 2 ( ) + a b 2 5 G 5 a = a( ) +i i G 6 a = a( ) G 7 = G 8 = G 9 = Table 5.1: Normal-form representatives of the nine four-qubit SLOCC classes defined in [85]. where 0 p 1, W 4 is the generalized W-state for 4 qubits, and W is defined by W 4 = 1 ( ). (5.59) 2 The states ψ p are permutationally invariant, which allows us to omit the subscripts defining the ordering of the subsystems, and to rewrite the strong monogamy inequality (5.46) as τ (1) 3 τ (2) + 3 (τ (3) ) 3 2. (5.60) We can bound the three-tangle τ (3) by the RDL method [75] described above. Denoting by ρ p the reduced state of any three qubits, its eigenvectors are 1 p = p 1 p ( ) (5.61) 1 + 2p 1 + 2p

77 71 and 2 p = p 1 p ( ). (5.62) 3 2p 3 2p In this particular case, the polynomial equation τ (3) ( 1 p + z 2 p ) = 0 is biquadratic and admits two doubly degenerate roots z = ± ξ, with ( ) 3 2p p ξ = 1 + 2p 1 p. (5.63) We can then construct the mixture [ 1p 1 p + ξ 2 p 2 p ] π = (1 + ξ), (5.64) with vanishing three-tangle, and define a pure state as in Eq. (5.56), which turns out to be simply φ p 1 p. This yields an upper bound to the three-tangle of ρ p as in Eq. (5.57); namely, τ (3)up = p(1 p) 2p (1 p). (5.65) Concerning the bipartite terms in the strong monogamy inequality for ψ p, we have τ (1) = 3 + p(1 p) (5.66) 4 τ (2) = 1 4 p(1 p) + p(1 p). (5.67) Putting all together, we get for the residual four-partite term of ψ p, τ (4) τ (4)low τ (1) 3τ (2) 3(τ (3)up ) 3 2 0, (5.68) which proves the conjectured strong monogamy inequality (5.46). 5.5 Four-Qubit Strong Monogamy: Results We have stressed many times that local quantum operations cannot affect the entanglement of the system. Therefore, it is intuitive to classify entangled states dividing them in invariant classes under this type of operations. In particular, the operation that are used are reversible stochastic local quantum operations assisted by classical communication, the so-called SLOCC operations [32]. The requirement here is that we consider LOCC operations without imposing that they can be achieved with unit certainty. It has been proved [32] that for pure states of three qubits there are 4 classes of states (separable, biseparable, GHZ, W), while for four qubits, there are infinitely many inequivalent SLOCC classes [35]. However, in [85], Verstraete et al.

78 72 Figure 5.5: (Lower bound to the residual four-tangle τ (4)low q 1 q 2 q 3 q 4 versus the parameter a (here assumed real) for the normal-form states: G 2 abc with c = b + 1 = a (red solid line), G 3 ab with b = a/4 (green dashed line), G4 ab with b = a/2 (blue dotted line), G 5 a (magenta dot-dashed line), G 6 a (black dot-dot-dashed line). The residuals stay nonnegative for general choices of the parameters a, b, c. give a particularly insightful classification into nine groups. They showed that, up to permutations of the four qubits, any pure state ψ can be obtained as ψ = (A 1 A 2 A 3 A 4 ) G x, (5.69) where {A k } SL(2, C) are SLOCC operations with det(a k ) = 1, and each G x denotes a normal-form family of states, representative of the corresponding x th class, with x = 1,..., 9, see table 5.1 for their definition. We verified the conjectured strong monogamy inequality (5.46) for the representatives of the normal-forms G x of all the nine classes. Precisely, we obtained suitable analytic upper bounds to the τ (3) terms in all the threepartitions, as presented in Table 5.2, and Combined these bounds with the one-tangles τ (1) (q i q j q k q l ) squared concurrences τ (2) q i q j.the lower bounds to the residual τ (4) q i q j q k q l, and with the we obtained are plotted in Fig. 5.5 for some typical instances of G x with x = 2,..., 6. The other cases are straightforward, in particular for G 1 and G 7 all the reduced three-tangles vanish, so the strong monogamy reduces to the well-known inequality (5.29). We complement this collection of analytical results with a numerical exploration of arbitrary four-qubit states ψ. In each of the nine classes we generated states with randomized parameters, applying random SLOCC operations on G x. In particular we used a Gaussian distribution to generate the matrix elements of SLOCC operations on each qubit, and a uniform distribution in a bounded interval to generate the complex parameters in the states G x. We tested 10 6 states per class and on each state we computed the lower bound τ (4)low q i q j q k q l using the semi-analytical method we showed in the previous section [75] to bound the three-tangles in all relevant three-qubit partitions. Overall, this amounts to tested data points across all the different

79 73 Bounds to the reduced three-tangles τ (3) 1 τ (3) q i q j q k = 0 2 τ (3) q i q j q k 4 a2 b 2 c 2 ( a 2 + b 2 +4 c 2 ) 2 3 τ (3) q 1 q 2 q 3 = τ (3) q 1 q 3 q 4 = 0, τ (3) q 1 q 2 q 4 = τ (3) q 2 q 3 q 4 4 a b (1+ a 2 + b 2 ) 2 4 τ (3) q i q j q k 2 a2 b 2 (2+3 a 2 + b 2 ) 2 5 τ (3) q 1 q 2 q 3 = τ (3) q 1 q 3 q 4 16 a 2, τ (3) (3+4 a 2 ) 2 q 1 q 2 q 4 = τ (3) 4 q 2 q 3 q 4 (3+4 a 2 ) 2 6 τ (3) q 1 q j q k = 0, τ (3) q 2 q 3 q 4 { a ( a 3 4) 2 7 τ (3) q i q j q k = 0 8 τ (3) q 1 q j q k 1, τ (3) 4 q 2 q 3 q 4 = 0 9 τ (3) q 1 q j q k = 0, τ (3) q 2 q 3 q 4 = 1 (2 a 2 +3) 2 a < 2 2/3 0 a 2 2/3 Table 5.2: Upper bounds to the three-tangle of marginal three-qubit partitions q i q j q k ; here a, b, c, d are complex parameters with nonnegative real part. classes (class-9 states are excluded since for them q 1 is separable from the rest). As fig. 5.6 shows, no negative values of τ (4)low were found, providing a strongly supportive evidence for the validity of the strong monogamy inequality (5.46) on arbitrary four-qubit states.

80 (4)low Figure 5.6: Lower bound to the residual four-tangle τqi qj qk ql versus the one-tangle (1) τqi (qj qk ql ) for random four-qubit pure states, with 4 partitions tested per state. Each point is gray-scaled according to the SLOCC class of the state, from 90% gray (darkest, class 1) to 20% gray (lightest, class 8). The solid line is saturated by GHZ states. All the data points are above the horizontal axis, verifying the strong monogamy inequality (5.46).