Quantum Entanglement: Detection, Classification, and Quantification

Transcription

1 Quantum Entanglement: Detection, Classification, and Quantification Diplomarbeit zur Erlangung des akademischen Grades,,Magister der Naturwissenschaften an der Universität Wien eingereicht von Philipp Krammer betreut von Ao. Univ. Prof. Dr. Reinhold A. Bertlmann Wien, Oktober 2005

2 CONTENTS 1. Introduction Basic Mathematical Description Spaces, Operators and States in a Finite Dimensional Hilbert Space Bipartite Systems Qubits Qutrits Positive and Completely Positive Maps Detection of Entanglement Introduction Pure States General States Nonoperational Separability Criteria Operational Separability Criteria Classification of Entanglement Introduction Free and Bound Entanglement Distillation of Entangled States Bound Entanglement Locality vs. Non-locality EPR and Bell Inequalities General Bell Inequality Bell Inequalities and the Entanglement Witness Theorem Quantification of Entanglement Introduction Pure States General States Entanglement of Formation

3 Contents Concurrence and Calculating the Entanglement of Formation for 2 Qubits Entanglement of Distillation Distance Measures Comparison of Different Entanglement Measures for the 2-Qubit Werner State Hilbert-Schmidt Measure and Entanglement Witness Introduction Geometrical Considerations about the Hilbert-Schmidt Distance The Bertlmann-Narnhofer-Thirring Theorem How to Check a Guess of the Nearest Separable State Examples Isotropic Qubit States Isotropic Qutrit States Isotropic States in Higher Dimensions Tripartite Systems Introduction Basics Pure States Detection of Entangled Pure States Equivalence Classes of Pure Tripartite States General States Equivalence Classes of General Tripartite States Tripartite Witnesses Conclusion

4 1. INTRODUCTION What is Quantum Entanglement? If we look up the word entanglement in a dictionary, we find something like state of being involved in complicated circumstances, the term also denotes an affair between two people. Thus in quantum mechanics we could describe quantum entanglement literally as a complicated affair between two or more particles. The first one to introduce the term was Erwin Schrödinger in Ref. [68]. Since this article was published in German, entanglement is a later translation of the word Verschränkung. Schrödinger does not refer to a mathematical definition of entanglement. He introduces entanglement as a correlation of possible measurement outcomes and states the following: Maximal knowledge of a whole system does not necessarily include knowledge of all of its parts, even if these are totally divided from each other and do not influence each other at the present time. Note that system is always a generalized expression for some physical realization; in this context a system of two or more particles is meant. Nowadays the definition of entanglement is a mathematical one and rather simple (see Chapter 2) however, the phenomenological description of entanglement is still difficult. Since J.S. Bell introduced his Bell Inequality [3] it has become clear that the correlations related to quantum entanglement can be stronger than merely classical correlations. Classical correlations are those that are explainable by a local realistic theory, and it was propagated by Einstein, Podolsky, and Rosen [31] that quantum mechanics should also be a local realistic theory (see Sec. 4.3). The mysteriousness inherent to quantum entanglement mainly comes from the fact that often in cannot be explained with a classical deterministic model [3, 4] and so underlines the new physics that comes with quantum mechanics and distinguishes it from classical physics. Why do we need quantum entanglement? What at first seemed to be a more philosophical investigation became of practical use in recent years. With the development of quantum information theory a new quantum way of information processing and communication was initiated which makes direct use of

5 1. Introduction 5 quantum entanglement and takes advantage of it (see, e.g., Refs. [16, 13, 45]). There are various tasks involving quantum entanglement that are an improvement to classical information theory, for example quantum teleportation and cryptography (see, e.g., Refs. [5, 17, 15, 32]). In the course of years computing has become and still becomes more and more efficient, information has to be encoded into less physical material. To be able to keep pace with the technological demand, quantum information theory could serve as the future concept of information processing and communication devices. It is therefore not only of philosophical but also of practical use to deepen and extend the description of quantum entanglement. Aim and Structure of this Work. The aim of this work is to provide a basic mathematical overview of quantum entanglement which includes the fundamental aspects of detecting, classifying, and quantifying entanglement. Several examples should give insight of the explicit application of the given theory. There is no emphasis on detailed proofs. Nevertheless some proofs that are useful to be explicitly mentioned and do not take too long are given, otherwise the reader is refered to other literature. The main part of the work is concerned with bipartite systems, these are systems that consist of two parts (i.e. particles in experimental application). The work is organized as follows: In Chapter 2 we start with basic mathematical concepts. Next, in Chapter 3, we address the problem of detecting entanglement; in Chapter 4 entanglement is classified according to certain properties, and in Chapter 5 we discuss several methods to quantify entanglement. In Chapter 6 we combine the concept of detecting and quantifying entanglement. Finally, in Chapter 7, we briefly take a look on tripartite systems.

6 2. BASIC MATHEMATICAL DESCRIPTION 2.1 Spaces, Operators and States in a Finite Dimensional Hilbert Space Operators act on the Hilbert space H of a quantum mechanical system, they make up a Hilbert Space themselves, called Hilbert-Schmidt space A. We are only interested in finite dimensional Hilbert spaces, so that in fact A can be regarded as a space of matrices, taking into account that in finite dimensions operators can be written in matrix form. A scalar product defined on A is (A, B A) A, B = TrA B, (2.1) with the corresponding Hilbert-Schmidt norm A = A, A A A. (2.2) Matrix Notation. Generally, any operator A A can be expressed as a matrix with the elements A ij = e i A e j, (2.3) where e i and e j are vectors of an arbitrary basis {e i } of the Hilbert Space. Of course the same holds for states, since they are operators. Definition of a State. An operator ρ is called state (or density operator or density matrix) if 1 Trρ = 1, ρ 0, (2.4) where ρ 0 means that ρ is a positive operator (more precise: positive semidefinite), that is, if all its eigenvalues are larger than or equal to zero. Positivity of ρ can be equivalently expressed as where P is any projector, defined by P 2 = P. 2 TrρP 0 P, (2.5) 1 Certainly the presented conditions refer to the matrix form of a state ρ. 2 Eq. (2.5) follows from the fact that the eigenvalues are nonnegative, since ρ can be written in appropriate matrix notation in which it is diagonal, where the eigenvalues are

7 2. Basic Mathematical Description 7 Remark. In early quantum mechanics (pure) states are represented as vectors ψ in Hilbert Space. This concept is widened with the introduction of mixed states, so that in general states are viewed as operators. If one is interested in pure states only, either the vector representation ψ or the operator representation ρ pure = ψ ψ can be used. Note that Eqs. (2.4) and (2.5) imply Trρ In particular we have Trρ 2 = 1 ρ is a pure state, Trρ 2 < 1 ρ is a mixed state. (2.6) 2.2 Bipartite Systems In all chapters but the last we will consider bipartite systems. Following the convention of quantum communication, the two parties are usually referred to as Alice and Bob. For bipartite systems the Hilbert space is denoted as H d 1 A Hd 2 B, where d 1 is the dimension of Alice s subspace and d 2 is Bob s, or just H A H B when there need not be a special indication to the dimensions. We may also drop for convenience the indices A and B, e.g. we will often consider the Hilbert Space H 2 H 2, states on this space are called 2-qubit states. Matrix Notation. In general, we can write a state ρ as a matrix according to Eq. (2.3). However, often we have to use a product basis, to guarantee that certain calculations etc. make sense. In this case for the matrix notation of a state ρ on H d 1 A Hd 2 B we have ρ mµ,nν = e m f µ ρ e n f ν. (2.7) Here {e i } and {f i } are bases of Alice s and Bob s subspaces. Reduced Density Matrices. The notation in a product basis is for example needed to calculate the reduced density matrices of a state ρ. These are obtained if Alice neglects Bob s system, or vice versa, which mathematically means she takes a partial trace of the density matrix, she traces out Bob s the diagonal elements. Now multiplying this diagonal matrix with a projector cannot give a matrix of negative trace, since projectors in matrix notation need to have positive diagonal entries. 3 This is because all eigenvalues have to be smaller than 1.

8 2. Basic Mathematical Description 8 system. The notation is ρ A = Tr B ρ, ρ B = Tr A ρ, (2.8) where ρ A denotes Alices reduced density matrix and ρ B Bob s. The matrix elements of the reduced density matrices are (ρ A ) mn = (ρ B ) µν = d 2 β=1 d 1 a=1 ρ mβ,nβ, ρ aµ,aν. (2.9) Definition of Entangled Pure States. A bipartite pure state is called entangled if it cannot be written as a single product of vectors which describe states of the subsystems, i.e. ψ prod = ψ A ψ B. (2.10) Such a state that is not entangled is called product state. General Definition of Entanglement. A state ρ is called separable if it can be written as a convex combination of product states, i.e. [76] ρ = i p i ρ i A ρ i B, 0 p i 1, i p i = 1. (2.11) All separable states are the elements of the set of separable states S. If a state is not separable in the sense of Eq. (2.11), then it is called entangled. Why this Definition of Separability? Naturally the question arises why exactly (2.11) is the definition of separability (as being the counterpart of entanglement). When it was introduced by Werner in Ref. [76] he gave a plausible physical reasoning: Werner differentiated between uncorrelated states and classically correlated states (which both were denoted later as separable states). An uncorrelated state is a product state that can be written as ρ = ρ A ρ B, (2.12)

9 2. Basic Mathematical Description 9 because then expectation values of joint measurements (denoted by operators A for Alice and B for Bob) on such a state factorize: A B = TrρA B = Tr ( ρ A ρ B) (A B) = Trρ A A Trρ B B. (2.13) Here the classical rule of multiplying probabilities occurs, and this corresponds to the fact that the measurements by Alice and Bob are independent of each other. For the classically correlated states one can think of the following physical preparation devices: Alice and Bob each have a device with a switch that can be set in different positions i = 1,..., n, n > 1. For each setting of the switch the devices prepare states ρ A i and ρ B i. Before the measurement, a random number between 1 and n is drawn, and the switches of the devices are set according to this number. Furthermore, each number i occurs with probability p i. Now the expectation value of a measurement A B will be a weighed sum of factorized expectation values: A B = = n p i Trρ A i ATrρ B i B i=1 n p i Tr ( ρ A i i=1 ρ B i ) (A B) =: TrρA B. (2.14) Here we defined ρ like in Eq. (2.11). With this definition of ρ we can write the expectation value as one obtained from a single state, and this state is called classically correlated. We say classically because the preparation of this state is done merely classical, and correlated because the expectation value no longer factorizes but has to be written as a weighed sum like Eq. (2.14). The definition (2.11) contains both the product and the classically correlated states, since here n 1, so the uncorrelated states are referred to as well if n = 1. Fraction. The fraction or fidelity of a state ρ with respect to a maximally entangled pure state ψ max is given by F ψmax := ψ max ρ ψ max (2.15) Eq. (2.15) is nothing but the probability that the resulting state of a projective measurement (in a basis where ψ max is one basis vector) is ψ max. So the range of possible values of F ψmax (ρ) is 0 F ψmax (ρ) 1.

10 2. Basic Mathematical Description 10 Isotropic States. We define an isotropic state ρ (d) α on a Hilbert Space H d H d as (see Refs. [39, 45]): ρ (d) α = α φ d + φ d α 1 1, α R, 1 d 2 d 2 1 α 1. (2.16) Here d is the dimension of the Hilbert space H d H d, the range of α is determined by the positivity of the state. The state φ d + is maximally entangled and given by φ d 1 d 1 + = d i i, (2.17) where { i } is a basis in H d. The state is called isotropic because it is invariant under any U U transformations [39] (U is a unitary operator, U is its complex conjugate) i=0 (U U )ρ (d) α (U U ) = ρ α. (2.18) The isotropic state (2.16) has the following properties [39]: 4 1 d 2 1 α 1 ρ (d) α separable, d d + 1 < α 1 ρ(d) α entangled. (2.19) Instead of the parameter α in Eq. (2.16) we can also define an equivalent isotropic state ρ F with the fraction F (2.15) as the parameter. In case of ψ max = φ d + (2.17) we write shortly Fφ+ := F. According to Eq. (2.15) we get F = φ d + ρ (d) α φ d 1 + α(d 2 1) + =, (2.20) d 2 or α = d2 F 1 d 2 1. (2.21) Inserting Eq. (2.21) into the definition (2.16) we get the equivalent form of an isotropic state (( ρ (d) F = d2 F 1 ) ) φ d d 2 1 d 2 + φ d (1 F ) (2.22) d 2 4 The entangled property of the isotropic state is prooved by using the reduction criterion (see Theorem 3.8) in Sec It is shown in Ref. [39] that for the remaining values of the parameter α the state can be written as a mixture of product states and thus is separable (see Eq. (2.11)).

11 2. Basic Mathematical Description Qubits Single Qubits. A qubit state ω, acting on H 2, can be decomposed in terms of Pauli matrices (we use the convention to sum over same indices): ω = 1 2 ( 1 + ni σ i), n i R, i n 2 i = n 1. (2.23) Note that for n 2 < 1 the state is mixed (corresponding to Trω 2 1) whereas for n 2 = 1 the state is pure (Trω 2 = 1). 2 Qubits. According to the notation (2.7) the density matrix of 2 qubits, acting on H 2 H 2, has the form ρ 11,11 ρ 11,12 ρ 11,21 ρ 11,22 ρ = ρ 12,11 ρ 12,12 ρ 12,21 ρ 12,22 ρ 21,11 ρ 21,12 ρ 21,21 ρ 21,22. (2.24) ρ 22,11 ρ 22,12 ρ 22,21 ρ 22,22 The matrix (2.24) is usually obtained by calculating its elements in the standard product basis (e 1 = f 1 = 0, e 2 = f 2 = 1 ) which has the properties { 0 0, 0 1, 1 0, 1 1 }, (2.25) i j = δ ij. (2.26) Alternatively, we can write any 2-qubit density matrix in a basis of the 4 4 matrices composed of the identity matrix and the Pauli matrices, ρ = 1 4 ( ai σ i 1 + b i 1 σ i + c ij σ i σ j), a i, b i, c ij R. (2.27) A product state ρ A ρ B has the form ρ A ρ B = 1 4 (1 1 + n iσ i 1 + m i 1 σ i + n i m j σ i σ j ), n i, m i R, n 1, m 1. (2.28) Any separable state (2.11) can be written as the convex combination of expressions (2.28), ρ sep = k p ( k n k i σ i 1 + m k i 1 σ i + n k i m k j σ i σ j), n k i, m k i R, n k 1, m k 1. (2.29)

12 2. Basic Mathematical Description 12 Bell Basis. A basis in H 2 H 2 is the Bell basis, which consists of 4 orthonormal maximally entangled pure states: ψ = 1 2 ( ) (2.30) ψ + = 1 2 ( ) (2.31) φ = 1 2 ( ) (2.32) φ + = 1 2 ( ). (2.33) Isotropic Qubit State. We can write a 2-qubit isotropic state ρ (2) F a mixture of the Bell states (2.30) - (2.33): ρ (2) F =: ρ F = F φ + φ F F 3 ψ ψ + 1 F 3 ψ + ψ + + (2.22) as φ φ, 0 F 1. (2.34) Werner State. A state we will often use in examples is the 2-qubit Werner state (introduced for general dimensions in [76] and for 2-qubits in this form in [62]) ρ α = α ψ ψ + 1 α 1 1, α 1. (2.35) Note that the interval for α follows from the necessity that Trρ = 1. The matrix notation of ρ α in the standard basis (2.25) is, according to Eq. (2.24): 1 α α α ρ α = α 1+α 0 0. (2.36) α Qutrits Single Qutrits. The description of qutrits is very similar to the one for qubits. A qutrit state ω on H 3 can be expressed in the matrix basis {1, λ 1, λ 2,..., λ 8 } with an appropriate set of coefficients {n i } ω = 1 (1 + ) 3 n i λ i, n i R, n 2 i = n 2 1. (2.37) 3 i

13 2. Basic Mathematical Description 13 The factor 3 is included for a proper normalization, i.e. Trω 2 1 (see also Refs. [2, 20]). The matrices λ i (i = 1,..., 8) are the eight Gell-Mann matrices i λ 1 = 1 0 0, λ 2 = i 0 0, λ 3 = 0 1 0, λ 4 = λ 7 =, λ 5 = i 0 i i i 0 0, λ 8 = 1 3, λ 6 = ,, (2.38) with properties Tr λ i = 0, Tr λ i λ j = 2 δ ij. Note that a matrix of Eq. (2.37) with an arbitrary set of coefficients {n i } is a density matrix only if it is positive - unlike the qubit case there exist sets {n i } for which the matrix is not a state, as can be seen in the following example [53]: Example. Let us consider a set of coefficients {n i } where all coefficients vanish except n 8. According to Eq. (2.37) the only possible values for this coefficient are n 8 = +1 or n 8 = 1. If we have n 8 = +1, then we get for a matrix A +1 formed like in Eq. (2.37) A +1 = 1 ( ) = (2.39) Although we have TrA +1 =1, A +1 is not a state because one eigenvalue, i.e. 1/3, is negative. On the other hand, if n 8 = 1, we find A 1 = 1 3 ( 1 ) 3 = , (2.40) which clearly is a state since TrA +1 =1 and A +1 0, we can write A 1 = ω to maintain the notation of Eq. (2.37).

14 2. Basic Mathematical Description 14 2 Qutrits. For 2-qutrit states (that is, bipartite qutrit states acting on H 3 H 3 ) the 9 9 matrix notation according to Eq. (2.7) is ρ = ρ 11,11 ρ 11,12 ρ 11,13 ρ 11,21 ρ 11,22 ρ 11,23 ρ 11,31 ρ 11,32 ρ 11,33 ρ 12,11 ρ 12,12 ρ 12,13 ρ 12,21 ρ 12,22 ρ 12,23 ρ 12,31 ρ 12,32 ρ 12,33 ρ 13,11 ρ 13,12 ρ 13,13 ρ 13,21 ρ 13,22 ρ 13,23 ρ 13,31 ρ 13,32 ρ 13,33 ρ 21,11 ρ 21,12 ρ 21,13 ρ 21,21 ρ 21,22 ρ 21,23 ρ 21,31 ρ 21,32 ρ 21,33 ρ 22,11 ρ 22,12 ρ 22,13 ρ 22,21 ρ 22,22 ρ 22,23 ρ 22,31 ρ 22,32 ρ 22,33 ρ 23,11 ρ 23,12 ρ 23,13 ρ 23,21 ρ 23,22 ρ 23,23 ρ 23,31 ρ 23,32 ρ 23,33 ρ 31,11 ρ 31,12 ρ 31,13 ρ 31,21 ρ 31,22 ρ 31,23 ρ 31,31 ρ 31,32 ρ 31,33 ρ 32,11 ρ 32,12 ρ 32,13 ρ 32,21 ρ 32,22 ρ 32,23 ρ 32,31 ρ 32,32 ρ 32,33 ρ 33,11 ρ 33,12 ρ 33,13 ρ 33,21 ρ 33,22 ρ 33,23 ρ 33,31 ρ 33,32 ρ 33,33. (2.41) Usually we calculate the elements in the standard product basis (e 1 = f 1 = 0, e 2 = f 2 = 1, e 3 = f 3 = 2 ) { 0 0, 0 1, 0 2, 1 0, 1 1, 1 2, 2 0, 2 1, 2 2 }. (2.42) The basis (2.42) has the properties (2.26). A 2-qutrit state can also be represented in a basis of 9 9 matrices consisting of the unit matrix 1 and the eight Gell-Mann matrices λ i, ρ = 1 ( ai λ i 1 + b i 1 λ i + c ij λ i λ j), a i, b i, c ij R. 9 (2.43) By the same argumentation as for qubits any separable 2-qutrit state is a convex combination of product states, ρ sep = 1 p k ( n ki λ i 1 + ) 3 m ki 1 λ i + 3 n ki m kj λ i λ j. 9 k (2.44) A linear map 2.3 Positive and Completely Positive Maps Λ : A 1 A 2 (2.45) maps operators from a space A 1 into a space A 2. Λ is called positive if it maps positive operators into positive operators, Λ(A) 0 A 0. (2.46)

15 2. Basic Mathematical Description 15 A positive map Λ is called completely positive if the map Λ 1 d : A 1 M d A 2 M d (2.47) is still a positive map for all d = 2, 3, 4...; 1 d is the identity matrix of the matrix space M d of all d d matrices.

16 3. DETECTION OF ENTANGLEMENT 3.1 Introduction In this chapter various methods are described that help deciding whether a given quantum mechanical state is entangled or not. We will see that for pure states the decision is rather easy. For mixed states the situation is more complicated. There is still no key method which could be applied to any state (arbitrary dimensions and number of particles) that always gives a result whether the state is entangled or not. Nevertheless there are some relatively simple methods for states on lower dimensional Hilbert spaces [57, 42, 39, 45]. We have to distinguish between two classes of methods of detecting entanglement: Nonoperational and operational separability criteria. We call a criterion nonoperational if there exists no recipe to perform the criterion on a given state, and operational if such a recipe indeed exists. Apart from that, separability criteria can be necessary or necessary and sufficient conditions for separability. A necessary condition for separability has to be fulfilled by every separable state. So if a state does not fulfill the condition, it has to be entangled - but if it fulfills it, we cannot be sure. On the other hand, a necessary and sufficient condition for separability can only be satisfied by separable states, if a given state fulfills a necessary and sufficient condition, than we can be sure that the state is separable. The chapter is organized as follows: In Sec. 3.2 we briefly discuss the results for pure states, in Sec. 3.3 we consider general states (pure and mixed states) - in particular we investigate nonoperational separability criteria in Sec , whereas in Sec operational criteria are discussed. We will see that for the 2-qubit case H 2 H 2 (and for H 2 H 3 or H 3 H 2 ) there exist operational separability criteria that are necessary and sufficient conditions for separability.

17 3. Detection of Entanglement Pure States We can check easily if a pure state ψ is entangled by looking at the reduced density matrices of ψ ψ : According to Eq. (2.10) the state is a product state if and only if the reduced density matrices are pure states. 1 Example. Let us consider the pure state ψ, where ψ is the singlet state (2.30). When written as a density matrix in the standard product basis (2.25) we get (see (2.24)) ψ ψ = 0 1/2 1/ /2 1/2 0. (3.1) Now we can calculate the reduced density matrices, according to Eqs. (2.8) and (2.9), ( ) 1/2 0 ρ A = ρ B =. (3.2) 0 1/2 We see that the above matrix is a mixed state, since (according to Eq. (2.6)) Trρ 2 A = Trρ2 B < 1. So we conclude that ψ is entangled. 3.3 General States If a state ρ is a mixed state (2.6) then the results of Sec. 3.2 are not valid. The following considerations are valid for mixed and pure states Nonoperational Separability Criteria The Entanglement Witness Theorem (EWT) The following theorem was introduced as a Lemma in Ref. [42], the term entanglement witness originates from Ref. [70]. For further discussion of the subject see, e.g., Refs. [45, 71, 19, 12, 11] Theorem 3.1 (EWT). A state ρ ent is entangled if and only if there exists a Hermitian operator A A, called entanglement witness, such that ρ ent, A = TrAρ ent < 0, ρ, A = TrAρ 0 ρ S. (3.3) 1 A similar method uses the Schmidt decomposition [67] of a pure state ψ (for details see, e.g., Ref. [45]).

18 3. Detection of Entanglement 18 Fig. 3.1: Geometric illustration of a plane in Euclidean space and the different values of the scalar product for states above ( b u ), within ( b p ) and under ( b d ) the plane. Geometric derivation. Theorem 3.1 can be derived via the Hahn-Banach Theorem of functional analysis; this is done in Ref. [42]. Here we want to illustrate how the theorem can be derived with help of the geometrical representation of the Hahn-Bahnach theorem, which states the following (see, e.g., Ref. [65]: Theorem 3.2. Let A be a convex, compact set, and let b / A. Then there exists a hyper-plane that separates b from the set A. First, let us consider the following geometric consideration: In Euclidean space a plane is defined by its orthogonal vector a. The plane separates vectors for which their scalar product with a is negative from vectors with positive scalar product, vectors in the plane have, of course, a vanishing scalar product with a (see Fig. 3.1). This can be compared with our situation: A scalar function ρ, A = 0 defines a hyperplane in the set of all states, and this plane separates up states ρ u for which ρ u, A < 0 from down states ρ d with ρ d, A > 0. States ρ p with ρ p, A = 0 are inside the hyperplane. According to the Hahn-Banach Theorem 3.2, we conclude that due to the convexity of the set of separable states, there always exists a plane that separates an entangled state from the set of separable states. An entanglement witness is optimal, i.e. A opt, if apart from Eqs. (3.1) there exists a separable state ρ S for which ρ, A opt = 0. (3.4) It is optimal in the sense that it defines a tangent plane to the set of separable states S and is called tangent functional for that reason [12]. It detects more entangled states than non optimal entanglement witnesses, see Fig. 3.2.

19 3. Detection of Entanglement 19 Fig. 3.2: Illustration of an optimal entanglement witness The Positive Map Theorem (PMT) In Ref. [42] it is shown that from the EWT (Theorem 3.1) another theorem can be derived: Theorem 3.3 (PMT). A bipartite state ρ is separable if and only if (1 Λ)ρ 0 positive maps Λ. (3.5) The fact that we have (1 Λ)ρ 0 for a separable state ρ can be seen easily [57]: Applying (1 Λ) to a separable state (2.11) gives (1 Λ)ρ = n p i ρ A i Λ(ρ B i ), (3.6) i=1 and since Λ is positive, Λ(ρ B i ) is as well, and so (I Λ)ρ is positive. In Ref. [42] the PMT is proved in the other direction (that a state ρ has to be separable if (1 Λ)ρ 0 positive maps Λ). To put it another way, Theorem 3.3 says that a state ρ ent is entangled if and only if there exists a positive map Λ, such that (1 Λ)ρ ent < 0. (3.7) Here < 0 is short for is not a positive operator. According to Eq. (2.47) this map cannot be completely positive. So it is clear that only not completely positive maps help to detect entangled states. Example. An example for a not completely positive map is the transposition T. To see this, it is enough to show that (1 T ) φ + φ + < 0, (3.8)

20 3. Detection of Entanglement 20 where φ + is defined in Eq. (2.33). Written in matrix notation (2.24) in the standard product basis (2.25) we have: 1/ /2 φ + φ + = (3.9) 1/ /2 We can check the positivity of the state by calculating the eigenvalues: These are {1, 0, 0, 0}, all are positive, as expected. Now what happens if we apply 1 T? We know that the transposition of a 2 2 matrix (A ij ) is simply done by interchanging the indices of the elements: T ((A ij )) =: (A T ij) = (A j i ). So 1 T means that only Bob s part is subjected to transposition, we speak of partial transposition. Only the Greek indices of the matrix elements (2.7) are interchanged: (1 T )(ρ mµ,nν ) =: (ρ T B mµ,nν ) = (ρ mν,nµ ). (3.10) Applying (3.10) on Eq. (2.24) we obtain (1 T ) ψ + ψ + : 1/ ( ψ + ψ + ) T B = 0 0 1/ / (3.11) /2 The eigenvalues of this operator are { 1/2, 1/2, 1/2, 1/2}. One is negative, so the resulting operator is not positive (and hence cannot be called state any longer). We see that T is not a completely positive map Operational Separability Criteria Bell Inequalities In the literature the term Bell inequalities (BIs) is predominantly used for inequalities that can be derived out of the assumption of a local realistic theory, and is violated by states that do not admit such a theory. Special BIs are often named differently, for example CHSH inequality. BIs are famous for showing that for many entangled states it is not possible to apply a local realistic description of measurement processes. For a more detailed discussion and references see Sec Apart from that, BIs can serve as necessary - but not sufficient - separability conditions: Every separable state has to satisfy a BI [76]. So if a state violates a BI, it must be entangled - but if it fulfills it, we cannot be sure.

21 3. Detection of Entanglement 21 The CHSH Inequality as a Seperability Criterion. The CHSH inequality was introduced in Ref. [23] and discussed as a separability criterion in Refs. [40, 70, 45, 47]. Theorem 3.4 (CHSH Criterion). Any 2-qubit separable state ρ has to satisfy the inequality ρ, 21 B 0, B = a σ ( b + b ) σ + a σ ( b b ) σ, (3.12) where a, a, b, b are any unit vectors in R 3 ; σ is the vector out of the three Pauli matrices, σ = (σ x, σ y, σ z ). If for a given state the inequality (3.12) is not fulfilled, then the state is entangled for sure. If it is fulfilled, then we cannot be sure. What at first does not look user friendly is the fact that in order to check if a given state ρ violates the inequality (3.12), we have to check many or even all measurement directions a, a, b, b. Of course we could also minimize over all directions, but in Ref. [40] a theorem is proved that allows to check a violation quite faster: Theorem 3.5. A 2-qubit state violates the CHSH inequality (3.12) for some operator B (some set of measurement directions a, a, b, b ) if and only if M(ρ) > 1. (3.13) Here M(ρ) is the sum of the two greater eigenvalues of a matrix U ρ. The matrix U ρ can be constructed in the following way: First we calculate the matrix elements of a matrix T ρ, (T ρ ) nm = Trρσ n σ m (n, m = 1, 2, 3, σ 1 corresponds to σ x, etc.). Then U ρ = T T ρ T ρ. Example. We want to examine if the Werner state (2.35) violates the CHSH inequality (3.12), and if yes, for what interval of the parameter α. The matrix notation (2.36) can be expressed in a basis of Pauli matrices (see Eq. (2.27)), ρ α = 1 4 (1 α σ σ), 1 3 p 1, (3.14) where we defined σ σ := σ x σ x + σ y σ y + σ z σ z. Written in this way, the matrix elements (T ρ ) nm can easily be calculated. When taking the trace, we remember that TrA B = TrATrB. (3.15) Since Trσ n = 0 n = x, y or z, only the diagonal terms (T ρ ) nn do not vanish, since here Tr(σ n σ n )(σ n σ n ) = 4. These are (T ρ ) nn = α 4 4 = α. (3.16)

22 3. Detection of Entanglement 22 So we have T = α α α, U = α α α 2. (3.17) Now we can calculate the sum of the two greater eigenvalues of U: M(ρ α ) = 2α 2. (3.18) According to Theorem 3.5, ρ α violates the CHSH inequality (3.12) if α > 1 2, (3.19) so we conclude that all Werner states with α > 1 2 are entangled for sure. Entropy Inequalities Other necessary separability criteria are inequalities that compare certain quantum entropies of a state and its reduced density matrix: S(ρ A ) S(ρ) and S(ρ B ) S(ρ) separable states ρ. (3.20) As usual, ρ A and ρ B are Alice s and Bob s reduced density matrices (see Eqs. (2.8) and (2.9)). The inequalities originated from an observation by Schrödinger [68] that an entangled state provides more information about the whole system than about the subsystems. If we associate entropy with the absence of information, then the inequalities (3.20) state the opposite, which is assumed to be a property of separable states. Indeed, for certain quantum entropies the correctness of the inequalities (3.20) has been shown [41, 46]. Here we want to discuss three of them: S 0 (ρ) = log R(ρ), (3.21) S 1 (ρ) = Trρ log ρ, (3.22) S 2 (ρ) = log Trρ 2, (3.23) where R(ρ) is the rank of the matrix ρ, i.e. the number of nonvanishing eigenvalues. The logarithm can be taken to any base, since for different bases, the logarithm functions differ only in some constant which cancels out in the inequality.

23 3. Detection of Entanglement 23 Example. As an example we want to check the inequalities for the Werner state ρ α (2.35). To do this, we first consider the matrix notation (2.36) and calculate the reduced density matrices. We get ( 1 ) 0 (ρ α ) A = (ρ α ) B = 2. (3.24) S 0. First we calculate the S 0 entropies (3.21). The rank of the reduced density matrix is 2, since it has two nonvanishing eigenvalues (can be seen directly from the matrix (3.24), since it is diagonal). In order to determine the rank of ρ α we need to calculate the eigenvalues of ρ α. These are λ 1 = λ 2 = λ 3 = 1 α, λ 4 = 1 + 3α. (3.25) 4 4 If α 1, all eigenvalues are greater than zero and therefore do not vanish. The rank of ρ α is 4. Comparing the S 0 entropies we get 2 4 S 0 ((ρ α ) A ) = S 0 ((ρ α ) B ) < S 0 (ρ α ), (3.26) which agrees with the entropy inequalities (3.20). Therefore we cannot say anything if or for what α the state is entangled. If, however, α = 1, then only λ 4 = 1, the other eigenvalues are 0. In this case the rank of ρ α is 1. By comparison of the ranks we get 2 1 S 0 ((ρ α ) A ) = S 0 ((ρ α ) B ) > S 0 (ρ α ), (3.27) which contradicts the inequalities (3.20). Thus only if α = 1, that is the special case in which the Werner state equals ψ ψ, we can say for sure that the state is entangled. S 1. The von Neumann entropy S 1 (3.22) is the most common quantum entropy used for many purposes. First we need to remember that functions acting on a matrix are defined by acting on the elements of the diagonalized matrix, that is, acting on the eigenvalues. When taking the trace, we can always write a state in diagonal matrix form, since the trace operation is independent of the choice of basis. Therefore Trρ log ρ = i λ i log λ i, (3.28) where the λ i s are the eigenvalues of the state ρ. Using Eq. (3.28) we get for the reduced density matrices S 1 (ρ A ) = S 1 (ρ B ) = log 1 2 = log 1 2 = log 2. (3.29)

24 3. Detection of Entanglement S red S 2 S ,7476 a 1 3@0,5774 Fig. 3.3: Plot of S 1, S 2 as functions of the parameter p and intersections with the entropies of the reduced density matrices S red = 1 And if we take the logarithm to the base 2, we obtain S 1 (ρ A ) = S 1 (ρ B ) = 1. (3.30) For the state ρ α we find ( ) 1 α 1 α S 1 (ρ α ) = 3 log α 1 + 3α log (3.31) 4 The entropy inequalities (3.20) are satisfied if S 1 (ρ α ) 1. Since we cannot solve the equation S 1 (ρ α ) 1 analytically, we plot the function S 1 (ρ α ) in dependence of α (see Fig. 3.3) and calculate the intersection with the entropy of both reduced density matrices numerically. We obtain a violation of the inequalities (3.20) for α > 0, 7476, which is a weaker condition than the CHSH inequality, since that gave a violation for α > 1 2 = 0, So the entropy inequalities with the S 1 or von Neumann entropy do not give a greater range of the parameter α where we can know for sure that the state is entangled. S 2. To calculate the S 2 entropy (3.23) we use S 2 (ρ) = log (Trρ 2 ) = log i λ 2 i, (3.32)

25 3. Detection of Entanglement 25 S 2? entangled CHSH? entangled S 1? entangled 0 0,5 1 Fig. 3.4: Comparison of the information gained about the Werner state ρ α with 3 different separability criteria: 2 entropy inequalities and the CHSH inequality a and obtain for the reduced density matrices (where it is useful again to use log 2 ) ( 1 S 2 (ρ A ) = S 2 (ρ B ) = log ) 1 = log = log 2 2 = 1, (3.33) and for the whole state we get ( ) 2 ( ) ) 2 1 p 1 + 3p S 2 (ρ α ) = log 2 (3 +. (3.34) 4 4 Now we can analytically solve the inequality S 2 (ρ p ) < 1 and find that for α > 1 3 the entropy inequalities (3.20) are violated. Hence for this value of α the state is entangled for sure (see Fig. 3.3). This is a stronger condition than the CHSH inequality, since 1 3 < 1 2 and so we got a larger range of the parameter with certain entanglement. In Ref. [41] it is shown that for all 2-qubit states the S 2 entropy inequalities are always stronger than the CHSH inequality. The gained information about the entanglement of the Werner state ρ α is illustrated in Figure 3.4. (To be precise, in all the figures of course the possible values of α could be extended to the value 1/3, for reasons of simplicity this is neglected there.) The Positive Partial Transpose (PPT) Criterion The PPT Criterion is very useful for 2-qubit systems, since it is an operational criterion and a necessary and sufficient condition for separability. It was

26 3. Detection of Entanglement 26 recognized as a necessary separability criterion in Ref. [57] and extended to a necessary and sufficient one for 2 qubits in Ref. [42]. Theorem 3.6 (PPT Criterion). A state ρ acting on H 2 H 2, H 3 H 2 or H 2 H 3 is separable if and only if its partial transposition is a positive operator, ρ T B = (1 T )ρ 0. (3.35) For states acting on higher dimensional Hilbert spaces, the criterion is only necessary for separability. We call any state ρ for which Eq. (3.35) is satisfied a PPT state. Proof. We have already seen in section that the transposition is a positive, but not completely positive map. In Eq. (3.6) we have seen that for any positive map Λ the operation (1 Λ)ρ on a separable ρ gives a positive operator. So of course for Λ = T this has to be true as well. But so far only a necessary condition for separability has been gained. This fact was already apprehended by Peres [57]. To prove that the criterion is also a sufficient one for H 2 H 2, H 3 H 2 or H 2 H 3 [42] we need a theorem by Størmer and Woronowitz [69, 80]: Theorem 3.7. Any positive map Λ that maps operators on Hilbert spaces H 2 H 2, H 3 H 2 or H 2 H 3 can be decomposed in the following way: Here Λ CP 1 and Λ CP 2 are completely positive maps. Λ = Λ CP 1 + Λ CP 2 T. (3.36) Now let us suppose we have a state for which (1 T )ρ 0, and we want to show that this fact is sufficient for separability, which means that the state has to be separable for sure. Since Λ CP 1 and Λ CP 2 are completely positive maps the following statement has to be true: or Using Theorem 3.7 we get (1 Λ CP 1 )ρ + (1 Λ CP 2 )(1 T )ρ 0 (3.37) (1 Λ CP 1 )ρ + (1 Λ CP 2 T )ρ 0. (3.38) (1 Λ)ρ 0. (3.39) This is nothing but the PMT Theorem 3.3, because for all positive maps Λ (with respect to the special Hilbert spaces mentioned above) we can find a decomposition (3.36) where the steps (3.37) and (3.38) can be done. The PMT Theorem is a necessary and sufficient condition for separability and so the proof is completed.

27 3. Detection of Entanglement 27 Example. We want to investigate the Werner state again. The partial transposition of the matrix (2.36) is, according to Eq. (3.10), 1 α α α ρ α = α (3.40) 4 α 1 α The eigenvalues of this matrix are λ 1 = λ 2 = λ 3 = 1 + α, λ 4 = 1 3α. (3.41) 4 4 The first three eigenvalues are positive for all possible parameters α. λ 4 can be negative, and we get, applying the PPT Criterion (Theorem 3.6): 1 3 α 1 3 ρ α is separable, 1 3 < α 1 ρ α is entangled. (3.42) It is interesting that the PPT Criterion gives a remarkable wider range of entanglement of the Werner state than the other necessary separability conditions discussed in the last paragraphs did. This becomes particularly obvious when looking at a graphical comparison of different separability criteria (see Fig. 3.5). The Reduction Criterion Another separability criterion whose properties are similar to the PPT criterion (Theorem 3.6) is the reduction criterion [39]: Theorem 3.8 (Reduction Criterion). A state ρ acting on H 2 H 2, H 3 H 2 or H 2 H 3 is separable if and only if ρ A 1 ρ 0. (3.43) For states acting on higher dimensional Hilbert spaces, the criterion is only necessary for separability. Here ρ A is Alice s reduced density matrix, as usual (see Eqs. (2.8), (2.9)); of course, we could equivalently write 1 ρ B ρ 0.

28 3. Detection of Entanglement 28 PPT separable entangled S 2? entangled CHSH? entangled S 1? entangled 0 0,5 1 Fig. 3.5: Comparison of the PPT criterion with other separability criteria for the 2-qubit Werner state ρ α : The PPT criterion clearly distinguishes between separable and entangled states and gives a wider range of entanglement that the other criteria. Proof. According to the PMT Theorem (3.3) we know that for a positive map Λ we have (1 Λ) ρ 0 (3.44) a if the state ρ is separable. Now we can take a particular positive 2 map, i.e. Λ(M) = TrM1 M, (3.45) where M is any quadratic matrix. If we insert the above Λ in Eq. (3.44), we get Theorem 3.8. In Ref. [39] it is shown that the reduction criterion is equivalent to the PPT criterion (3.6) for H 2 H 2, H 2 H 3 or H 3 H 2 and thus is a necessary and sufficient criterion for those cases. Remark. In Ref. [39] it is proved that in higher dimensions, a map (3.45) can be decomposed in the way of Eq. (3.36). Now if the reduction criterion (Theorem 3.8) is violated, then of course (3.44) is violated too. If we look at Eq. (3.39), we see that the only way it can be violated is a violation of the PPT criterion. So the reduction criterion is not stronger than the PPT criterion (it does not detect more entangled states). 2 Proof of positivity: If we write Λ(M) in its diagonal form Λ(M) d, for a positive M we have (λ i are the eigenvalues of M, M d is the diagonalized M) Λ(M) d = i λ i1 M d. The diagonal elements of this matrix are the eigenvalues µ j of Λ(M), µ j = i λ i λ j = i j λ i 0, and so Λ(M) 0.

29 3. Detection of Entanglement 29 Example 1. We examine the Werner state ρ α (2.35) in matrix notation (2.36) again. We got for the reduced density matrix (3.24): ( 1 ) 0 (ρ α ) A = 2 = (3.46) And furthermore we obtain (ρ α ) A 1 = (3.47) If we want to apply the reduction criterion (Theorem 3.8), we calculate the diagonal matrix ((ρ α ) A 1 ρ α ) d, because then the eigenvalues are the diagonal elements. We find with the help of Eq. (3.47) ((ρ α ) A 1 ρ α ) d = ((ρ α ) A 1) d (ρ α ) d = (ρ α) d. (3.48) We conclude from Eq. (3.25) that the diagonalized Werner state is 1 α α (ρ α ) d = α (3.49) 4 1+3α So Eq. (3.48) becomes ((ρ α ) A 1 ρ α ) d = 1+α α α α (3.50) The eigenvalue 1 3p 4 can be negative for some range of the parameter α, so we obtain 1 3 α 1 3 ρ α is separable, 1 3 < α 1 ρ α is entangled, (3.51) which is exactly the same result as Eq. (3.42) in connection with the PPT criterion.

30 3. Detection of Entanglement 30 Example 2. The following example illustrates that for states on Hilbert spaces of more general dimensions, the reduction criterion (Theorem 3.8) can be more useful than the PPT criterion. The state of interest is the isotropic state ρ (d) α (2.16) of any dimension d 2. We first calculate the reduced density matrix (ρ (d) α ) A = Tr B ρ (d) α = αtr B φ d + φ d + 1 α + Tr d 2 B 1 1, (3.52) and because the reduced density matrix of the maximally entangled pure state φ d + has to be the maximally mixed state 11 of the subsystem, we d obtain (ρ (d) α ) A = Tr B ρ (d) α = α d α d 1 = 1 d 1. (3.53) The term of interest for the reduction criterion is (ρ (d) α ) A 1 ρ (d) α = 1 d 1 1 α φ d + φ d + 1 α 1 1. (3.54) d 2 Like in the first example we can diagonalize the whole term (3.54), ( (ρ (d) α ) A 1 ρ (d) α )d = α + d α ( ) φ d d 2 + φ d +. (3.55) d Since φ d + φ d + is a pure state, the diagonal matrix always has one element equal to 1 and all others equal to 0. So with help of Eq. (3.55) we find the eigenvalues λ 1 = α(1 d2 ) + d 1, λ d 2 2,..., λ d = d 1 + α (3.56) d 2 of (ρ (d) α ) A 1 ρ (d) α. The eigenvalues λ 2,..., λ d are positive for all possible values of α and d 2. The eigenvalue λ 1 is, however, negative for some values of α and we have 1 d + 1 < α 1 ρ(d) α is entangled. (3.57) In Ref. [39] it is shown that for the other possible values of α the state can always be written as a mixture of product states, and so 1 d 2 1 α 1 d + 1 ρ (d) α is separable. (3.58) Finally, we want to formulate Eqs. (3.57) and (3.58) with the fraction F instead of α, since we know that the notations (2.16) and (2.22) are equivalent.

31 3. Detection of Entanglement 31 We insert Eq. (2.21) in Eqs. (3.57) and (3.58) and find 1 d < F 1 ρ(d) F is entangled, 0 F 1 d ρ(d) F is separable. (3.59)

32 4. CLASSIFICATION OF ENTANGLEMENT 4.1 Introduction Not every entangled state has the same properties. There are different classes of entanglement, according to special properties. We can, e.g, classify the entangled states via the possibility to assign a local hidden variables (LHV) model to them (in this context see, e.g., Refs. [3, 23, 70, 58, 10]). Another classification is the distillability of entangled states (if one can obtain a maximal entangled pure state out of a mixed entangled state via local operations and classical communication (LOCC)). The distillation of mixed entangled states was introduced in Ref. [7], for further application of the subject see, e.g., Refs. [8, 27, 45]. Distillable entangled states are called free entangled and non-distillable entangled states are called bound entangled [44]. The chapter is organized as follows: The concept of distillation and the classification connected with it is discussed in Sec In Sec. 4.3 we investigate LHV models under general viewpoints, that is, Bell s original idea is extended to more general considerations (more general measurements, etc.). 4.2 Free and Bound Entanglement Distillation of Entangled States A Problem in Quantum Communication Let us think of the following problem: Alice and Bob want to do quantum communication, e.g., teleportation. Thus Alice produces 2-qubit singlet states ψ (2.30) and sends one particle from each pair to Bob. But the channel she uses for her transmission is noisy, so when Bob receives his particle, Alice and Bob share no pure singlet state ψ any longer, but some mixed state ρ. Can they, by any means, obtain the singlet states again? The answer is yes [7], for some mixed states ρ, Alice and Bob can do local operations and classical communication (LOCC) to recover from a given number of the same mixed states ρ a smaller number of (nearly) maximally entangled

33 4. Classification of Entanglement 33 singlets ψ. Note the word nearly in the last sentence. It means that with a finite number n of input states ρ, we can distill a smaller number k (with some probability p k ) of states ρ dist out of them that have a higher fidelity F ψ (ρ dist ) (2.15) than the input states ρ. If we apply the same distillation protocol to the distilled states ρ dist again, we obtain fewer states ρ dist2 with a higher fidelity F ψ (ρ dist2 ) than the states ρ dist. So we can get output states ρ out with an arbitrarily high fidelity F ψ (ρ out ) by applying the same protocol again and again. However, for some protocols, (e.g., the BBPSSW protocol [7]) in the limit of infinitely many input states ρ, 1 the distillation rate R dist (ρ) of distilled output states per input state (asymptotic distillation rate) tends to zero. Nevertheless there are distillation protocols [7, 8] for which R dist (ρ) does not tend to zero, but to some positive constant c R, k R dist (ρ) = lim n n = c. (4.1) The maximal possible distillation rate that can be achieved out of input states ρ and with any distillation protocol is called entanglement of distillation [8] E dist (ρ) = max LOCC R dist(ρ) (4.2) and is used as an entanglement measure (see Chapter 5). The BBPSSW Distillation Protocol The first distillation protocol was introduced in Ref. [7] by Bennett, Brassard, Popescu, Schumacher, Smolin and Wootters, and is thus called BBPSSW protocol. It works for all entangled 2-qubit states ρ for which a maximally entangled state ψ max exists such that 2 F ψmax (ρ) > 1/2, (4.3) where F ψmax (ρ) is the fraction given in Eq. (2.15). Note that if a state ρ has the property (4.3) then it cannot have a fraction higher than 1/2 with respect to any other pure state. The protocol itself consists of the following steps: 1 That means we can apply the protocol infinitely many times, since we have an infinite source of input pairs. So F ψ (ρ out ) 1. 2 The BBPSSW protocol is suitable for general states that satisfy the mentioned properties. There also exist distillation (more precise: concentration) protocols for pure states only [6] and it can be shown that all entangled pure states are distillable.

34 4. Classification of Entanglement First, the state ρ is subjected to a suitable local unitary transformation U A U B that transforms it into a state ρ 1 with a fraction F φ+ =: F > 1/2, where φ + is the state defined in Eq. (2.33) (i.e. the maximally entangled state (2.17) with d = 2). Such a transformation is always possible [45]. ρ ρ 1 = (U A U B )ρ(u A U B ). (4.4) 2. Next, Alice and Bob perform a random U U transformation on the state, where U is any unitary transformation and U is its complex conjugate (Alice performs a random U, then tells Bob, who performs U ). This transforms the state into a isotropic state ρ F (2.34) [45]: ρ 1 ρ F = du(u U )ρ 1 (U U ). (4.5) The transformation (4.5) leaves F invariant, F (ρ 1 ) = F (ρ F ). 3. Let us consider that Alice and Bob share two pairs of particles, each pair is in the state ρ F. This means that Alice holds two particles, and Bob as well. Each of them now applies a so-called XOR-operation to her / his particles. A XOR-operation is defined as U XOR a b = a (a + b)mod 2, (4.6) where a, b = 0 or 1 and x mod 2 means that if x 2, we have to subtract 2 from x so many times until we have x < 2 (thus in our case we have (a + b)mod 2 = 0 if a + b = 2). Here a is called source, b is called target. We obtain the state ρ that is a state of two pairs: ρ F ρ F ρ = U XOR (ρ F ρ F )U XOR (4.7) 4. In the next step Alice and Bob measure the spin of the target pair along the z-axis. If their outcomes are parallel (both measure 0 or both measure 1 ), then the source pair is kept. We calculate the resulting state of the source pair via performing a projection according to the measurement and tracing out the target pair, ( ( ) ( ) ) 1 ρ ρ P ρ 1 P := Tr target Tr ( ) ( ), (4.8) 1 P ρ 1 P where P = The factor Tr ( 1 P ) ρ ( 1 P ) gives the probability that Alice and Bob measure parallel spins and is needed for the normalization of the state (Trρ = 1).