Quantum Entanglement: Detection, Classification, and Quantification

Home , Density matrix, Entropy of entanglement, LOCC, Qutrit, Separable state, Shannon (unit)

Diplomarbeit zur Erlangung des akademischen Grades ,,Magister der Naturwissenschaften’’ an der Universitat¨ Wien

eingereicht von Philipp Krammer

betreut von Ao. Univ. Prof. Dr. Reinhold A. Bertlmann

Wien, Oktober 2005 CONTENTS

1. Introduction ...... 4

2. Basic Mathematical Description ...... 6 2.1 Spaces, Operators and States in a Finite Dimensional Hilbert Space ...... 6 2.2 Bipartite Systems ...... 7 2.2.1 Qubits ...... 11 2.2.2 Qutrits ...... 12 2.3 Positive and Completely Positive Maps ...... 14

3. Detection of Entanglement ...... 16 3.1 Introduction ...... 16 3.2 Pure States ...... 17 3.3 General States ...... 17 3.3.1 Nonoperational Separability Criteria ...... 17 3.3.2 Operational Separability Criteria ...... 20

4. Classification of Entanglement ...... 32 4.1 Introduction ...... 32 4.2 Free and Bound Entanglement ...... 32 4.2.1 Distillation of Entangled States ...... 32 4.2.2 Bound Entanglement ...... 36 4.3 Locality vs. Non-locality ...... 42 4.3.1 EPR and Bell Inequalities ...... 42 4.3.2 General Bell Inequality ...... 44 4.3.3 Bell Inequalities and the Entanglement Witness Theorem 49

5. Quantification of Entanglement ...... 52 5.1 Introduction ...... 52 5.2 Pure States ...... 52 5.3 General States ...... 54 5.3.1 Entanglement of Formation ...... 54 Contents 3

5.3.2 Concurrence and Calculating the Entanglement of For- mation for 2 Qubits ...... 56 5.3.3 Entanglement of Distillation ...... 58 5.3.4 Distance Measures ...... 59 5.3.5 Comparison of Diﬀerent Entanglement Measures for the 2-Qubit Werner State ...... 64

6. Hilbert-Schmidt Measure and Entanglement Witness .. 66 6.1 Introduction ...... 66 6.2 Geometrical Considerations about the Hilbert-Schmidt Distance 66 6.3 The Bertlmann-Narnhofer-Thirring Theorem ...... 68 6.4 How to Check a Guess of the Nearest Separable State . . . . . 70 6.5 Examples ...... 72 6.5.1 Isotropic Qubit States ...... 72 6.5.2 Isotropic Qutrit States ...... 74 6.5.3 Isotropic States in Higher Dimensions ...... 76

7. Tripartite Systems ...... 79 7.1 Introduction ...... 79 7.2 Basics ...... 79 7.3 Pure States ...... 81 7.3.1 Detection of Entangled Pure States ...... 81 7.3.2 Equivalence Classes of Pure Tripartite States ...... 81 7.4 General States ...... 87 7.4.1 Equivalence Classes of General Tripartite States . . . . 87 7.4.2 Tripartite Witnesses ...... 88

8. Conclusion ...... 90 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 Schrödingerin Ref. [68]. Since this article was published in German, ‘entanglement’ is a later transla- tion of the word ‘Verschränkung’.Schrödingerdoes 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 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 improve- ment to classical information theory, for example quantum teleportation and cryptography (see, e.g., Refs. [5, 17, 15, 32]). In the course of years comput- ing 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. 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) hA, Bi = TrA†B, (2.1) with the corresponding Hilbert-Schmidt norm p kAk = hA, Ai A ∈ A . (2.2)

Matrix Notation. Generally, any operator A ∈ A can be expressed as a matrix with the elements

Aij = hei| A |eji , (2.3) where ei and ej are vectors of an arbitrary basis {ei} 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 TrρP ≥ 0 ∀P, (2.5) where P is any projector, defined by P 2 = P . 2 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 2. Basic Mathematical Description 7

Remark. In early quantum mechanics (pure) states are represented as vectors |ψi 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 |ψi or the operator representation ρpure = |ψi hψ| can be used.

Note that Eqs. (2.4) and (2.5) imply Trρ2 ≤ 1.3 In particular we have

Trρ2 = 1 ⇒ ρ is a pure state , Trρ2 < 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 d1 d2 HA ⊗ HB , where d1 is the dimension of Alice’s subspace and d2 is Bob’s, or just HA ⊗ HB 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 H2 ⊗ H2, 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 d1 d2 a state ρ on HA ⊗ HB we have

ρmµ,nν = hem ⊗ fµ| ρ |en ⊗ fνi . (2.7)

Here {ei} and {fi} 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. 2. Basic Mathematical Description 8 system. The notation is

ρA = TrBρ ,

ρB = TrAρ , (2.8) where ρA denotes Alices reduced density matrix and ρB Bob’s. The matrix elements of the reduced density matrices are

Xd2 (ρA)mn = ρmβ,nβ , β=1 Xd1 (ρB)µν = ρaµ,aν . (2.9) a=1

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.

|ψprodi = |ψAi ⊗ |ψBi . (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] X X i i ρ = pi ρA ⊗ ρB, 0 ≤ pi ≤ 1, pi = 1 . (2.11) i i 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 = ρA ⊗ ρB , (2.12) 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: ¡ ¢ hA ⊗ Bi = TrρA ⊗ B = Tr ρA ⊗ ρB (A ⊗ B) = TrρAA TrρBB. (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 A B switch the devices prepare states ρi and ρi . 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 pi. Now the expectation value of a measurement A ⊗ B will be a weighed sum of factorized expectation values: Xn A B hA ⊗ Bi = piTrρi ATrρi B i=1 Xn ¡ A B¢ = piTr ρi ⊗ ρi (A ⊗ B) i=1 =: 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 |ψmaxi is given by

Fψmax := hψmax| ρ |ψmaxi (2.15)

Eq. (2.15) is nothing but the probability that the resulting state of a projective measurement (in a basis where |ψmaxi is one basis vector) is |ψmaxi. So the range of possible values of Fψmax (ρ) is 0 ≤ Fψmax (ρ) ≤ 1. 2. Basic Mathematical Description 10

(d) d d Isotropic States. We deﬁne an isotropic state ρα on a Hilbert Space H ⊗H as (see Refs. [39, 45]): ¯ ® ¯ 1 − α 1 ρ(d) = α ¯φd φd ¯ + 1 ⊗ 1, α ∈ R, − ≤ α ≤ 1 . (2.16) α + + d2 d2 − 1

d d Here d is the dimension of the Hilbert space H ⊗H¯ ®, the range of α is deter- ¯ d mined by the positivity of the state. The state φ+ is maximally entangled and given by ¯ ® Xd−1 ¯ d 1 φ+ = √ |ii ⊗ |ii , (2.17) d i=0 where {|ii} is a basis in Hd. 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)

∗ (d) ∗ † (U ⊗ U )ρα (U ⊗ U ) = ρα . (2.18)

The isotropic state (2.16) has the following properties [39]: 4

− 1 ≤ α ≤ 1 ⇒ ρ(d) separable , 2 d + 1 α d − 1 (2.19) 1 < α ≤ 1 ⇒ ρ(d) entangled . d + 1 α Instead of the parameter α in Eq. (2.16) we can also deﬁne an equivalent isotropic state¯ ® ρF with the fraction F (2.15) as the parameter. In case of ¯ d |ψmaxi = φ+ (2.17) we write shortly Fφ+ := F . According to Eq. (2.15) we get ¯ ¯ ® 1 + α(d2 − 1) F = φd ¯ ρ(d) ¯φd = , (2.20) + α + d2 or d2F − 1 α = . (2.21) d2 − 1 Inserting Eq. (2.21) into the deﬁnition (2.16) we get the equivalent form of an isotropic state µµ ¶ ¶ d2 1 ¯ ® ¯ 1 ρ(d) = F − ¯φd φd ¯ + (1 − F ) (2.22) F d2 − 1 d2 + + d2

4 The entangled property of the isotropic state is prooved by using the reduction criterion (see Theorem 3.8) in Sec. 3.3.2. 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)). 2. Basic Mathematical Description 11

2.2.1 Qubits Single Qubits. A qubit state ω, acting on H2, can be decomposed in terms of Pauli matrices (we use the convention to sum over same indices): 1 ¡ ¢ X ω = 1 + n σi , n ∈ R, n2 = |~n| ≤ 1 . (2.23) 2 i i i i

Note that for |~n|2 < 1 the state is mixed (corresponding to Trω2 ≤ 1) whereas for |~n|2 = 1 the state is pure (Trω2 = 1).

2 Qubits. According to the notation (2.7) the density matrix of 2 qubits, acting on H2 ⊗ H2, has the form   ρ11,11 ρ11,12 ρ11,21 ρ11,22  ρ ρ ρ ρ  ρ =  12,11 12,12 12,21 12,22  . (2.24)  ρ21,11 ρ21,12 ρ21,21 ρ21,22  ρ22,11 ρ22,12 ρ22,21 ρ22,22

The matrix (2.24) is usually obtained by calculating its elements in the standard product basis (e1 = f1 = |0i, e2 = f2 = |1i)

{|0i ⊗ |0i , |0i ⊗ |1i , |1i ⊗ |0i , |1i ⊗ |1i} , (2.25) which has the properties hi|ji = δij . (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, 1 ¡ ¢ ρ = 1 ⊗ 1 + a σi ⊗ 1 + b 1 ⊗ σi + c σi ⊗ σj , a , b , c ∈ R . (2.27) 4 i i ij i i ij

A product state ρA ⊗ ρB has the form

1 1 1 i 1 1 i i j ρA ⊗ ρB = 4 ( ⊗ + niσ ⊗ + mi ⊗ σ + nimjσ ⊗ σ ) , ni, mi ∈ R, |~n| ≤ 1, |~m| ≤ 1 . (2.28)

Any separable state (2.11) can be written as the convex combination of expressions (2.28), P ¡ ¢ ρ = p 1 1 ⊗ 1 + nkσi ⊗ 1 + mk1 ⊗ σi + nkmkσi ⊗ σj , sep k k 4 i ¯ ¯ i¯ ¯ i j k k R ¯ ~k¯ ¯ ~k¯ ni , mi ∈ , ¯n ¯ ≤ 1, ¯m ¯ ≤ 1 . (2.29) 2. Basic Mathematical Description 12

Bell Basis. A basis in H2 ⊗ H2 is the Bell basis, which consists of 4 orthonormal maximally entangled pure states: 1 |ψ−i = √ (|0i ⊗ |1i − |1i ⊗ |0i) (2.30) 2 1 |ψ+i = √ (|0i ⊗ |1i + |1i ⊗ |0i) (2.31) 2 1 |φ−i = √ (|0i ⊗ |0i − |1i ⊗ |1i) (2.32) 2 1 |φ+i = √ (|0i ⊗ |0i + |1i ⊗ |1i) . (2.33) 2

(2) Isotropic Qubit State. We can write a 2-qubit isotropic state ρF (2.22) as a mixture of the Bell states (2.30) - (2.33):

1 − F 1 − F ρ(2) =: ρ = F |φ i hφ | + |ψ i hψ | + |ψ i hψ | + F F + + 3 − − 3 + + 1 − F + |φ i hφ | , 0 ≤ F ≤ 1 . (2.34) 3 − −

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]) 1 − α 1 ρ = α |ψ i hψ | + 1 ⊗ 1, − ≤ α ≤ 1 . (2.35) α − − 4 3 Note that the interval for α follows from the necessity that Trρ = 1. The matrix notation of ρα in the standard basis (2.25) is, according to Eq. (2.24):

 1−α  4 0 0 0  1+α −α   0 4 2 0  ρα =  −α 1+α  . (2.36) 0 2 4 0 1−α 0 0 0 4

2.2.2 Qutrits Single Qutrits. The description of qutrits is very similar to the one for qubits. A qutrit state ω on H3 can be expressed in the matrix basis {1, λ1, 2 8 λ , . . . , λ } with an appropriate set of coeﬃcients {ni} 1 ³ √ ´ X ω = 1 + 3 n λi , n ∈ R , n2 = |~n|2 ≤ 1 . (2.37) 3 i i i i 2. Basic Mathematical Description 13

√ The factor 3 is included for a proper normalization, i.e. Trω2 ≤ 1 (see also Refs. [2, 20]). The matrices λi (i = 1, ..., 8) are the eight Gell-Mann matrices       0 1 0 0 −i 0 1 0 0 λ1 =  1 0 0  , λ2 =  i 0 0  , λ3 =  0 −1 0  , 0 0 0 0 0 0 0 0 0       0 0 1 0 0 −i 0 0 0 λ4 =  0 0 0  , λ5 =  0 0 0  , λ6 =  0 0 1  , 1 0 0 i 0 0 0 1 0     0 0 0 1 0 0 λ7 =  0 0 −i  , λ8 = √1  0 1 0  , (2.38) 3 0 i 0 0 0 −2 with properties Tr λi = 0, Tr λiλj = 2 δij. Note that a matrix of Eq. (2.37) with an arbitrary set of coeﬃcients {ni} is a density matrix only if it is positive - unlike the qubit case there exist sets {ni} 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 {ni} where all coefficients vanish except n8. According to Eq. (2.37) the only possible values for this coefficient are n8 = +1 or n8 = −1. If we have n8 = +1, then we get for a matrix A+1 formed like in Eq. (2.37)   2 0 0 1 ³ √ ´ 3 1  2  A+1 = + 3 = 0 3 0 . (2.39) 3 1 0 0 − 3

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 n8 = −1, we ﬁnd   0 0 0 1 ³ √ ´ A = 1 − 3 =  0 0 0  , (2.40) −1 3 0 0 1 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). 2. Basic Mathematical Description 14

2 Qutrits. For 2-qutrit states (that is, bipartite qutrit states acting on H3 ⊗ H3) the 9 × 9 matrix notation according to Eq. (2.7) is   ρ11,11 ρ11,12 ρ11,13 ρ11,21 ρ11,22 ρ11,23 ρ11,31 ρ11,32 ρ11,33    ρ12,11 ρ12,12 ρ12,13 ρ12,21 ρ12,22 ρ12,23 ρ12,31 ρ12,32 ρ12,33     ρ13,11 ρ13,12 ρ13,13 ρ13,21 ρ13,22 ρ13,23 ρ13,31 ρ13,32 ρ13,33     ρ21,11 ρ21,12 ρ21,13 ρ21,21 ρ21,22 ρ21,23 ρ21,31 ρ21,32 ρ21,33    ρ =  ρ22,11 ρ22,12 ρ22,13 ρ22,21 ρ22,22 ρ22,23 ρ22,31 ρ22,32 ρ22,33  .    ρ23,11 ρ23,12 ρ23,13 ρ23,21 ρ23,22 ρ23,23 ρ23,31 ρ23,32 ρ23,33     ρ31,11 ρ31,12 ρ31,13 ρ31,21 ρ31,22 ρ31,23 ρ31,31 ρ31,32 ρ31,33   ρ32,11 ρ32,12 ρ32,13 ρ32,21 ρ32,22 ρ32,23 ρ32,31 ρ32,32 ρ32,33  ρ33,11 ρ33,12 ρ33,13 ρ33,21 ρ33,22 ρ33,23 ρ33,31 ρ33,32 ρ33,33 (2.41) Usually we calculate the elements in the standard product basis (e1 = f1 = |0i, e2 = f2 = |1i, e3 = f3 = |2i) © |0i ⊗ |0i , |0i ⊗ |1i , |0i ⊗ |2i , |1i ⊗ |0i , |1i ⊗ |1i , ª |1i ⊗ |2i , |2i ⊗ |0i , |2i ⊗ |1i , |2i ⊗ |2i . (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 λi, 1 ¡ ¢ ρ = 1 ⊗ 1 + a λi ⊗ 1 + b 1 ⊗ λi + c λi ⊗ λj , a , b , c ∈ R . 9 i i ij i i ij (2.43) By the same argumentation as for qubits any separable 2-qutrit state is a convex combination of product states, X 1 ³ √ √ ´ ρ = p 1 ⊗ 1 + 3 nk λi ⊗ 1 + 3 mk 1 ⊗ λi + 3 nkmk λi ⊗ λj . sep k 9 i i i j k (2.44)

2.3 Positive and Completely Positive Maps

A linear map Λ: A1 → A2 (2.45) maps operators from a space A1 into a space A2. Λ is called positive if it maps positive operators into positive operators,

Λ(A) ≥ 0 ∀A ≥ 0 . (2.46) 2. Basic Mathematical Description 15

A positive map Λ is called completely positive if the map

Λ ⊗ 1d : A1 ⊗ Md → A2 ⊗ Md (2.47) is still a positive map for all d = 2, 3, 4 ...; 1d is the identity matrix of the matrix space Md of all d × d matrices. 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. 3.3.1, whereas in Sec. 3.3.2 operational criteria are discussed. We will see that for the 2-qubit case H2 ⊗H2 (and for H2 ⊗H3 or H3 ⊗H2) there exist operational separability criteria that are necessary and sufficient conditions for separability. 3. Detection of Entanglement 17

3.2 Pure States

We can check easily if a pure state |ψi is entangled by looking at the reduced density matrices of |ψi hψ|: 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−i, where |ψ−i is the singlet state (2.30). When written as a density matrix in the standard product basis (2.25) we get (see (2.24))   0 0 0 0  0 1/2 −1/2 0  |ψ i hψ | =   . (3.1) − −  0 −1/2 1/2 0  0 0 0 0

Now we can calculate the reduced density matrices, according to Eqs. (2.8) and (2.9), µ ¶ 1/2 0 ρ = ρ = . (3.2) A B 0 1/2 We see that the above matrix is a mixed state, since (according to Eq. (2.6)) 2 2 TrρA = TrρB < 1. So we conclude that |ψ−i 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.

3.3.1 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 ρent is entangled if and only if there exists a Hermitian operator A ∈ A, called entanglement witness, such that

hρent,Ai = TrAρent < 0 , hρ, Ai = TrAρ ≥ 0 ∀ρ ∈ S. (3.3)

1 A similar method uses the Schmidt decomposition [67] of a pure state |ψi (for details see, e.g., Ref. [45]). 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 (~bu), within (~bp) and under ~ (bd) 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 hρ, Ai = 0 de- ﬁnes a hyperplane in the set of all states, and this plane separates ‘up’ states ρu for which hρu,Ai < 0 from ‘down’ states ρd with hρd,Ai > 0. States ρp with hρp,Ai = 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. Aopt, if apart from Eqs. (3.1) there exists a separable stateρ ˜ ∈ S for which

hρ,˜ Aopti = 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. 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 Xn 1 A B ( ⊗ Λ)ρ = piρi ⊗ Λ(ρi ), (3.6) i=1

B and since Λ is positive, Λ(ρi ) 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 ρent is entangled if and only if there exists a positive map Λ, such that

(1 ⊗ Λ)ρent < 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+i hφ+| < 0 , (3.8) 3. Detection of Entanglement 20

where |φ+i is deﬁned in Eq. (2.33). Written in matrix notation (2.24) in the standard product basis (2.25) we have:   1/2 0 0 1/2  0 0 0 0  |φ i hφ | =   . (3.9) + +  0 0 0 0  1/2 0 0 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 (Aij) is simply done by interchanging the indices of the T 1 elements: T ((Aij)) =: (Aij) = (Aj i). So ⊗ 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 TB ( ⊗ T )(ρmµ,nν) =: (ρmµ,nν) = (ρmν,nµ) . (3.10)

Applying (3.10) on Eq. (2.24) we obtain (1 ⊗ T ) |ψ+i hψ+|:   1/2 0 0 0  0 0 1/2 0  (|ψ i hψ |)TB =   . (3.11) + +  0 1/2 0 0  0 0 0 1/2

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.

3.3.2 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. 4.3. 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. 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

hρ, 21 − Bi ≥ 0,B = ~a · ~σ ⊗ (~b +~b0) · ~σ + ~a0 · ~σ ⊗ (~b −~b0) · ~σ, (3.12)

where ~a,~a0,~b,~b0 are any unit vectors in R3; ~σ is the vector out of the three Pauli matrices, ~σ = (σx, σy, σz). 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,~a0,~b,~b0. 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,~a0,~b,~b0) if and only if

M(ρ) > 1 . (3.13)

Here M(ρ) is the sum of the two greater eigenvalues of a matrix Uρ. The matrix Uρ can be constructed in the following way: First we calculate the nm n m 1 matrix elements of a matrix Tρ,(Tρ) = Trρσ ⊗ σ (n, m = 1, 2, 3, σ x T corresponds to σ , etc.). Then Uρ = Tρ Tρ.

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)), 1 1 ρ = (1 − α~σ ⊗ ~σ) , − ≤ p ≤ 1 , (3.14) α 4 3 where we deﬁned ~σ ⊗ ~σ := σx ⊗ σx + σy ⊗ σy + σz ⊗ σz. Written in this way, nm the matrix elements (Tρ) can easily be calculated. When taking the trace, we remember that TrA ⊗ B = TrATrB. (3.15) n nn Since Trσ = 0 ∀n = x, y or z, only the diagonal terms (Tρ) do not vanish, since here Tr(σn ⊗ σn)(σn ⊗ σn) = 4. These are −α (T )nn = · 4 = −α . (3.16) ρ 4 3. Detection of Entanglement 22

So we have     −α 0 0 α2 0 0 T =  0 −α 0  ,U =  0 α2 0  . (3.17) 0 0 −α 0 0 α2

Now we can calculate the sum of the two greater eigenvalues of U:

2 M(ρα) = 2α . (3.18)

According to Theorem 3.5, ρα violates the CHSH inequality (3.12) if 1 α > √ , (3.19) 2 so we conclude that all Werner states with α > √1 are entangled for sure. 2

Entropy Inequalities Other necessary separability criteria are inequalities that compare certain quantum entropies of a state and its reduced density matrix:

S(ρA) ≤ S(ρ) and S(ρB) ≤ S(ρ) ∀ separable states ρ . (3.20)

As usual, ρA and ρB are Alice’s and Bob’s reduced density matrices (see Eqs. (2.8) and (2.9)). The inequalities originated from an observation by Schr¨odinger[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:

S0(ρ) = log R(ρ) , (3.21)

S1(ρ) = −Trρ log ρ , (3.22) 2 S2(ρ) = − log Trρ , (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. 3. Detection of Entanglement 23

Example. As an example we want to check the inequalities for the Werner state ρα (2.35). To do this, we ﬁrst consider the matrix notation (2.36) and calculate the reduced density matrices. We get µ ¶ 1 2 0 (ρα)A = (ρα)B = 1 . (3.24) 0 2

S0. First we calculate the S0 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 ρα we need to calculate the eigenvalues of ρα. These are 1 − α 1 + 3α λ = λ = λ = , λ = . (3.25) 1 2 3 4 4 4 If α 6= 1, all eigenvalues are greater than zero and therefore do not vanish. The rank of ρα is 4. Comparing the S0 entropies we get

2 ≤ 4 ⇒ S0 ((ρα)A) = S0 ((ρα)B) < S0(ρα) , (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 λ4 = 1, the other eigenvalues are 0. In this case the rank of ρα is 1. By comparison of the ranks we get

2 ≥ 1 ⇒ S0 ((ρα)A) = S0 ((ρα)B) > S0(ρα) , (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−i hψ−|, we can say for sure that the state is entangled.

S1. The ’von Neumann entropy’ S1 (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 X −Trρ log ρ = − λi log λi , (3.28) i

where the λis are the eigenvalues of the state ρ. Using Eq. (3.28) we get for the reduced density matrices 1 1 1 S (ρ ) = S (ρ ) = −2 · log = − log = log 2 . (3.29) 1 A 1 B 2 2 2 3. Detection of Entanglement 24

S1 1.5 S2

Sred 1

0.5 a 0.2 0.4 0.6 0.8 1 0,7476 1 3 @ 0,5774

Fig. 3.3: Plot of S1, S2 as functions of the parameter p and intersections with the entropies of the reduced density matrices Sred = 1

And if we take the logarithm to the base 2, we obtain

S1(ρA) = S1(ρB) = 1 . (3.30)

For the state ρα we ﬁnd µ ¶ 1 − α 1 − α 1 + 3α 1 + 3α S (ρ ) = −3 log − log . (3.31) 1 α 4 2 4 4 2 4

The entropy inequalities (3.20) are satisﬁed if S1(ρα) ≥ 1. Since we cannot solve the equation S1(ρα) ≥ 1 analytically, we plot the function S1(ρα) 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 = 0, 7071. So the entropy inequalities with the S or von 2 1 Neumann entropy do not give a greater range of the parameter α where we can know for sure that the state is entangled.

S2. To calculate the S2 entropy (3.23) we use X 2 2 S2(ρ) = − log (Trρ ) = − log λi , (3.32) i 3. Detection of Entanglement 25

S2 ? entangled

CHSH ? entangled

S1 ? entangled

0 a0,5 1

Fig. 3.4: Comparison of the information gained about the Werner state ρα with 3 diﬀerent separability criteria: 2 entropy inequalities and the CHSH inequality

and obtain for the reduced density matrices (where it is useful again to

use log2) µ ¶ 1 1 1 S (ρ ) = S (ρ ) = − log + = − log = log 2 = 1 , (3.33) 2 A 2 B 2 4 4 2 2 2 and for the whole state we get Ã µ ¶ µ ¶ ! 1 − p 2 1 + 3p 2 S (ρ ) = − log 3 + . (3.34) 2 α 2 4 4

Now we can analytically solve the inequality S2(ρp) < 1 and ﬁnd that for α > √1 the entropy inequalities (3.20) are violated. Hence for this 3 value of α the state is entangled for sure (see Fig. 3.3). This is a stronger condition than the CHSH inequality, since √1 < √1 and so we got a 3 2 larger range of the parameter with certain entanglement. In Ref. [41] it is shown that for all 2-qubit states the S2 entropy inequalities are always stronger than the CHSH inequality.

The gained information about the entanglement of the Werner state ρα 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 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 H2 ⊗ H2, H3 ⊗ H2 or H2 ⊗ H3 is separable if and only if its partial transposition is a positive operator, ρTB = (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 3.3.1 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 H2 ⊗ H2, H3 ⊗ H2 or H2 ⊗ H3 [42] we need a theorem by Størmer and Woronowitz [69, 80]: Theorem 3.7. Any positive map Λ that maps operators on Hilbert spaces H2 ⊗ H2, H3 ⊗ H2 or H2 ⊗ H3 can be decomposed in the following way:

CP CP Λ = Λ1 + Λ2 ◦ T. (3.36) CP CP Here Λ1 and Λ2 are completely positive maps. 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 CP CP the state has to be separable for sure. Since Λ1 and Λ2 are completely positive maps the following statement has to be true: 1 CP 1 CP 1 ( ⊗ Λ1 )ρ + ( ⊗ Λ2 )( ⊗ T )ρ ≥ 0 (3.37) or 1 CP 1 CP ( ⊗ Λ1 )ρ + ( ⊗ Λ2 ◦ T )ρ ≥ 0 . (3.38) Using Theorem 3.7 we get (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. 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  4 0 0 2  1+α   0 4 0 0  ρα =  1+α  . (3.40) 0 0 4 0 −α 1−α 2 0 0 4 The eigenvalues of this matrix are 1 + α 1 − 3α λ = λ = λ = , λ = . (3.41) 1 2 3 4 4 4

The ﬁrst three eigenvalues are positive for all possible parameters α. λ4 can be negative, and we get, applying the PPT Criterion (Theorem 3.6): 1 1 − ≤ α ≤ ⇒ ρ is separable , 3 3 α 1 < α ≤ 1 ⇒ ρ is entangled . (3.42) 3 α 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 H2 ⊗H2, H3 ⊗H2 or H2 ⊗ H3 is separable if and only if

ρA ⊗ 1 − ρ ≥ 0 . (3.43)

For states acting on higher dimensional Hilbert spaces, the criterion is only necessary for separability.

Here ρA is Alice’s reduced density matrix, as usual (see Eqs. (2.8), (2.9)); of course, we could equivalently write 1 ⊗ ρB − ρ ≥ 0. 3. Detection of Entanglement 28

PPT separable entangled

S2 ? entangled

CHSH ? entangled

S1 ? entangled

0 a0,5 1

Fig. 3.5: Comparison of the PPT criterion with other separability criteria for the 2-qubit Werner state ρα: 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) if the state ρ is separable. Now we can take a particular positive2 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 H2 ⊗ H2, H2 ⊗ H3 or H3 ⊗ H2 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, forP a positive M 1 we have (λi are the eigenvalues of M, Md is the diagonalized M) Λ(M)d = Pi λi − Md. TheP diagonal elements of this matrix are the eigenvalues µj of Λ(M), µj = i λi − λj = i6=j λi ≥ 0, and so Λ(M) ≥ 0. 3. Detection of Entanglement 29

Example 1. We examine the Werner state ρα (2.35) in matrix notation (2.36) again. We got for the reduced density matrix (3.24): µ ¶ 1 0 1 2 1 (ρα)A = 1 = . (3.46) 0 2 2 And furthermore we obtain 1 (ρ ) ⊗ 1 = 1 ⊗ 1 . (3.47) α A 2 If we want to apply the reduction criterion (Theorem 3.8), we calculate the diagonal matrix ((ρα)A ⊗ 1 − ρα)d, because then the eigenvalues are the diagonal elements. We ﬁnd with the help of Eq. (3.47) 1 ((ρ ) ⊗ 1 − ρ ) = ((ρ ) ⊗ 1) − (ρ ) = 1 ⊗ 1 − (ρ ) . (3.48) α A α d α A d α d 2 α d We conclude from Eq. (3.25) that the diagonalized Werner state is

 1−α  4 0 0 0  1−α   0 4 0 0  (ρα)d =  1−α  . (3.49) 0 0 4 0 1+3α 0 0 0 4 So Eq. (3.48) becomes

 1+α  4 0 0 0  0 1+α 0 0  1  4  ((ρα)A ⊗ − ρα)d =  1+α  . (3.50) 0 0 4 0 1−3α 0 0 0 4

1−3p The eigenvalue 4 can be negative for some range of the parameter α, so we obtain 1 1 − ≤ α ≤ ⇒ ρ is separable , 3 3 α 1 < α ≤ 1 ⇒ ρ is entangled , (3.51) 3 α which is exactly the same result as Eq. (3.42) in connection with the PPT criterion. 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 (d) isotropic state ρα (2.16) of any dimension d ≥ 2. We first calculate the reduced density matrix ¯ ® ¯ 1 − α (ρ(d)) = Tr ρ(d) = αTr ¯φd φd ¯ + Tr 1 ⊗ 1 , (3.52) α A B α B + + d2 B and because¯ ® the reduced density matrix of the maximally entangled pure ¯ d 1 1 state φ+ has to be the maximally mixed state d of the subsystem, we obtain α 1 − α 1 (ρ(d)) = Tr ρ(d) = 1 + 1 = 1 . (3.53) α A B α d d d The term of interest for the reduction criterion is 1 ¯ ® ¯ 1 − α (ρ(d)) ⊗ 1 − ρ(d) = 1 ⊗ 1 − α ¯φd φd ¯ − 1 ⊗ 1 . (3.54) α A α d + + d2 Like in the first example we can diagonalize the whole term (3.54), ¡ ¢ α + d − 1 ¡¯ ® ¯¢ (ρ(d)) ⊗ 1 − ρ(d) = 1 ⊗ 1 − α ¯φd φd ¯ . (3.55) α A α d d2 + + d ¯ ® ¯ ¯ d d ¯ Since φ+ φ+ 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

α(1 − d2) + d − 1 d − 1 + α λ = , λ , . . . , λ = (3.56) 1 d2 2 d d2

(d) (d) of (ρα )A ⊗ 1 − ρα . The eigenvalues λ2, . . . , λd are positive for all possible values of α and d ≥ 2. The eigenvalue λ1 is, however, negative for some values of α and we have 1 < α ≤ 1 ⇒ ρ(d) is entangled . (3.57) d + 1 α 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 1 − ≤ α ≤ ⇒ ρ(d) is separable . (3.58) d2 − 1 d + 1 α 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. 3. Detection of Entanglement 31

We insert Eq. (2.21) in Eqs. (3.57) and (3.58) and ﬁnd 1 < F ≤ 1 ⇒ ρ(d) is entangled , d F 1 0 ≤ F ≤ ⇒ ρ(d) is separable . (3.59) d F 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. 4.2. 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

4.2.1 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−i (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−i 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 4. Classification of Entanglement 33

singlets |ψ−i. 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 pk) of states ρdist out of them that have a higher ﬁdelity

Fψ− (ρdist) (2.15) than the input states ρ. If we apply the same distillation protocol to the distilled states ρdist again, we obtain fewer states ρdist2 with a higher ﬁdelity Fψ− (ρdist2) than the states ρdist. So we can get ‘output’ states ρout with an arbitrarily high ﬁdelity

Fψ− (ρout) by applying the same protocol again and again. However, for some protocols, (e.g., the BBPSSW protocol [7]) in the limit 1 of inﬁnitely many input states ρ, the distillation rate Rdist(ρ) of distilled output states per input state (asymptotic distillation rate) tends to zero. Nevertheless there are distillation protocols [7, 8] for which Rdist(ρ) does not tend to zero, but to some positive constant c ∈ R, k Rdist(ρ) = lim = c . (4.1) n→∞ n 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]

Edist(ρ) = max Rdist(ρ) (4.2) LOCC 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 2 entangled state |ψmaxi exists such that

Fψmax (ρ) > 1/2 , (4.3)

where Fψmax (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 Fψ− (ρout) → 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. 4. Classification of Entanglement 34

1. First, the state ρ is subjected to a suitable local unitary transformation

UA ⊗ UB that transforms it into a state ρ1 with a fraction Fφ+ =: F > 1/2, where |φ+i 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 → ρ1 = (UA ⊗ UB)ρ(UA ⊗ UB) . (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 ρF (2.34) [45]: Z ∗ ∗ † ρ1 → ρF = dU(U ⊗ U )ρ1(U ⊗ U ) . (4.5)

The transformation (4.5) leaves F invariant, F (ρ1) = F (ρF ). 3. Let us consider that Alice and Bob share two pairs of particles, each pair is in the state ρF . 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

UXOR |ai ⊗ |bi = |ai ⊗ |(a + b)mod 2i , (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 |ai is called ‘source’, |bi is called ‘target’. We obtain the stateρ ˜ that is a state of two pairs:

† ρF ⊗ ρF → ρ˜ = UXOR(ρF ⊗ ρF )UXOR (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 |0i or both measure |1i), 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, Ã ¡ ¢ ¡ ¢ ! 1 1 0 ⊗ Pk ρ˜ ⊗ Pk ρ˜ → ρ := Trtarget ¡ ¢ ¡ ¢ , (4.8) Tr 1 ⊗ Pk ρ˜ 1 ⊗ Pk ¡ ¢ ¡ ¢ where Pk = |00i h00|+|11i h11|. The factor Tr 1 ⊗ Pk ρ˜ 1 ⊗ Pk gives the probability that Alice and Bob measure parallel spins and is needed for the normalization of the state (Trρ0 = 1). 4. Classification of Entanglement 35

Fig. 4.1: Plot of the ﬁdelity g(F ) of the distilled state ρ0

Now if we calculate the steps described above in detail, we finally find for the fidelity F 0 in dependence of the fidelity F of the input states ρ,

F 2 + 1 (1 − F )2 F 0(F ) := hφ | ρ0 |φ i = 9 . (4.9) + + 2 2 5 2 F + 3 F (1 − F ) + 9 (1 − F ) Let us take a look at a plot of the functions g(F ) := F 0(F ) and f(F ) := F in Fig. 4.1: Only for F > 1/2 we always have g(F ) > f(F ). So only if we start with a state ρ for which F > 1/2, we can increase the fidelity by iterating the process 1 - 4. What about the number k of output states ρout after l iterations of the protocol 1 - 4? According to the protocol, we get np k = , (4.10) 2 l where n is the number of input pairs and p is the probability that in each “round” we get the desired outcome of step 4 (and hence is the product of the probabilities to get parallel spins after measurement). If we want to reach a fidelity F 0 = 1, we have to iterate the process infinitely often. That also means we need an infinite supply of input pairs, and the probability p tends to zero. So for the asymptotic distillation rate (4.1) we have (with Eq. (4.10)) Rdist(ρ) → 0 for l → ∞, p → 0 . (4.11) However, there exist protocols slightly different to the BBPSSW protocol, which give a nonzero asymptotic rate for all 2-qubit entangled states with

Fψmax > 1/2 (see Refs. [7, 8]). 4. Classification of Entanglement 36

Distillable Entangled States Entangled 2-qubit States. In Ref. [43] it is shown that with a special LOCC operation called ‘ﬁltering’, one can obtain (with a certain probability of suc- cess) from an entangled 2-qubit state with Fψmax ≤ 1/2 a state with F > 1/2, which then can be subjected to the BBPSSW protocol (see last section). So we can state the following theorem: Theorem 4.1. Every entangled 2-qubit state can be distilled.

Entangled Isotropic States. Let us suppose that Alice and Bob apply a (d) projective operation P ⊗ P to an isotropic state ρα (2.16), where

P = |0i h0| + |1i h1| . (4.12)

We can also say that Alice and Bob measure the state of their particles, and they only keep their pair if they get |0i or |1i. The resulting state is a 2- (2) qubit isotropic state ρα , where we normalized the outcome of the operation (2) (d) 1 according to Trρα = 1. If ρα is entangled, that is, we have d+1 < α ≤ 1, (2) 1 then for the resulting state ρα we get 3 < α ≤ 1, so the 2-qubit isotropic state is entangled too. This state is distillable, since all entangled 2-qubit (2) states are distillable. If we use the equivalent form ρF (2.34) of the 2-qubit 1 isotropic state, then, according to Eq. (3.59), we have 2 < F ≤ 1, and so the resulting state can be distilled with the BBPSSW protocol without any prior ﬁltering. So for entangled isotropic states we state the following theorem: Theorem 4.2. Any entangled isotropic state can be distilled.

States that Violate the Reduction Criterion. It is shown in Ref. [39] that by applying a suitable ﬁltering operation on a state that violates the reduction criterion (Theorem 3.8), we obtain (with a certain probability) a state that has a fraction F > 1/d, and this state can then be transformed via a random ∗ (d) U ⊗ U -transformation (4.5) into an entangled isotropic state ρF (2.22), which can be distilled. So we can say that Theorem 4.3. Any state that violates the reduction criterion can be distilled.

4.2.2 Bound Entanglement Entangled PPT states Interestingly, there exist entangled states that cannot be distilled. Any entangled state that is not distillable is called bound entangled, whereas distillable 4. Classification of Entanglement 37 entangled states are called free entangled [44]. In particular we have the following theorem: Theorem 4.4. A PPT state (i.e. a state that remains positive under partial transposition) cannot be distilled. Theorem 4.4 can be proved in diﬀerent ways. One way [44] uses yet another theorem from which Theorem 4.4 can be derived. Here we want to sketch a proof from Ref. [45]: This proof is done in two steps, ﬁrst, it is shown that any LOCC on PPT states result in PPT states [44]. Second, it is shown that for a PPT state ρPPT we always have 1 F (ρ ) ≤ , (4.13) PPT d

(see also Ref. [64]) so that we can never achieve a fraction F (ρPPT ) near 1, therefore ρPPT cannot be distilled. Bound entanglement causes many important consequences in quantum information, for example irreversibility of a quantum mechanical operation [74]: Alice and Bob can create out of some pure entangled state a (mixed) bound entangled state. So once they did this, they cannot distill the pure state out of the bound entangled state again. Another consequence is the following: One can prove [46] that any bound entangled state has to satisfy the S0 entropy inequality (3.20), (3.21). So this inequality is also a necessary condition not only for separability, but also for bound entanglement. To prove that a bound entangled state has to satisfy the S0 entropy one can show that [46] any state violating the inequality also has to violate the reduction criterion, and according to Theorem 4.3 we know that such a state is distillable.

Do there exist bound entangled NPT states? We have already learned in the last section that all entangled PPT states are not distillable (bound entangled). Now the question arises if there exist entangled states that are not positive under partial transposition (NPT), but nevertheless are not distillable. There have not been any rigorously conclusive results yet, but there is a strong implication that bound entangled NPT states exist [29, 28]. Fig. 4.2 illustrates the gained results.

Example of Bound Entanglement We want to investigate the following 2-qutrit state (introduced in this form in Ref. [45] and based on matrices of Ref. [69]): 2 ¯ ® ¯ β 5 − β ρ = ¯φ3 φ3 ¯ + σ σ , 0 ≤ β ≤ 5 , (4.14) β 7 + + 7 + 7 − 4. Classification of Entanglement 38

2-qubit states

separable states free entangled states

PPT states NPT states general states bound entangled states

separable states free entangled states

Fig. 4.2: Illustration of entanglement and distillability. Since all entangled 2-qubit states are distillable and NPT, we have a clear distinction in this case. For general states, however, there are entangled PPT states (bound entangled) and maybe bound entangled NPT states, which are those outside the “box” of the free entangled states. Note that this is not a geometric representation of sets of states. 4. Classification of Entanglement 39 where, according to Eq. (2.17) ¯ ® 1 ¯φ3 = √ (|0i ⊗ |0i + |1i ⊗ |1i + |2i ⊗ |2i) , (4.15) + 3 and

1 σ+ = 3 (|01i h01| + |12i h12| + |20i h20|) , (4.16) 1 σ− = 3 (|10i h10| + |21i h21| + |02i h02|) . (4.17) If we write the state (4.14) in matrix notation (2.41) in the standard basis (2.42) we obtain

 2 2 2  21 0 0 0 21 0 0 0 21  0 5−β 0 0 0 0 0 0 0   21   0 0 β 0 0 0 0 0 0   21   β   0 0 0 21 0 0 0 0 0   2 2 2  ρβ =  21 0 0 0 21 0 0 0 21  . (4.18)  5−β   0 0 0 0 0 21 0 0 0   5−β   0 0 0 0 0 0 21 0 0   β  0 0 0 0 0 0 0 21 0 2 2 2 21 0 0 0 21 0 0 0 21

A check of the eigenvalues of the matrix (4.18) gives the result that ρβ ≥ 0 for 0 ≤ β ≤ 5, and this is why we limited the range of β in Eq. (4.14). Now let us check the eigenvalues λ1, λ2, . . . , λ9 of the partially transposed TB state ρβ . We ﬁnd 2 λ = λ = λ = 1 2 3 21 1 ³ p ´ λ = λ = λ = 5 − 41 − 20β + 4β2 4 5 6 42 1 ³ p ´ λ = λ = λ = 5 + 41 − 20β + 4β2 . (4.19) 7 8 9 42

With the exception of λ4(= λ5 = λ6), the eigenvalues (4.19) are positive. Looking at λ4 we ﬁnd

λ4 < 0 for 0 ≤ β < 1 ,

λ4 ≥ 0 for 1 ≤ β ≤ 4 ,

λ4 < 0 for 4 < β ≤ 5 . (4.20) Because for 2-qutrit states the PPT criterion (Theorem 3.6) is only necessary for separability, from Eq. (4.20) we know that ρβ is entangled for sure if 4. Classification of Entanglement 40

0 ≤ β < 1 or 4 < β ≤ 5 ; but for 1 ≤ β ≤ 4 the state is PPT and we cannot be certain if the state is separable or entangled. If we want to ﬁnd out if somewhere within this range of β the state is entangled, we have to use another method than the PPT criterion - e.g. the positive map Theorem 3.3. According to the PMT, if for some positive map Λ and for some β ∈ [1, 4] the expression (1⊗Λ)ρβ is negative, then ρβ is entangled and PPT, and thus bound entangled according to Theorem 4.4.

Remark. A positive map Λ for which (1 ⊗ Λ)ρ < 0 and ρTB ≥ 0 cannot be decomposable like Eq. (3.36), because we argued in the proof of the PPT criterion (Theorem 3.6) in Sec. 3.3.2 that if a map Λ is decomposable and ρTB ≥ 0, then this fact is equivalent to (1⊗Λ)ρ ≥ 0, which is a contradiction to the premises.

Clearly the diﬃculty lies in ﬁnding a suitable positive map Λ. The following map3 turns out to be useful:     a11 a12 a13 a11 + a22 −a12 −a13 Λ  a21 a22 a23  =  −a21 a22 + a33 −a23  . (4.21) a31 a32 a33 −a31 −a32 a33 + a11

Proof that Λ (4.21) is positive. In Ref. [21] it is argued that a map Λ is positive if the corresponding biquadratic form     x1 y1 T ¡ T ¢ f(x, y) := y ·Λ x · x ·y, x =  x2  , y =  y2  , xi, yi ∈ R (4.22) x3 y3 is positive for all x, y. Inserting our Λ from Eq. (4.21) we obtain

2 2 2 2 2 2 f(x, y) = x1y1 + x2y2 + x3y3 − 2x2x3y2y3 − 2x1x3y1y3 − 2x1x2y1y2 + 2 2 2 2 2 2 + x3y2 + x2y1 + x1y3 . (4.23) We can search for minima of this function and ﬁnd that a global minimum is f = 0. So f(x, y) ≥ 0 ∀x, y and we proved that Λ (4.21) is positive. In Figure 4.3 the function f(x, y) is plotted for x2 = x3 = y2 = y3 = 0.

3 Note that the map presented in Ref. [45] is slightly diﬀerent to the map (4.21). The map of Ref. [45] does not give evidence of bound entanglement. Furthermore, in Ref. [45] the reader is referred to Ref. [21] in order to check the positivity of the map introduced in Ref. [45]. The map presented and proved to be positive in Ref. [21] is, however, slightly diﬀerent to the map of Ref. [45] and to the map (4.21) (the map of Ref. [21] would not give any evidence for bound entanglement either). 4. Classification of Entanglement 41

Fig. 4.3: Plot of the function f(x1, x2 = 0, x3 = 0, y1, y2 = 0, y3 = 0). We can see that the global minimum f = 0 is not taken at a single point but for many diﬀerent values of x1 and y1.

Now let us calculate (1 ⊗ Λ)ρβ. Since Λ is applied only partially, we apply it to the nine 3 × 3 sectors the matrix (4.18) can be divided into. We get

 7−β 2 2  21 0 0 0 − 21 0 0 0 − 21  0 5 0 0 0 0 0 0 0   21   0 0 2+β 0 0 0 0 0 0   21   2+β   0 0 0 21 0 0 0 0 0  (1 ⊗ Λ) ρ =  2 7−β 2  . (4.24) β  − 21 0 0 0 21 0 0 0 − 21   5   0 0 0 0 0 21 0 0 0   5   0 0 0 0 0 0 21 0 0   2+β  0 0 0 0 0 0 0 21 0 2 2 7−β − 21 0 0 0 − 21 0 0 0 21 The eigenvalues of the above matrix are 5 λ = λ = λ = 1 2 3 21 3 − β λ = 4 21 9 − β λ = λ = 5 6 21 2 + β λ = λ = λ = . (4.25) 7 8 9 21 4. Classification of Entanglement 42

0 2 3 4 5 b ? separable bound free entangled entangled

Fig. 4.4: Illustration of the various properties of the state ρβ (4.14). The question mark says that in this area we do not have enough information, we only know that for 0 ≤ β < 1 the state is NPT and therefore entangled.

All eigenvalues are positive (within the allowed region of β), except for λ4 we have λ4 < 0 for 3 < β ≤ 5 . (4.26)

So indeed for the above range of the parameter β we have (1⊗Λ)ρβ < 0 and, since ρβ is PPT for 1 ≤ β ≤ 4, the state is PPT and entangled, or bound entangled, for 3 < β ≤ 4 . (4.27)

In Ref. [45] it is shown that for 2 ≤ β ≤ 3 the state ρβ is separable and for 4 < β ≤ 5 the state is free entangled (because it can be projected onto an entangled 2-qubit state, and thus is distillable, see Theorem 4.1.) A graphical illustration of what we learned about the state ρβ (4.14) is shown in Fig. 4.4.

4.3 Locality vs. Non-locality

4.3.1 EPR and Bell Inequalities The issue began with the famous ‘EPR-paradox’ in 1935 [31]. Actually Ein- stein, Podolsky and Rosen did not formulate a paradox, but rather their own interpretation of quantum mechanics. They came to the conclusion that quantum mechanics is incomplete; that there have to be intrinsic properties of quantum mechanical objects which determine the outcome of measurements. In order to illustrate their viewpoint they stated a gedankenexperiment, in which a source emits two entangled particles in opposite direction. Let us here consider Bohm’s variant of the experiment [14] where the source emits two spin 1/2 particles in a singlet state |ψ−i (2.30). Note that the vector |0i denotes “spin up” and |1i stands for “spin down”. EPR considered four requirements which they considered necessary to be fulﬁlled by any physical theory: 4. Classification of Entanglement 43

(i) Perfect (anti-)correlation. If we measure the spins of both particles in the same direction, we can be sure that we will get antiparallel spins.

(ii) Locality. Performing a measurement on one particle cannot inﬂuence the other particle (at least information cannot be transmitted faster than the speed of light) because they are spatially separated.

(iii) Reality. If in an experiment one can exactly predict the value of a physical quantity without inﬂuencing the system, then there has to be an ‘element of reality’ that corresponds to this quantity.

(iv) Completeness. A complete physical theory has to represent any elements of reality involved.

Translating the above requirements to our ‘gedanken’ experiment we can conclude the following: Once we measure the spin of one particle, we instan- taneously know what the outcome of the measurement of the other particle will be. If we consider that the second measurement is performed immedi- ately after the ﬁrst, that is, information could not have been transmitted from one particle to the other viewing the speed of light as the maximum possible speed, then, due to locality (ii) the particles cannot inﬂuence one another. Thus, according to reality (iii) there has to be an ‘element of reality’ corresponding to measurement outcomes that should be included in quantum theory. Now for a long time the question if quantum theory could be completed with such an element of reality remained open. In 1964 J. S. Bell showed [3] that if one strictly follows EPR’s requirements (i)-(iv), then the mysterious ‘element of reality’ corresponds to so-called local hidden variables assigned to pairs of particles, which predetermine the outcome of spin measurements in arbitrary directions. He considered two spin 1/2 particles (which we would call a 2-qubit state in quantum information) in a pure state and set up the famous Bell inequality (BI) which every 2-qubit state should satisfy if it admits his local hidden variable theory (LHV). The BI involves expectation values of spin measurements. To his own surprise, he found that for some spin measurement directions the singlet state |ψ−i (2.30) violates the BI. That means that for this state local hidden variables cannot be assigned to the particles telling them how to behave in measurements. So we cannot help accepting some kind of non-locality of quantum mechanics, there is some ‘spooky action at distance’ that makes the particle which is measured after the other behave in the anti-correlated way. There is no need to believe in some faster-than- light information exchange, but for sure quantum correlations are stronger than classical correlations in a barely comprehensible way. 4. Classification of Entanglement 44

There have been many variations and extensions of Bell’s inequality for- mulated until now. An example is the CHSH inequality (3.12) [23], already mentioned in Sec. 3.3.2. If an entangled state does not violate a specific kind of BI, it is not at all sure that it does not violate some other kind. There have been many efforts to give a more generalized formulation of Bell inequalities. In Ref. [76] Werner showed that some bipartite entangled states, the Werner states (for 2-qubits see Eq. (2.35)), do not violate an inequality derived by assuming general projective measurements of Alice and Bob. In this case one can definitely use a LHV model to describe the process. Nevertheless, if one does not restrict the measurements to projective ones but to the most general measurements, so-called positive operator valued measurements POVMs (see, e.g., Ref. [56]), then the Werner states do indeed violate a kind of BI (shown in Ref. [63]). In this work a state is called ‘local’ only if it does not violate any possible kind of BI, or, equivalently, if it does not violate the most general expression of a BI (which we will call general Bell inequality) even after subjection to any LOCC. If any BI is violated, then a state is called ‘non-local’. The question arises whether non-locality is a necessary feature of all entangled states, or if there exist ‘local’ entangled states for which there exist LHVs according to general measurements. It is in particular useful to determine if an entangled state is local, since in quantum information a state admitting a LHV theory is not useful; such a state could be replaced by classical bits according to the LHV model [18]. It is known that any pure entangled state violates a BI (e.g. the CHSH inequality after applying a particular LOCC to it, see Ref. [35]). For mixed states the situation is not clear (yet). All distillable (see Sec. 4.2) entangled states violate a Bell inequality, since they can be transformed into pure entangled states by LOCC. So the question reduces to the following one (see, e.g., Ref. [77]):

Do there exist local bound entangled states?

There is still no answer to this question, but nevertheless as a step toward solving the problem we want to state a most general formulation of Bell inequalities in the following section.

4.3.2 General Bell Inequality This formulation of a general Bell inequality follows mostly Ref. [70] as well as Ref. [58], the basics to this references can be found in Refs. [34, 59, 60]. Alice and Bob can perform any general measurements. We denote them 4. Classification of Entanglement 45 as A A A B B B Alice: M1 ,M2 ,...,Mn Bob: M1 ,M2 ,...,Mm . (4.28) They are general because the outcomes of each measurement are described by operators

A A A A Mi : Ei,1,Ei,2,...,Ei,p(i) ; B B B B Mj : Ej,1,Ej,2,...,Ej,q(i) ; (4.29) here p(i) is the number of possible outcomes of the i-th measurement for Alice, and equivalently for Bob. The operators (4.29) correspond to POVM measurements (see, e.g., [56]) and satisfy the condition

Xp(i) A 1 A Ei,k = ,Ei,k ≥ 0 (4.30) k=1

(and equivalently for Bob) but are not necessarily orthonormal like in the case of projectors Qi,k, which always satisfy Qi,kQi,r = δkr. We can calculate the following probabilities for a (bipartite) state ρ: ¡ ¢ P A = Tr EA ⊗ 1 ρ (4.31) i,k ¡ i,k ¢ B 1 B Pj,l = Tr ⊗ Ei,k ρ (4.32) A,B ¡ A B ¢ Pi,k;j,l = Tr Ei,k ⊗ Ei,k ρ . (4.33) Eq. (4.31) gives the probability that Alice measures in the i-th measurement the k-th outcome, with Eq. (4.32) we obtain the probability that Bob measures in the j-th measurement the l-th outcome, and if we want to calculate the probability that Alice measures the k-th and Bob the l-th outcome in a joint measurement of the i-th and j-th measurement, we use Eq. (4.33). For reasons of clarity we can write all probabilities together in a ‘probability vector’ of a state ρ corresponding to measurements (4.28), ³ ´ ~ ~ A,B ~ A ~ B Pρ = P , P , P , (4.34)

~ A,B ~A ~B A,B A B where P , P and P contain all probabilities Pi,k;j,l,Pi,k and Pj,l ((4.31) - (4.33)) corresponding to the various combinations of measurements / outcomes for joint measurements, Alice’s measurements and Bob’s measurements. Hidden variables ‘instruct’ the system which outcome a certain measurement should give. That is, a specific hidden variable λi defines an instruction 4. Classification of Entanglement 46 vector with entries 0 or 1, which give the probabilities that in a certain measurement a particular outcome is realized. There are instruction vectors for the measurements of Alice, Bob, or both, denoted as B~ A , B~ B , B~ A,B. For λi λi λi example if Alice has 2 possible measurements with 2 outcomes each, the instruction vector B~ A defined by one hidden variable λ would be, e.g., λi i ~ A Bλi = (0, 1; 1, 0) , (4.35) where the first two entries give the probabilities for the realization of outcomes 1 and 2 of the first measurement; here it is determined that with certainty outcome 2 is realized; and the last two entries describe the second measurement equivalently. Of course Bob’s instruction vector B~ B has λi a similar form, according to the number of his possible measurements and outcomes. It is important that the instruction vector for measurements of both Alice and Bob takes the assumption of locality into account. That means it is assumed that the measurements are independent from each other and we can write B~ A,B = B~ A ⊗ B~ B . (4.36) λi λi λi ~ The ‘total’ instruction vector Bλ is given by ³ ´ ~ ~ A,B ~ A ~ B Bλi = Bλ , Bλ , Bλ . (4.37)

For example, if Alice has 2 possible measurements with 2 outcomes each and Bob 1 measurement with 3 outcomes, we have, e.g., ~ Bλi = ((0, 1; 1, 0) ⊗ (0, 1, 0) , (0, 1; 1, 0) , (0, 1, 0)) . (4.38) ~ ~ Note that the vectors Bλi (4.37) and Pρ (4.34) have the same number of entries, since there is the same number of possible combinations of measurements and outcomes. A LHV theory assigns a probability to each possible instruction vector ~ ~ Bλi , so that a LHV probability vector PLHV of the whole LHV theory is written as a convex combination of instruction vectors [59, 60], X X ~ ~ PLHV = qiBλi , qi ≥ 0, qi = 1 . (4.39) i i ~ The set of all possible LHV theory vectors PLHV form a convex cone LLHV (M), where we use the expression (M) to clarify that the set is in general diﬀerent for diﬀerent possible measurements (4.28) of Alice and Bob. Now we can formulate the following theorem: 4. Classification of Entanglement 47

Theorem 4.5. A bipartite state ρ can be described by a LHV theory with respect to a particular ensemble of measurements (4.28) if and only if ~ Pρ ∈ LLHV (M) . (4.40)

Proof. We say that a state ρ can be described by a LHV theory if there ~ ~ ~ ~ exists a LHV theory vector PLHV such that Pρ = PLHV . If Pρ ∈ LLHV (M), ~ ~ then we can write Pρ as a convex combination of instruction vectors Bλi , ~ ~ ~ thus there exists a LHV theory vector PLHV such that Pρ = PLHV . In the ~ ~ ~ other direction it is clear that if we have Pρ = PLHV then we can write Pρ as ~ ~ a convex combination of vectors Bλi and therefore Pρ ∈ LLHV (M).

In Ref. [70] it is shown that indeed all separable states are elements of LLHV (M). What about the entangled states? There exists a useful Lemma, called the Minkowski-Farkas Lemma (see, e.g., Ref. [66]), that gives a condition for a vector not being an element of a convex cone, and which applied to our case is of the following form: ~ Lemma 4.1. The probability vector Pρ˜ of a state ρ˜ is not an element of ~ LLHV (M) if and only if there exists a ‘Farkas vector’ F , such that ~ ~ ~ ~ F · Pρ˜ < 0 and F · Bλi ≥ 0 ∀λi . (4.41)

In general the Farkas vector F~ can have any real components; however, in Ref. [58] it is shown that it suﬃces to consider integers only. From Lemma 4.1 and Theorem 4.5 we can induce a general Bell inequality. If we have a Farkas ~ ~ ~ vector F for which F · Bλi ≥ 0 ∀λi, we can also say that X X ~ ~ qiF · Bλi ≥ 0 , ∀qi ≥ 0, qi = 1 , (4.42) i i and with help of Eq. (4.39) we obtain ~ ~ ~ F · PLHV ≥ 0 ∀PLHV (4.43) as a general Bell inequality. We can claim the following theorem: Theorem 4.6. For all Farkas vectors F~ that imply ~ ~ ~ F · PLHV ≥ 0 ∀PLHV (4.44) this inequality is a general Bell inequality for some measurement ensemble (4.28). 4. Classification of Entanglement 48

~ Since the probability vector PρLHV of a state ρLHV that can be described by a LHV theory (regarding a particular ensemble of measurements) has to ~ be represented by a LHV theory vector PLHV (according to Theorem 4.5), for all such states we have ~ ~ F · PρLHV ≥ 0 , (4.45) where F~ is a Farkas vector F~ that implies Eq. (4.44). We say that a stateρ ˜ violates the general Bell inequality (4.44) if

~ ~ F · Pρ˜ < 0 , (4.46) and, according to the Minkowski-Farkas Lemma 4.1 and Theorem 4.5, such a state can in general not be described by a LHV theory.

Example. As an example we consider the CH inequality [22]. This is an inequality for 2-qubit states where the measurements of Eq. (4.28) are the spin measurements (equivalent to the measurements of Eq. (3.12))

A A ~0 M1 = ~a · ~σ, M2 = a · ~σ, B ~ B ~0 M1 = b · ~σ, M2 = b · ~σ. (4.47)

We only have to consider probabilities to measure the outcome +1 (in suitable units), so that we write the components of a LHV probability vector (4.39) as (here, e.g., for the joint measurement of Alice measuring the spin along ~a and Bob along b~0 with outcomes +1)

A,B LHV (PLHV )a,+1;b0,+1 =: Pab0 , (4.48) and similar for the single probabilities. The CH inequality is of the form

LHV LHV LHV LHV LHV LHV Pa − Pab + Pb0 − Pa0b0 + Pa0b − Pab0 ≥ 0 . (4.49) or, shortly written, ~ ~ F · PLHV ≥ 0 ∀PLHV , (4.50) where F~ is a vector which has appropriate entries 0, 1 or −1. We see that the CH inequality (4.49) is equivalent to the general Bell inequality (4.44) for the measurement ensemble (4.47) and one particular Farkas vector F~ . 4. Classification of Entanglement 49

Remark. If a state violates the CH inequality (4.49) (or any other Bell inequality) it can in general not be described by a LHV theory and is therefore called ‘non-local’ (and is of course entangled). If it does not violate the inequality we cannot deﬁnitely say that the state can be described by a LHV theory, not even with respect to the regarded measurement ensemble, since we only checked one possible Farkas vector and not all possible Farkas vectors. Furthermore, we should note that in the literature often the expression ‘local state’ is connected with a particular ensemble of measurements - it is meant that for a particular ensemble of measurements we can apply a LHV theory. Nevertheless it might be the case that with other measurements a non-locality of the state is revealed. To be accurate only states that satisfy all general Bell inequalities (with all possible Farkas vectors and all possible measurement ensembles), even if they are subjected to any prior LOCC, should be called ‘local’.

4.3.3 Bell Inequalities and the Entanglement Witness Theorem Does there exist a connection between the general Bell inequality (Theo- rem 4.6) and the entanglement witness Theorem 3.1? The answer is yes, but in this section we will see that given a violation of the general Bell inequality (4.44) for a certain entangled (and non-local) state ρent, we can construct an entanglement witness for this state. But we cannot, in general, construct a violation of a general Bell inequality out of a given entanglement witness for an entangled state.

Construction of an Entanglement Witness out of a Violation of the General Bell Inequality We consider a violation of the general Bell inequality (4.46) for an entangled ~ state ρent and a particular Farkas vector F . We denote the components of the Farkas vector similar to the probability vector (4.34), ³ ´ ³ ´ ¡ ¢ ¡ ¢ ~ ~ A,B ~ A ~ B ~ A,B A,B ~ A A ~ B B F = F , F , F , F = Fi,k;j,l , F = Fi,k , F = Fj,l , (4.51) and deﬁne an operator X X X A,B A B A A 1 B1 B A := Fi,k;j,lEi,k ⊗ Ej,l + Fi,kEi,k ⊗ + Fj,l ⊗ Ej,l , (4.52) i,k,j,l i,k j,l

A where the operators Ei,k are those introduced in Eq. (4.29). With Eqs. (4.31) - (4.33) we calculate for any stateρ ˜ ~ ~ F · Pρ˜ = TrAρ˜ . (4.53) 4. Classification of Entanglement 50

The violation of the general Bell inequality (4.46) clearly corresponds to TrAρent < 0. Since we already know that any separable state ρ can be ~ ~ described by a LHV theory we have Pρ = PρLHV and therefore, according to Eq. (4.45) ~ ~ F · Pρ = TrAρ ≥ 0 ∀ separable states ρ . (4.54) Thus we have

hρent,Ai = TrAρent < 0 , hρ, Ai = TrAρ ≥ 0 ∀ρ ∈ S, (4.55) which is exactly the entanglement witness Theorem 3.1.

Example. We want to construct an entanglement witness out of a violation of the CH inequality (4.49). The operators (4.29) corresponding to the outcome +1 of the possible measurements (4.47) are projectors, for example the projector for a measurement along ~a is 1 EA = EB =: Q = (1 + ~a · ~σ) , (4.56) a,+1 a,+1 +1 2 and equivalently for the other measurement directions. We obtain the entanglement witness [70] with help of Eq. (4.52), where we sum over the measurement directions ~a, a~0,~b, b~0 (we do not have to sum over all possible outcomes, since terms for the outcome −1 do not matter because the component of F~ is 0 for those cases): A = 21 − B. (4.57) Here B is the operator (called ‘Bell-CHSH operator’) deﬁned in Eq. (3.12). Comparing the situation with the CHSH inequality (3.12) from Sec. 3.3.2, we see that the operator A (4.57) is not only an entanglement witness for all states violating the CH inequality (4.49), but also for all states that violate the CHSH inequality.

Can We Construct a General Bell Inequality out of an Entanglement Witness? In the last section we have seen that starting with a violation of a general Bell inequality (4.44), (4.46) we can construct an entanglement witness (4.52). But what is the situation if we want to ﬁnd a general Bell inequality given an entanglement witness (and a violation of the inequality for the state ρent that is detected by the entanglement witness)? In general we cannot do this. This is due to the following: If we have an entanglement witness A for a particular 4. Classification of Entanglement 51

entangled state ρent, we know that for all separable states ρ we have TrAρ ≥ 0 ~ ~ or F · Pρ ≥ 0 according to Eq. (4.53). The problem is that for a general Bell ~ ~ inequality (4.44) we have to ensure that F · PLHV ≥ 0 for all possible LHV ~ theory vectors PLHV . But among those vectors there might exist vectors that do not correspond to quantum mechanical probability vectors. So the ~ ~ ~ ~ condition F · Pρ ≥ 0 does in general not imply the condition F · PLHV ≥ 0. In Ref. [70] it is shown that we can indeed always induce a general Bell inequality from an entanglement witness if we do not allow LHV theory vectors that do not correspond to quantum mechanical probability vectors. Interestingly, the fact that we cannot in general construct a (violation of a) general Bell inequality out of an entanglement witness (which exists for every entangled state) leads to the possibility of local (bound) entangled states. 5. QUANTIFICATION OF ENTANGLEMENT

5.1 Introduction

So far we have seen how it is possible to detect entanglement and classify it according to certain properties of the entangled state. In quantum information entangled states are useful for performing various tasks (see, e.g., Refs. [13, 45, 32, 5, 17, 15]), and it is known that some entangled states are better suited than others. So it is of interest to somehow quantify or measure the entanglement of states. We will see that for pure states we have a useful measure that can be calculated for all states on a finite dimensional Hilbert space Hd1 ⊗Hd2 - the entropy measure1 [8, 6] - it is the von Neumann entropy of the reduced density matrices. Unfortunately, the von Neumann entropy turns out to be a bad entanglement measure for mixed states. Here we have to define other entanglement measures (see, e.g., Refs. [8, 79, 73, 72, 78, 55]), and often it is very hard to calculate them, only for lower dimensional systems there exist algebraic ‘recipe’ methods for calculating a measure. Addition- ally, often it is not clear which entanglement measure is better suited than others, and still we do not know which properties are more and which are less important to be satisfied by a measure. The chapter is organized as follows: In Sec. 5.2 we start with quantifying the entanglement of pure states and introduce the entropy measure, in Sec. 5.3 we discuss various entanglement measures for general (pure or mixed) states: Entanglement of formation, concurrence, entanglement of distillation, relative entropy of entanglement and the Hilbert-Schmidt measure.

5.2 Pure States

For pure states ρ = |ψi hψ| a good and convenient entanglement measure is the von Neumann entropy (3.22) of the state’s reduced density matrices (2.8), (2.9); i.e. the entropy measure [8, 6]

EvN (ρ) = S(ρA) = S(ρB) , (5.1)

1 Some references use the term ‘entanglement of entropy’. 5. Quantification of Entanglement 53

where we dropped the index ‘1’ in the von Neumann entropy S1 introduced in Eq. (3.22). It is useful to take the logarithm in the deﬁnition of the von Neumann entropy to the base d if we consider a Hilbert space Hd ⊗ Hd, because then for all states we have

0 ≤ EvN (ρ) ≤ 1 . (5.2) Here the left limit 0 is achieved if the pure state is a product state, |ψi = |ψAi ⊗ |ψBi (this is independent of the choice of logarithm basis), and 1 is achieved for maximally entangled states (since for those states the reduced density matrices are maximally mixed states). There are some characteristic properties of EvN that fulﬁll natural expectations of an entanglement measure [8, 6]:

(i) EvN is additive: ⊗n EvN (ρ ) = nE(ρ) , (5.3) where ρ⊗n is short for ρ ⊗ ρ ⊗ ... ⊗ ρ with ρ appearing n times.

(ii) EvN does not change under local unitary transformations, i.e.

† EvN ((UA ⊗ UB)ρ(UA ⊗ UB) ) = EvN (ρ) (5.4)

2 (iii) After LOCC the expectation value of EvN cannot have increased: X piEvN (ρi) ≤ EvN (ρ) , (5.5) i

where the states ρi are the residual states after LOCC that occur with probability pi. We know that there are protocols in which from a large number of less entangled input states we can obtain a smaller number of singlet output states (2.30), i.e. the ‘distillation’ or concentration of entanglement for pure states [6]. This is also the reason why why added ‘expectation value’ in property (iii), since in principle we can obtain only with a certain probability from one state a state with a higher value of entropy measure. Considering the other way round, we can also get from a small number of input singlets a ‘larger’ number of less entangled output states. In general the asymptotic (limit of inﬁnitely many input states) rate of output states ρout per input state ρin is [6] m E (ρ ) lim out = vN in . (5.6) n→∞ nin EvN (ρout)

2 See also footnote to Eq. (5.28) 5. Quantification of Entanglement 54

Here nin is the number of input states and mout is the number of output states. If we consider the case in which the output states are non maximally entangled and the input states are singlets, we have (since EvN (|ψ−i hψ−|) = 1) moutEvN (ρout) = nin . (5.7)

So we can identify the entropy measure of the state EvN (ρout) as the minimum ‘number’ of singlets required to prepare the state ρout (mout = 1).

5.3 General States

5.3.1 Entanglement of Formation Mathematical Deﬁnition For general states (pure and mixed states) a logical extension of the entropy measure is the entanglement of formation EF . We can deﬁne it in three steps [8]:

(i) The entanglement of formation of a pure state ρpure is the entropy measure (see Sec. 5.2),

EF (ρpure) = EvN (ρpure) . (5.8)

(ii) The© entanglementª of formation of an ensemble of pure states ² := i i pi, ρpure , where pi are the probabilities for the states ρpure to occur in the ensemble, is X X i i EF (²) = piEF (ρpure) = piEvN (ρpure) . (5.9) i i

(iii) The entanglement of formation of a mixed state ρ is the minimum of the entanglement of formation of all possible ensembles ² realizing ρ,3

EF (ρ) = min EF (²) (5.10) ²

Practical Justiﬁcation of the Deﬁnition Similar to the entropy measure for pure states, the entanglement of formation EF (ρ) of a state ρ can be viewed as the minimum number of singlets needed to prepare a state ρ via LOCC in the asymptotic sense (we also say entanglement

3 P i Remember that a mixed state ρ can be written as ρ = i piρpure, but this decomposition is not unique. 5. Quantification of Entanglement 55 cost) [8]4. Why is this the case? We have already seen in the last section that in the case of pure states Alice and Bob need EvN (ρpure) singlets to© create anª i arbitrary pure state ρpure. So in order to prepare an ensemble ² := pi, ρpure P i (see (ii) in the deﬁnition above) they need i piEvN (ρpure) = EF (²) singlets. Obviously the minimum number of singlets needed to create the state ρ is the minimum of the entanglement of formation of all possible ensembles ², which is EF (ρ), according to (iii). However, one might object the following to the above reasoning: What if 0 0 Alice and Bob prepare a state ρ with a cost EF (ρ ) and then apply LOCC to increase the entanglement of formation so that they obtain a state ρ for which 0 EF (ρ) > EF (ρ ) ? (5.11) No, they cannot5, because we can proof that the expected entanglement of formation is not increasing under LOCC. The proof is done in Ref. [8] and uses the assumption that any LOCC can be separated into the following basic operations (here, e.g., by Alice, but equivalently by Bob):

(a) Alice appends an ancillary system to her system (which has no prior entanglement to Bob’s part of the system).

(b) Alice performs a unitary transformation.

(d) Alice ‘throws away’ (means: traces out) part of her system.

Example. We can demonstrate that the entanglement of formation (5.9) of diﬀerent ensembles realizing the same mixed state ρ is in general diﬀerent: 1 1 Consider the maximally mixed 2-qubit state 2 . This state can be prepared as an equal mixture of four orthogonal product states, so the entanglement of formation of such an ensemble is 0. On the other hand, we could also prepare the state as an equal mixture of the four Bell states, where the entanglement of formation for this ensemble is 1 (all Bell states have entropy measure 1 and all appear with probability 1/4).

4 This statement involves the conviction that any LOCC operation can be separated into several basic operations (a) - (d), mentioned later on. Although intuitively and physically evident, there is no strict mathematical proof that this can be done. Therefore some authors distinguish between entanglement of formation and entanglement cost. 5 There could be an increase of entanglement of formation only with some probability (e.g. distillation), but for the justification of the definition (5.8) - (5.10) it suffices to consider expectation values. 5. Quantification of Entanglement 56

5.3.2 Concurrence and Calculating the Entanglement of Formation for 2 Qubits The definition of entanglement of formation for mixed states, Eq. (5.10), is clear and rather simple. Nevertheless the calculation is in general hard to do; it is often difficult to determine all possible ensembles of pure states that realize a given mixed state. For 2-qubit states, however, there exists an operational method to calculate the entanglement of formation [79]. This is done via introducing another function of a 2-qubit density matrix ρ, the so-called concurrence. We will see that the concurrence itself can be used as an entanglement measure. First, we have to start with some definitions. The spin flip operation on a pure state |ψi is given by ¯ E ¯ ˜ ∗ |ψi → ¯ψ = σy |ψ i , (5.12) where ‘*’ denotes complex conjugation. The operation is called spin flip, because it ‘flips’ the spin of the |0i and |1i states:

σy |0i = i |1i , σy |1i = −i |0i . (5.13) The spin ﬂip operation on 2-qubit density matrices ρ is of the form ∗ ρ → ρ˜ = (σy ⊗ σy) ρ (σy ⊗ σy) , (5.14) where the matrix notation has to be in the standard product basis (2.25). The following theorems concerning entanglement of formation and concurrence are of importance (introduced in Refs. [8, 79]): Theorem 5.1. The entanglement of formation of a 2-qubit state ρ is a function of the concurrence C, µ √ ¶ 1 + 1 − C2 E (ρ) = E (C(ρ)) = H . (5.15) F F 2 Here H is the Shannon entropy function

H(x) = −x log2 x − (1 − x) log2 (1 − x) . (5.16) Theorem 5.2. The concurrence C of a 2-qubit pure state |ψi is ¯D E¯ ¯ ¯ C(|ψi) = ¯ ψ|ψ˜ ¯ , (5.17) and of a general 2-qubit state ρ it is

C(ρ) = max {0, µ1 − µ2 − µ3 − µ4} , (5.18) where the µis are the squareroots of the eigenvalues of the matrix ρ · ρ˜ in decreasing order. 5. Quantification of Entanglement 57

Fig. 5.1: Plot of entanglement of formation EF as a function of concurrence C

The proofs of Theorems 5.1 and 5.2 can be found in Refs. [8, 79]. In Fig. 5.1 we can see a plot of the entanglement of formation EF as a function of the concurrence C: As C increases from 0 to 1, EF increases monotonically from 0 to 1 - so we can consider C as an entanglement measure itself. If we are only interested in the comparison of the ‘amount’ of entanglement of diﬀerent states, the concurrence does its job as an entanglement measure as good as the entanglement of formation; and we do not need to perform the extra calculation of Theorem 5.1.

Example. We want to determine the entanglement of formation and concurrence for the Werner state ρα (2.35) of 2 qubits. To obtain the concurrence C according to Eq. (5.18) we ﬁrst have to calculate the spin ﬂipped Werner stateρ ˜α using Eq. (5.14). Using the matrix notation (2.36) (in the standard basis, as needed) of the Werner state and   0 0 0 −1  0 0 1 0  σ ⊗ σ =   (5.19) y y  0 1 0 0  −1 0 0 0 we get ρ˜α = ρα . (5.20) Now we need to calculate the square roots of the eigenvalues of ρ · ρ˜. In our case we have to calculate the square roots of the eigenvalues of ρ2, which are nothing but the eigenvalues λi of the Werner state (3.25) themselves. We calculate (where we ordered the eigenvalues in decreasing order, we have 5. Quantification of Entanglement 58

0.8 C (ra )

0.6

E ( ) 0.4 F ra

0.2

a 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 5.2: Plot of concurrence C(ρα) and entanglement of formation EF (ρα) of the Werner state ρα for values of α where ρα is entangled

λ4 ≥ λ3 = λ2 = λ1 ) 3α − 1 µ − µ − µ − µ = λ − λ − λ − λ = , (5.21) 1 2 3 4 4 3 2 1 2 and so for the concurrence we have, according to Eq. (5.18),

C(ρα) = 0 for −1/3 ≤ α ≤ 1/3 , 3α − 1 C(ρ ) = for 1/3 < α ≤ 1 . (5.22) α 2 We use Theorem 5.1 and obtain the entanglement of formation 1³ ³p ´ 1 ³ p ´ E (ρ ) = 3 + 6p − 9p2 − 2 log 2 − 3 + 6p − 9p2 − F α 4 2 4 ³ p ´ 1 ³ p ´´ − 2 − 3 + 6p − 9p2 log 2 + 3 + 6p − 9p2 (5.23) 2 4 Comparing the expressions (5.22) and (5.23) we can see that the concurrence has a much simpler form which makes it more convenient to work with. The results are in perfect agreement with the results of section 3.3.2, where we found out that the Werner state ρα is entangled only for 1/3 < α ≤ 1. The plot of concurrence (5.22) and entanglement of formation (5.23) for the entangled Werner state (Fig. 5.2) reveals that the entanglement of formation is always smaller or equal to the concurrence.

5.3.3 Entanglement of Distillation We have already introduced the entanglement of distillation in Sec. 4.2.1. It is given by [8] Edist(ρ) = max Rdist(ρ) , (5.24) LOCC 5. Quantification of Entanglement 59 and is the maximal possible rate of distilled output singlet states per input state ρ (in the limit of inﬁnitely many input states) that can be achieved by a distillation protocol. As an entanglement measure the value of Edist(ρ) varies between 0 and 1 which allows a comparison with the previous introduced measures concurrence and entanglement of formation. However, Edist(ρ) is ‘nonoperational‘, it is in general not known what the ‘best’ distillation protocol for a given entangled state ρ is.

5.3.4 Distance Measures In Refs. [73, 72] the following properties are believed to be necessary for any entanglement measure E 6:

(i) E(ρsep) = 0 ∀ separable states ρsep . (5.25)

(ii) The entanglement measure should be invariant under local unitary operations: ³ ´ † E(ρ) = E (UA ⊗ UB) ρ (UA ⊗ UB) . (5.26)

(iii) (a) The entanglement measure should not increase under any complete positive map (which corresponds to a general local physical operation and classical communication that can be performed): Ã ! X † E ViρVi ≤ E(ρ) , (5.27) i

P † where Vi is of the form Vi,A ⊗ Vi,B and i ViVi = 1. (b) The expectation value of the entanglement measure after selective operations7 should not increase:   X ³ ´ † † ViρiVi Tr ViρiV E  ³ ´ ≤ E(ρ) . (5.28) i † i Tr ViρiVi

6 In Ref. [73] the properties stated are (i), (ii) and (iii)(a), whereas in Ref. [72] we ﬁnd (i), (ii) and (iii)(b); here we follow Refs. [78, 55] and say that all properties (i), (ii) and (iii)(a) and (iii)(b) should be satisﬁed. 7 ‘selective’ means that one is interested in the outcome of a³ certain´ operation, for † example a measurement that occurs with a certain probability Tr ViρiVi . 5. Quantification of Entanglement 60

Remark. Note that the properties (5.25) - (5.28) are similar to the properties (5.3) - (5.5) in connection with the entropy measure, but they do not include the property of additivity (5.3). In principle we would not have to mention property (ii) (Eq. (5.26)), since it follows from Eq. (5.27), nevertheless it is useful to emphasize that the entanglement measure should not change under local basis transformations.

A distance function of two states ρ1, ρ2 is written as

d(ρ1, ρ2) . (5.29)

We deﬁne a distance measure D(¯ρ) of a stateρ ¯ as the minimal distance (5.29) of a stateρ ¯ to the set S of all separable states [73],

D(¯ρ) := min d(ρ, ρ¯) . (5.30) ρ∈S

In Refs. [73, 72] various suﬃcient conditions for distance functions (5.29) are given that guarantee, if fulﬁlled, that the distance measure (5.30) has the properties (5.25) - (5.28). There are several possible realizations of the distance function (5.29). Here we want to concentrate on two of them.

Relative Entropy of Entanglement

The relative entropy of entanglement DRE [73, 72] is a distance measure (5.30) that uses a distance function given by [50, 51]

dRE(ρ1, ρ2) := Tr (ρ2 (log2 ρ2 − log2 ρ1)) . (5.31) and so we have, according to Eq. (5.30),

DRE(¯ρ) := min dRE(ρ, ρ¯) . (5.32) ρ∈S

It is shown in Refs. [73, 72] that the relative entropy of entanglement indeed has the properties (5.25) - (5.28). In general, to compute Eq. (5.31) for any two states ρ1, ρ2 we use (similar to the calculations for the entropy inequalities in Sec. 3.3.2) X Tr (ρ2 (log2 ρ2 − log2 ρ1)) = (λi (log2 λi − log2 µi)) , (5.33) i where µi are the eigenvalues of the state ρ1 and λi those of ρ2. 5. Quantification of Entanglement 61

Example. What is the quantum relative entropy of the 2-qubit Werner state ρα (2.35)? First, we need to know the separable state ρ0,α for which Eq. 5.30 ent minimizes for the entangled Werner states ρα , so that we have

ent DRE(ρα ) = dRE(ρ0,α, ρα) (5.34)

In Ref. [73] it is determined that ρ0,α is independent of the parameter α and is given by8 ρ0,α =: ρ0 = ρ1/3 , (5.35) that is the Werner state for α = 1/3. The eigenvalues λi of ρα are given in Eq. (3.25), the eigenvalues µi of ρ0 (5.35) are, inserting α = 1/3 in Eq. (3.25), 1 1 µ = µ = µ = , µ = . (5.36) 1 2 3 6 4 2 We obtain X ent DRE(ρα ) = dRE(ρ0, ρα) = (λi (log2 λi − log2 µi)) µ µ i ¶¶ 1 − α 1 − α = 3 log + log 6 + 4 2 4 2 µ ¶ 1 + 3α 1 + 3α + log + 1 4 2 4

= a log2 a + (1 − a) log2 (1 − a) + 1 , (5.37)

1+3α where we deﬁned a := 4 , which is the notation used in Ref. [73]. We have, in correspondence to the results of Eq. (3.42),

DRE(ρ1/3) = 0 ,DRE(ρα) → 1 for α → 1 , (5.38) which is illustrated in Fig. 5.3.

Hilbert-Schmidt Measure Another distance function is given by the Hilbert-Schmidt distance, using the norm (2.2), dHS(ρ1, ρ2) = kρ1 − ρ2k , (5.39)

8 In Ref. [73] the nearest separable state for all states that are mixtures of the Bell states (2.30) - (2.33) is calculated, so we need to write ρα (2.35) as a mixture of Bell states, apply the results of Ref. [73] and again translate it into our notation with the parameter α. 5. Quantification of Entanglement 62

Fig. 5.3: Plot of the relative entropy of entanglement DRE(ρα) of the Werner state ρα for values of α where ρα is entangled and the distance measure in connection to the Hilbert-Schmidt distance (5.39), the Hilbert-Schmidt measure of a stateρ ¯, is [78]

D(¯ρ) = min dHS(ρ, ρ¯) = min kρ − ρ¯k . (5.40) ρ∈S ρ∈S

Although the Hilbert-Schmidt measure is much more convenient to handle in calculations than the relative entropy of entanglement, and despite of other advantages (see Chapter 6), it is still not clear if it fulﬁlls the properties (5.27) and (5.28). In Ref. [73] a suﬃcient condition for a distance measure to satisfy the property (5.27) is stated,

d(Θρ1, Θρ2) ≤ d(ρ1, ρ2) , (5.41) where Θ is any completely positive trace preserving map (see Sec. 2.3). In Ref. [78] it is conjectured that Eq. (5.41) is indeed fulﬁlled in the case of the Hilbert-Schmidt distance (5.39). Nevertheless, in Ref. [55] it is shown that the proof presented in Ref. [78] does not hold, and a counterexample is presented. However, there has not been any indication yet that there exist states with a Hilbert-Schmidt measure that does not have the properties (5.27) and (5.28).

Example. Let us investigate the Werner state ρα (2.35) again and determine the Hilbert-Schmidt measure. In Refs. [78, 12] the nearest separable state (for ent which Eq. (5.40) minimizes) to an entangled Werner state ρα (see Eq. (3.42)) 5. Quantification of Entanglement 63

Fig. 5.4: Plot of the Hilbert-Schmidt measure DHS(ρα) of the Werner state ρα for values of α where ρα is entangled is given by9 ρ0 = ρ1/3 . (5.42) Interestingly, it is the same state as for the relative entropy of entanglement (see Eq. (5.35)) and is independent of the parameter α. So the Hilbert- Schmidt measure (5.40) is ° ° ent ° ent° DHS(ρα ) = ρ0 − ρα . (5.43)

The explicit calculation can be done in diﬀerent ways. One way is to use the notation (2.27), like it is done (for the isotropic state) in Sec. 6.5. Here we ent want to use the diagonalized form of ρ0 − ρα , because then we have s ° ° X ent ° ent° 2 DHS(ρα ) = ρ0 − ρα = (µi − λi) , (5.44) i where µi are the eigenvalues of ρ0, see Eq. (5.36), and λi are the eigenvalues of ρα, see Eq. (3.25). Inserting the values of Eqs. (5.36) and (3.25) we ﬁnd √ µ ¶ 3 1 D (ρent) = α − . (5.45) HS α 2 3

ent For a plot of the Hilbert-Schmidt measure for ρα see Fig. 5.4. 9 We could also use the method in Sec. 6.4 to determine the nearest separable state, although there it is done for the isotropic qubit state, for the Werner state the procedure is nearly the same, we only have changed signs on the left-hand side of Eq. (6.34). 5. Quantification of Entanglement 64

5.3.5 Comparison of Diﬀerent Entanglement Measures for the 2-Qubit Werner State In the previous sections we calculated various entanglement measures for the 2-qubit Werner state ρα (2.35). Now we want to compare the concurrence (5.22), the entanglement of formation (5.23), the relative entropy of entanglement (5.37) and the Hilbert-Schmidt measure (5.45) for this particular state. First, we have to recognize that all of this entanglement measures vary between 0 and 1, with the exception of the Hilbert-Schmidt measure. In order to be able to properly compare the Hilbert-Schmidt measure with the other measures, we have to ‘normalize’ Eq. (5.45); namely, we obtain the ent normalized Hilbert-Schmidt measure by dividing DHS(ρα ) by the value of ent DHS(ρ1 ) and obtain µ ¶ 3 1 D (ρent) = α − (5.46) HSN α 2 3

Remarkably this is exactly the same value as we obtained for the concurrence ent C(ρα ) in Eq. (5.22), so that we have

ent ent DHSN(ρα ) = C(ρα ) . (5.47)

In Fig. 5.5 the entanglement measures are graphically compared. 5. Quantification of Entanglement 65

D HSN (ra ) = C (ra ) 0.8 E ( ) r 0.6

D (ra ) 0.4

0.2

a 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 5.5: Comparison of the concurrence C(ρα), the entanglement of formation EF(ρα), the relative entropy of entanglement DRE(ρα) and the ‘normalized’ Hilbert-Schmidt measure DHSN(ρα) of the Werner state ρα for values of α where ρα is entangled 6. HILBERT-SCHMIDT MEASURE AND ENTANGLEMENT WITNESS

6.1 Introduction

In this chapter we investigate the connection between the Hilbert-Schmidt measure of entanglement (5.40) and the concept of entanglement witnesses (see Theorem 3.1), it follows Refs. [12, 11]. This is of importance in many aspects, e.g., it becomes evident that the procedures of detecting and measuring entanglement are closely connected to each other. Furthermore, as it is often diﬃcult to construct an entanglement witness, a method for constructing an entanglement witness is given. The chapter is organized as follows: In Sec. 6.2 we start with geometrical considerations comparing Euclidean space geometry with Hilbert-Schmidt space geometry. Next, in Sec. 6.3 we derive a useful theorem related to the connection of the Hilbert-Schmidt measure to a so-called ‘generalized Bell inequality’1. Then we state a lemma concerning the detection of the nearest separable state to a given entangled state in Sec. 6.4, and ﬁnally, in Sec. 6.5 we discuss examples.

6.2 Geometrical Considerations about the Hilbert-Schmidt Distance

We can write the Hilbert-Schmidt distance (5.39) of any two states ρ1, ρ2 ∈ A as ¿ À ® ρ1 − ρ2 ¯ dHS(ρ1, ρ2) = kρ1 − ρ2k = ρ1 − ρ2, = ρ1 − ρ2, C , (6.1) kρ1 − ρ2k where we deﬁne the operator ρ − ρ C¯ := 1 2 . (6.2) kρ1 − ρ2k

1 Mind that the ‘generalized Bell inequality’ introduced in this context is diﬀerent from the general Bell inequality of Theorem 4.6. A more detailed discussion of the subject is given later on. 6. Hilbert-Schmidt Measure and Entanglement Witness 67

Fig. 6.1: Illustration of Eqs. (6.7) and (6.8): The scalar product hρl − ρ1, ρ1 − ρ2i is negative because the projection (ρl − ρ1)k onto ρ1 − ρ2 points in the opposite direction to ρ1 − ρ2. On the other side, hρr − ρ1, ρ1 − ρ2i is positive for states ρr, because then the projection (ρr − ρ1)k points in the same direction as ρ1 − ρ2.

Instead of C¯ we may also choose C := C¯ + c 1 (c ∈ C) because we have ® ® ¯ ¯ dHS(ρ1, ρ2) = ρ1 − ρ2, C = ρ1 − ρ2, C + hρ1 − ρ2, c 1i = hρ1 − ρ2,Ci , (6.3) 2 where we used hρ1 − ρ2, 1i = Trρ1 − Trρ2 = 0 . For convenience we ﬁx c to hρ , ρ − ρ i c = − 1 1 2 , (6.4) kρ1 − ρ2k and obtain ρ − ρ − hρ , ρ − ρ i 1 C = 1 2 1 1 2 . (6.5) kρ1 − ρ2k

Analogously to Euclidean space we deﬁne a hyperplane P that includes ρ1 and is orthogonal to ρ1 − ρ2 as the set of all states ρp satisfying 1 hρp − ρ1, ρ1 − ρ2i = 0 . (6.6) kρ1 − ρ2k

For all states on one side of the plane, let us call them ‘left-hand’ states ρl, we have 1 hρl − ρ1, ρ1 − ρ2i < 0 , (6.7) kρ1 − ρ2k whereas the states on the other side, the ‘right-hand’ states ρr are given by 1 hρr − ρ1, ρ1 − ρ2i > 0 . (6.8) kρ1 − ρ2k 2 This value of c will turn out to be useful. 6. Hilbert-Schmidt Measure and Entanglement Witness 68

For an illustration see Fig. 6.1. We can re-write Eqs. (6.6), (6.7), and (6.8) with help of operator C by using ¿ À ρ − ρ hρ , ρ − ρ i hρ , Ci = ρ , 1 2 − 1 1 2 hρ , 1i kρ1 − ρ2k kρ1 − ρ2k 1 = hρ − ρ1, ρ1 − ρ2i . (6.9) kρ1 − ρ2k Then the plane P is determined by

hρp ,Ci = 0 , (6.10) and the ‘left-hand’ and ‘right-hand’ states satisfy the inequalities

hρl ,Ci < 0 and hρr ,Ci > 0 . (6.11)

6.3 The Bertlmann-Narnhofer-Thirring Theorem

According to Ref. [12] we call part of Eq. (3.3), i.e.

hρ, Ai ≥ 0 ∀ρ ∈ S, (6.12) a generalized Bell inequality. In this context ‘generalized’ means that it detects entanglement in the mathematical sense and not non-locality. Thus it does not serve as a criterion to determine if a state admits a LHV theory like the usual Bell inequalities [3, 23] do. Pay attention that it is diﬀerent to the ‘general Bell inequality’ (4.44) discussed in Sec. 4.3.2; there the inequality can detect non-locality like usual Bell inequalities, but has a general form (arbitrary measurements, etc.; see, e.g., Refs. [70, 10]). As mentioned, this general Bell inequality (from Sec. 4.3.2) does not necessarily detect entanglement. Nevertheless, every entangled state violates the inequality (6.12) for an appropriate entanglement witness A. We can re-write Eq. (3.3) as

hρ, Ai − hρent,Ai ≥ 0 ∀ρ ∈ S. (6.13)

The maximal violation of the GBI is deﬁned by µ ¶

B(ρent) = max min hρ, Ai − hρent,Ai , (6.14) A, kA−a1k≤1 ρ∈S 6. Hilbert-Schmidt Measure and Entanglement Witness 69 where the maximum is taken over all possible entanglement witnesses A, suitably normed3. Interestingly, one can ﬁnd connections between the Hilbert-Schmidt measure and the concept of entanglement witnesses. In particular, there exists the following equivalence stated in the Bertlmann-Narnhofer-Thirring Theo- rem [12]:

Theorem 6.1. The Hilbert-Schmidt measure of an entangled state equals the maximal violation of the GBI:

D(ρent) = B(ρent) . (6.15)

Proof. We want to prove the Theorem in a diﬀerent way as in Ref. [12]. For an entangled state ρent the minimum of the Hilbert-Schmidt distance – the Hilbert-Schmidt measure – is attained for some state ρ0 since the norm is continuous and the set S is compact

min kρ − ρentk = kρ0 − ρentk . (6.16) ρ∈S

In Eqs. (6.3) and (6.5) we identify ρ1 = ρ0 and ρ2 = ρent and with C given by Eq. (6.5) we obtain the Hilbert-Schmidt measure

dHS(ρ0, ρent) = D(ρent) = hρ0,Ci − hρent,Ci . (6.17)

In Eq. (6.17) the operator C has to be an optimal entanglement witness for the following reason: The state ρ0 lies on the boundary of the set of all separable states S and the hyperplane deﬁned by hρp ,Ci = 0 is orthogonal to ρ0 −ρent. Because ρ0 is the nearest separable state to ρent the plane has to be tangent to the set S (see Fig. 6.2). Eqs. (6.10), (6.11) imply the inequalities (3.3), therefore it follows that C is an optimal entanglement witness4

ρ0 − ρent − hρ0, ρ0 − ρenti 1 Aopt = C = , (6.18) kρ0 − ρentk which we use to rewrite the Hilbert-Schmidt measure (6.17)

D(ρent) = hρ0,Aopti − hρent,Aopti . (6.19)

3 In the Pauli matrices notation (with a decomposition into 4×4 matrices like in (2.27)) of an operator A, a is the real coeﬃcient related to the 1 ⊗ 1 term. For more details see [12, 11]. 4 Note that in general (that is, with arbitrary states ρ1 and ρ2) the operator C (6.5) is not an entanglement witness. 6. Hilbert-Schmidt Measure and Entanglement Witness 70

Fig. 6.2: Illustration of the Bertlmann-Narnhofer-Thirring Theorem

Since we have max (− hρent,Ai) = − hρent,Aopti , (6.20) A where A is restricted by kA − a1k ≤ 1 and hρ0,Aopti = 0 , we obtain

D(ρent) = hρ0,Aopti − hρent,Aopti = max (hρ0,Ai − hρent,Ai) A, kA−a1k≤1 µ ¶

= max min hρ, Ai − hρent,Ai = B(ρent) , (6.21) A, kA−a1k≤1 ρ∈S which completes the proof. Similar methods for constructing an entanglement witness can be found in Ref. [61]; for other approaches see, e.g., Refs. [49, 38, 48].

6.4 How to Check a Guess of the Nearest Separable State

Given an entangled state ρent, for the Hilbert-Schmidt measure we have to calculate the minimal distance to the set of separable states S, Eq. (5.40). In general it is not easy to ﬁnd the correct state ρ0 which minimizes the distance (for speciﬁc procedures, see, e.g., Refs. [26, 82, 81]). However, we can use an operator like in Eq. (6.5) for checking a good guess of ρ0. How does it work? Let us start with an entangled state ρent and let us callρ ˜ the guess of the nearest separable state. From previous considerations (Eqs. (6.5), (6.6) and (6.10)) we know that the operator

ρ˜ − ρ − hρ,˜ ρ˜ − ρ i 1 C˜ = ent ent (6.22) kρ˜ − ρentk 6. Hilbert-Schmidt Measure and Entanglement Witness 71

Fig. 6.3: Illustration why C˜ cannot be an entanglement witness ifρ ˜ is not the nearest separable state. The hatched area is the one were the condition hρ, C˜i ≥ 0 ∀ρ ∈ S is violated.

deﬁnes a hyperplane which is orthogonal toρ ˜− ρent and includesρ ˜. Now we state the following lemma:

Lemma 6.1. A state ρ˜ is equal to the nearest separable state ρ0 if and only if C˜ is an entanglement witness.

Proof. We already know from Sec. 6.3 that ifρ ˜ is the nearest separable state then the operator C˜ is an entanglement witness. So we need to prove the opposite: If C˜ is an entanglement witness the stateρ ˜ has to be the nearest separable state ρ0. We prove it indirectly. Ifρ ˜ is not the nearest separable state then kρent − ρ˜k does not give the minimal distance to S; the plane ˜ deﬁned by hρp, Ci = 0 is not tangent to S and thus the existence of ‘left- ˜ ˜ hand’ separable states ρsep satisfying hρsep, Ci < 0 follows. That means C cannot be an entanglement witness (inequalities (3.3) are not fulﬁlled), see Fig. 6.3.

Remark. Of course it is in general not easy to apply Lemma 6.1 to determine the nearest separable state, since in general it is diﬃcult to check whether the operator C˜ (6.22) is an entanglement witness. For some cases (see Sec. 6.5), however, it is much more easier to apply Lemma 6.1 than using other methods to determine the nearest separable state. If C˜ is indeed an entanglement witness, then, because it is tangent to S, ˜ it is optimal and can be written as C = Aopt , exactly like Eq. (6.18). It is the operator for which the GBI is maximally violated. 6. Hilbert-Schmidt Measure and Entanglement Witness 72

6.5 Examples

6.5.1 Isotropic Qubit States In Ref. [12] the 2-qubit Werner state has been studied5 – here we consider the isotropic state in 2 dimensions (acting on H2 ⊗ H2, it is obtained for d = 2 in Eq. (2.16)) ¯ ® ¯ 1 − α 1 ρ = α ¯φ2 φ2 ¯ + 1 , − ≤ α ≤ 1 , (6.23) α + + 4 3 ¯ ® ¯ 2 where φ+ := |φ+i (2.33). In matrix notation in the standard product basis (2.25) we get   1 + α 0 0 α  4 2   1 − α   0 4 0 0  ρα =  1 − α  , (6.24)  0 0 4 0  α 1 + α 2 0 0 4 whereas in terms of the Pauli matrices basis (2.27) the state can be expressed by 1 ρ = (1 + α Σ) , (6.25) α 4 with the deﬁnition

Σ := σx ⊗ σx − σy ⊗ σy + σz ⊗ σz . (6.26)

We know that ρα is (recall Eq. (2.19)) 1 1 1 for − ≤ α ≤ separable , for < α ≤ 1 entangled . (6.27) 3 3 3 To compute the Hilbert-Schmidt measure (5.40) for an entangled isotropic ent ent ent state ρα we need to calculate D(ρα ) = minρ∈S kρ − ρα k , that is, we need to ﬁnd the nearest separable state ρ0 to the entangled state in order to obtain ent ent D(ρα ) = kρ0 − ρα k . From the separability condition (6.27) we see that the state with α = 1/3 lies on the boundary between separable and entangled isotropic states. Thus our guess for all isotropic entangled qubit states is (and we call itρ ˜): µ ¶ 1 1 ρ˜ = ρ = 1 + Σ . (6.28) 1/3 4 3

5 We also calculated the Hilbert-Schmidt measure for the 2-qubit Werner state in Sec. 5.3.4. 6. Hilbert-Schmidt Measure and Entanglement Witness 73

Now we have to check that the operator C˜ (6.22) is an entanglement witness (see Lemma 6.1). For this purpose we calculate the expressions µ ¶ √ µ ¶ 1 1 ° ° 3 1 ρ˜ − ρent = − α Σ with °ρ˜ − ρent° = α − , (6.29) α 4 3 α 2 3 √ (note that kΣk = 2 3) and µ ¶ ® 1 1 ρ,˜ ρ˜ − ρent = Trρ ˜(˜ρ − ρent) = − α . (6.30) α α 4 3

Then the operator C˜ is explicitly given by ρ˜ − ρent − hρ,˜ ρ˜ − ρenti 1 1 ˜ α α √ 1 C = ent = ( − Σ) . (6.31) kρ˜ − ρα k 2 3

We need to examine that C˜ is an entanglement witness, i.e., we check the inequalities (3.3). For the entangled state (where α > 1/3) we get √ µ ¶ D E 3 1 ρent, C˜ = Tr ρentC˜ = − α − < 0 . (6.32) α α 2 3

So the ﬁrst condition is satisﬁed. The second one, the positivity of hρ, C˜i for all separable states ρ, can be seen in the following way: With notation (2.29) for ρsep the scalar product is D E X ¡ ¢ ˜ 1 k k k k k k ρsep, C = pk √ 1 − nxmx + nymy − nz mz , k 2 3 q ¡ ¢ ¯ ¯ ¯ ¯ k 2 k 2 k 2 ¯ k¯ ¯ k¯ (nx) + ny + (nz ) =: ~n ≤ 1, ~m ≤ 1 . (6.33) We have to show that

k k k k k k −nxmx + nymy − nz mz ≥ −1 , (6.34) then the right-hand side of Eq. (6.33) remains always positive. (The convex sum of positive terms stays positive.) From the property ¯ ¯ ¯ ¯ ¯ ¯ ¯~nk · ~mk¯ ≤ ¯~nk¯ ¯~mk¯ ≤ 1 or − 1 ≤ ~nk · ~mk ≤ 1 , (6.35) we ﬁnd indeed that Eq. (6.34) is satisﬁed,

k k k k k k k k k k k k k k −nxmx + nymy − nz mz ≥ −nxmx − nymy − nz mz = − ~n · ~m ≥ −1 , (6.36) 6. Hilbert-Schmidt Measure and Entanglement Witness 74 which completes the proof that hρ, C˜i ≥ 0 ∀ρ ∈ S. So C˜ represents an entanglement witness ˜ 1 Aopt = C = √ (1 − Σ) , (6.37) 2 3 and our guess for the nearest separable state was correct,ρ ˜ = ρ0 .

The Hilbert-Schmidt measure for the entangled isotropic state is determined by Eq. (6.29), √ µ ¶ ° ° 3 1 D(ρent) = °ρ − ρent° = α − . (6.38) α 0 α 2 3 It only remains to check the Bertlmann-Narnhofer-Thirring Theorem 6.1. ent Thus we calculate the maximal violation B(ρα ) (6.14) of the GBI. The maximum is attained for the optimal entanglement witness Aopt and the minimum for the nearest separable state ρ0 . Then Eq. (6.32) determines the ent value of B(ρα ) (recall that hρ0,Aopti = 0) √ µ ¶ ® 3 1 B(ρent) = − ρent,A = α − . (6.39) α α opt 2 3 ent ent So, indeed D(ρα ) = B(ρα ) , the Hilbert-Schmidt measure equals the maximal violation of the GBI.

6.5.2 Isotropic Qutrit States Eq. (2.16) deﬁnes the isotropic qutrit state for d = 3, i.e. ¯ ® ¯ 1 − α 1 ρ = α ¯φ3 φ3 ¯ + 1 , − ≤ α ≤ 1 , (6.40) α + + 9 8 where ¯ ® 1 ³ ´ ¯φ3 = √ |0i ⊗ |0i + |1i ⊗ |1i + |2i ⊗ |2i . (6.41) + 3 In matrix notation in the standard product basis (2.42) we have  1+2α α α  9 0 0 0 3 0 0 0 3  1−α   0 9 0 0 0 0 0 0 0   1−α   0 0 9 0 0 0 0 0 0   1−α   0 0 0 9 0 0 0 0 0   α 1+2α α  ρα =  3 0 0 0 9 0 0 0 3  . (6.42)  1−α   0 0 0 0 0 9 0 0 0   1−α   0 0 0 0 0 0 9 0 0   1−α  0 0 0 0 0 0 0 9 0 α α 1+2α 3 0 0 0 3 0 0 0 9 6. Hilbert-Schmidt Measure and Entanglement Witness 75

In the Gell-Mann matrices representation (2.43) the state ρα can be expressed as (see also Ref. [20] 6) µ ¶ 1 3α ρ = 1 + Λ , (6.43) α 9 2 with the deﬁnition

Λ := λ1 ⊗λ1 −λ2 ⊗λ2 +λ3 ⊗λ3 +λ4 ⊗λ4 −λ5 ⊗λ5 +λ6 ⊗λ6 −λ7 ⊗λ7 +λ8 ⊗λ8 . (6.44) From Eq. (2.19) we know that 1 1 −8 ≤ α ≤ 4 ⇒ ρα separable , (6.45) 1 4 < α ≤ 1 ⇒ ρα entangled . By the same argument as in the qubit case we guess the nearest separable state to the state (6.43), µ ¶ 1 3 ρ˜ = ρ = 1 + Λ . (6.46) 1/4 9 8

Again, to check our guess we examine if the operator C˜ (6.22) is an entanglement witness. We need the following expressions µ ¶ √ µ ¶ 1 1 ° ° 2 2 1 ρ˜ − ρent = − α Λ with °ρ˜ − ρent° = α − , (6.47) α 6 4 α 3 4 √ (where kΛk = 4 2) and µ ¶ ® 2 1 ρ,˜ ρ˜ − ρent = Trρ ˜(˜ρ − ρent) = − α . (6.48) α α 9 4

Then C˜ (6.22) is explicitly given by µ ¶ 1 3 C˜ = √ 1 − Λ . (6.49) 3 2 4

Now let us check the entanglement witness conditions (3.3) for C˜: √ µ ¶ D E 2 2 1 ρent, C˜ = Tr ρentC˜ = − α − < 0 . (6.50) α α 3 4

6 In Ref. [20] the projectors |ii hj| (i, j = 0, 1, 2) are expressed as linear combinations of the Gell-Mann matrices to obtain this form of the isotropic state ρα. 6. Hilbert-Schmidt Measure and Entanglement Witness 76

So the ﬁrst condition is satisﬁed since α > 1/4; for the second one we obtain (with help of Eq. (2.44)) D E X ¡ ˜ 1 k k k k k k k k k k ρsep, C = pk √ 1 − n m + n m − n m − n m + n m 3 2 1 1 2 2 3 3 4 4 5 5 k ¢ − nkmk + nkmk − nkmk , ¯ 6 ¯ 6 ¯7 7¯ 8 8 ¯~nk¯ ≤ 1, ¯~mk¯ ≤ 1 . (6.51)

Since the inequalities (6.35) apply here as well we have

k k k k k k k k k k −n1m1 + n2m2 − n3m3 − n4m4 + n5m5 − k k k k k k − n6m6 + n7m7 − n8m8 ≥ −~nk · ~mk ≥ −1 , (6.52) ˜ ˜ so that hρsep, Ci ≥ 0 . Indeed, C represents an entanglement witness and we identify µ ¶ ˜ 1 3 Aopt = C = √ 1 − Λ andρ ˜ = ρ0 . (6.53) 3 2 4 With Eq. (6.47) the Hilbert-Schmidt measure is √ µ ¶ ° ° 2 2 1 D(ρent) = °ρ − ρent° = α − , (6.54) α 0 α 3 4

ent and by the same argumentation as for qubits the maximal violation B(ρα ) (6.14) of the GBI is determined by Eq. (6.50) √ µ ¶ ® 2 2 1 B(ρent) = − ρent,A = α − . (6.55) α α opt 3 4

ent ent So again, D(ρα ) = B(ρα ) , we see that the Bertlmann-Narnhofer-Thirring Theorem 6.1 is satisﬁed.

6.5.3 Isotropic States in Higher Dimensions Finally, we want to show how we can generalize our isotropic qubit and qutrit d results to arbitraryn dimensions.o A general state on H can be written in a matrix basis 1, γ1, . . . , γd2−1 as Ã r ! 1 d(d − 1) X ω = 1 + n γi , n2 =: |~n|2 ≤ 1 . (6.56) d 2 i i i 6. Hilbert-Schmidt Measure and Entanglement Witness 77

q d(d−1) 2 We have included the factor 2 for the correct normalization (Trω ≤ 1) and the matrices γi have the properties

Tr γi = 0 , Tr γiγj = 2 δij . (6.57)

Considering the tensor product space Hd ⊗ Hd the notation of separable states is a straight forward extension to Eqs. (2.29) and (2.44) µ r X 1 d(d − 1) ρ = p 1 ⊗ 1 + nk γi ⊗ 1 sep k d2 2 i k r ¶ d(d − 1) d(d − 1) + , mk 1 ⊗ γi + nkmk γi ⊗ γj . (6.58) 2 i 2 i j

We express a d-dimensional isotropic state (on Hd ⊗ Hd) – a generalization of the isotropic qubit state (6.25) and qutrit state (6.43) – as µ ¶ 1 d 1 ρ = 1 + α Γ , − ≤ α ≤ 1 , (6.59) α d2 2 d2 − 1 where we define dX2−1 i i Γ := ci γ ⊗ γ , ci = ±1 . (6.60) i=1 d The factor 2 in Eq. (6.59) is due to normalization. The splitting of ρα into entangled and separable states is given by Eq. (2.19). There is strong evidence that expression (6.59) with definition (6.60) co- incides with the isotropic state definition (2.16), which we introduced in the beginning, for all dimensions d. That means, there exist d2 − 1 matrices γi with properties (6.57), which form a basis together with the identity 1 for all d2 × d2 matrices. They describe the quantum state in the isotropic way (6.59), (6.60) and can be expressed as suitable linear combinations of density matrices according to the standard basis notation. In this way a generalization of our previous results is possible and can be obtained by calculations very similar to the ones for qubits and qutrits (see Sec. 6.5.1 and Sec. 6.5.2). In particular, using the same notations as before, we find the following expressions for the nearest separable state ρ0, ent the Hilbert-Schmidt measure D(ρα ) and the optimal entanglement witness Aopt : µ ¶ 1 d ρ0 = ρ 1 = 1 + Γ , (6.61) d+1 d2 2(d + 1) 6. Hilbert-Schmidt Measure and Entanglement Witness 78

√ µ ¶ ° ° d2 − 1 1 D(ρent) = °ρ − ρent° = α − , (6.62) α 0 α d d + 1 µ ¶ d − 1 d Aopt = √ 1 − Γ . (6.63) d d2 − 1 2(d − 1) The maximal violation of the GBI gives √ µ ¶ ® d2 − 1 1 B(ρent) = − ρent,A = α − , (6.64) α α opt d d + 1

ent ent thus we see that again D(ρα ) = B(ρα ) and Theorem (6.1) is satisﬁed.

Remark. For the limit of inﬁnite dimensions, d → ∞ , the distance or the maximal violation of GBI approaches the parameter α , which matches the fact that the region where the isotropic state is separable shrinks to zero (see in this connection Refs. [82, 81]). 7. TRIPARTITE SYSTEMS

7.1 Introduction

So long we have only been viewing bipartite systems. Foundations of quantum information and much of its theory nowadays is related to bipartite systems where much has been achieved. Nevertheless, tripartite or multipartite systems are of interest as well. One example is the GHZ Theorem [37, 36, 52], which is an ‘extended’ Bell’s Theorem [3, 4] for tripartite systems: Where Bell’s Theorem says that quantum mechanics contradicts LHV theories via expectation values, the GHZ theorem predicts a contradiction that can in principle be verified with only one experiment with spin measurements in the same direction (Bell’s Theorem does not exhibit a contradiction for measurements in the same direction). Of course, tri- and multipartite systems are of interest for practical reasons in quantum information too (see, e.g., Refs. [25, 33, 54]). In this chapter we want to concentrate on tripartite systems. The entanglement is in this case defined as a straight forward extension of the bipartite case. It is essentially different that for tripartite states there exists a a priori classification of entanglement (apart from the question of distillability and admission of a LHV theory, discussed in Chapter 4). First steps toward this revelation were done in Refs. [9, 75]; in Ref. [30] an exact description of 3-qubit entanglement for pure states was introduced. The chapter is organized as follows: First, in Sec. 7.2 we give a basic mathematical description of tripartite systems. In Sec. 7.3 we examine a classification of pure entangled tripartite states, whereas in Sec. 7.4 we discuss general (pure or mixed) states. In this context we introduce the concept of tripartite witnesses which is similar to the concept of entanglement witnesses (see Sec. 3.3.1).

7.2 Basics

A tripartite system consists of three subsystems that are described by Hilbert di spaces H of dimension di, so that the whole system is described by a Hilbert 7. Tripartite Systems 80

d1 d2 d3 space HA ⊗ HB ⊗ HC . The indices ‘A’, ‘B’ and ‘C’ are often neglected. E.g. a 3-qubit system H2 ⊗ H2 ⊗ H2 consists of 3 2-dimensional subsystems.

Matrix Notation. A general matrix notation of a tripartite state ρ is Eq. (2.3). If we use a product basis, the matrix elements of a state ρ on Hd1 ⊗Hd2 ⊗Hd3 are

ρmA mB mC , nA nB nC = hemA ⊗ fmB ⊗ gmC | ρ |enA ⊗ fnB ⊗ gnC i . (7.1)

d1 d2 d3 Here {ei}, {fi} and {gi} are bases of the Hilbert spaces HA , HB and HC .

Reduced Density Matrices. The reduced density matrices are obtained by tracing out subsystems. If one subsystem is traced out, the resulting density matrix is a bipartite one. By again tracing out another subsystem we obtain a one particle density matrix. The notation of the three reduced (one-particle) density matrices is

ρA = TrB,C ρ ,

ρB = TrA,C ρ ,

ρC = TrA,B ρ , (7.2) and the matrix elements of the reduced density matrices are:

Xd3 Xd2

(ρA)mA nA = ρmA b c , nA b c , c=1 b=1 Xd3 Xd1

(ρB)mB nB = ρa mB c , a nB c , c=1 a=1 Xd2 Xd1

(ρC )mC nC = ρa b mC , a b nC . (7.3) b=1 a=1

Deﬁnition of Entangled Pure States. The deﬁnition of entanglement for tripartite systems is a logical extension of the bipartite case in Sec. 3.2. A tripartite pure state is called ‘entangled’ if it cannot be written as a single product of vectors which describe states of the subsystems, i.e.

|ψprodi = |ψAi ⊗ |ψBi ⊗ |ψC i . (7.4)

Such a state that is not entangled is called ‘product’ state. 7. Tripartite Systems 81

General Definition of Entanglement. In a quite similar way as we defined the separability of general (pure or mixed) bipartite states we can define the separability (and hence entanglement) of tripartite states: A state ρ is called ‘separable’ if it can be written as a convex combination of product states, i.e. X X i i i ρ = pi ρA ⊗ ρB ⊗ ρC , 0 ≤ pi ≤ 1, pi = 1 . (7.5) i i All separable states are the elements of the set of separable states S. If a state is not separable in the sense of Eq. (7.5), then it is called ‘entangled’.

3 Qubits. For 3 qubits, where d1 = d2 = d3 = 2, the matrix with elements (7.1) is a 8 × 8 matrix of the form   ρ111,111 ρ111,112 ρ111,121 ρ111,122 ρ111,211 ρ111,212 ρ111,221 ρ111,222    ρ112,111 ρ112,112 ρ112,121 ρ112,122 ρ112,211 ρ112,212 ρ112,221 ρ112,222     ρ121,111 ρ121,112 ρ121,121 ρ121,122 ρ121,211 ρ121,212 ρ121,221 ρ121,222     ρ122,111 ρ122,112 ρ122,121 ρ122,122 ρ122,211 ρ122,212 ρ122,221 ρ122,222  ρ =    ρ211,111 ρ211,112 ρ211,121 ρ211,122 ρ211,211 ρ211,212 ρ211,221 ρ211,222     ρ212,111 ρ212,112 ρ212,121 ρ212,122 ρ212,211 ρ212,212 ρ212,221 ρ212,222   ρ221,111 ρ221,112 ρ221,121 ρ221,122 ρ221,211 ρ221,212 ρ221,221 ρ221,222  ρ222,111 ρ222,112 ρ222,121 ρ222,122 ρ222,211 ρ222,212 ρ222,221 ρ222,222 (7.6) where we usually use the standard product basis e1 = f1 = g1 = |0i, e2 = f2 = g2 = |1i that has the known properties (2.26).

7.3 Pure States

7.3.1 Detection of Entangled Pure States To check if a pure state is entangled we have to check the reduced density matrices (7.2), (7.3): The state is a product state if all reduced density matrices are pure states.

7.3.2 Equivalence Classes of Pure Tripartite States The following classiﬁcation of pure tripartite states can be found in Ref. [30]. We say that a state |ψi can be converted into the state |φi with some non vanishing probability if a local operator A ⊗ B ⊗ C exists such that |ψi → |φi = A ⊗ B ⊗ C |ψi . (7.7) According to the following theorem all pure tripartite states can be divided into diﬀerent equivalence classes: 7. Tripartite Systems 82

Theorem 7.1. Two pure tripartite states |ψi, |φi belong to the same equivalence class if there exists a local operator A ⊗ B ⊗ C that relates the states in the following way:

|ψi → |φi = A ⊗ B ⊗ C |ψi and |φi → |ψi = A−1 ⊗ B−1 ⊗ C−1 |ψi . (7.8)

An operator that admits the above relations is called invertible local operator (ILO).

Remark. It is shown in Ref. [30] that for pure states any LOCC that transforms one state into another with some nonzero probability can be expressed with an operator A ⊗ B ⊗ C. Furthermore, if and only if there exists an ILO relating two pure states, then the states can be obtained from each other with some nonzero probability via LOCC. So in general we can say that all states that can be obtained from each other via LOCC (with some nonzero probability) belong to the same equivalence class.

The diﬀerent equivalence classes for 3-qubits are, according to Theorem 7.8:

(i) Class A-B-C, Product States. We have already mentioned at the beginning of this section that a tripartite pure state is a product state if all of its reduced density matrices are pure states. By local unitary operations any product state |ψprodi = |ψAi⊗|ψBi⊗|ψC i can be obtained from the state |prodi = |0i ⊗ |0i ⊗ |0i (7.9) and the other way round, thus there always exists an ILO that relates two product states in the way of Theorem 7.8, so that all product states belong to the same equivalence class. We say that the state |prodi (7.9) is the representative of the Class A-B-C.

(ii) Classes A-BC, B-CA and C-AB, States with Bipartite Entanglement. If one reduced density matrix of a pure state is a pure state, and the other two are mixed states, then we speak of tripartite states with bipartite entanglement. This means that only two parts of the system are entangled with each other, and these form a product state with the remaining part. Any such pure state can be reversibly transformed by a suitable unitary transformation into the state

2 2 |ψbipi = |0i ⊗ (α |0i ⊗ |0i + β |1i ⊗ |1i) α, β ∈ R, α + β = 1 . (7.10) 7. Tripartite Systems 83

The above state can be obtained with a suitable ILO from the state (and the other way round) 1 |bipi = √ |0i ⊗ (|0i ⊗ |0i + |1i ⊗ |1i) , (7.11) 2 which is the representative of this class, so that according to Theo- rem 7.8 all pure states of the classes A-BC, B-CA and C-AB belong to the same equivalence class.

(iii) States with Tripartite Entanglement. Pure states whose reduced density matrices are all mixed states have ‘true’ tripartite entanglement, i.e. entanglement between all three parts of the state. However, not all such states belong to the same equivalence class - they split into the following two classes:

(a) GHZ Class. All states of this equivalence class can be written as a sum of only two product terms. Such states can be obtained from or transformed into the state (with a suitable local unitary transformation) ¡ ¢ iφ |ψGHZi = γ α |0i ⊗ |0i ⊗ |0i + βe |φAi ⊗ |φBi ⊗ |φC i α, β ∈ R, α2 + β2 = 1, 0 ≤ φ < 2π , (7.12)

where γ is a normalization factor and the states |φAi, |φBi, and |φC i are arbitrary superpositions of the states |0i and |1i. There exists an ILO [30] relating the state |ψGHZi with the representative state 1 |GHZi = √ (|0i ⊗ |0i ⊗ |0i + |1i ⊗ |1i ⊗ |1i) (7.13) 2

(b) W Class. Pure states of this class cannot be written with less then three product terms. All of this states are related via a local unitary transformation to the state

|ψWi = a |0i ⊗ |0i ⊗ |1i + b |0i ⊗ |1i ⊗ |0i + + c |1i ⊗ |0i ⊗ |0i + d |0i ⊗ |0i ⊗ |0i (7.14)

which is actually a state with only three product terms since we can write

|ψWi = a |0i ⊗ |0i ⊗ |1i + b |0i ⊗ |1i ⊗ |0i + + (c |1i + d |0i) ⊗ |0i ⊗ |0i (7.15) 7. Tripartite Systems 84

The representative of this class is the state 1 |Wi = √ (|0i ⊗ |0i ⊗ |1i + |0i ⊗ |1i ⊗ |0i + |1i ⊗ |0i ⊗ |0i) 3 (7.16) since it is related to the state |ψWi (7.14) with a suitable ILO [30]. How do we know that there do not exist any ILOs relating states of diﬀerent equivalent classes (i) - (iii)? First, it can be seen [30] that any possible ILO cannot change the rank of the reduced density matrices1, which at once implies that class (i) and (ii) are inequivalent. Additionally, any ILO conserves the number of product terms needed to express a state2. Therefore the GHZ and W class are inequivalent.

Possible State Transformations under General LOCC. The equivalence classes (i) - (iii) are invariant under reversible LOCC (i.e. under ILOs). But what about non invertible local operators? It is shown in Ref. [30] that if an operator A ⊗ B ⊗ C is invertible, then at least one of the operators A, B, and C must have rank 1. So for states belonging to the GHZ or W class, the rank of at least one reduced density matrix has to be diminished (and so corresponds to a pure state). That means that there cannot be any LOCC that transforms a GHZ class state into a W class state - but there are non invertible operations that transform a GHZ class or W class state into a A-BC, B-CA, C-AB or A-B-C class state. Furthermore, noninvertible operations can transform a A-BC, B-CA, or C-AB class state into a A-B-C class state (see Tab. 7.1).

Entanglement Measures for Pure Tripartite States. What entanglement measures can be used to quantify the entanglement of tripartite states? One measure is the entropy measure, introduced in Sec. 5.2. Since this measure is deﬁned for bipartite states, we can interpret a tripartite state as a bipartite one with, e.g., ‘A’ being one part of the system and ‘B,C’ being the other part. In this manner we can calculate the von Neumann entropy (3.22) of the reduced density matrix ρA and similarly for the other reduced density matrices. If, for example, we have bipartite entanglement between ‘B’ and ‘C’, then the von Neumann entropy of ρB and ρC will have a nonzero value

1 Remember that the rank of a matrix is equal to the number of its non vanishing eigenvalues. So the reduced density matrices can have rank 2, which corresponds to a mixed state, or rank 1, which corresponds to a pure state. 2 Appending an ILO A ⊗ B ⊗ C to a sum of product terms leaves them linearly independent since the operator has to be invertible (A−1 etc. exists). 7. Tripartite Systems 85

Class LOCC into other classes SA SB SC τ A-B-C - 0 0 0 0 A-BC A-B-C 0 > 0 > 0 0 C-AB A-B-C > 0 > 0 0 0 B-AC A-B-C > 0 0 > 0 0 GHZ A-B-C, A-BC, B-CA, C-AB > 0 > 0 > 0 > 0 W A-B-C, A-BC, B-CA, C-AB > 0 > 0 > 0 0

Tab. 7.1: Features of the diﬀerent equivalent classes: Possible transformations via LOCC into other classes, the von Neumann entropies of the reduced density matrices SA, SB, SC , and the 3-tangle τ.

between 0 and 1, but ρA = 0. So the entropy measure helps to distinguish between the classes A-B-C, A-BC, B-CA and C-AB. However, GHZ class and W class states cannot be distinguished via the entropy measure, which in those cases has a nonzero value for all three reduced density matrices. For this reason another entropy measure is introduced: The so-called 3-tangle [24]. This measure is especially useful for states with ‘true’ tripartite entanglement since it is 0 for all W class states but nonzero for all GHZ class states.

Operational Method to Determine the Class of a Pure Tripartite State. Of course we would like to know a recipe for deciding to which equivalence class a given pure tripartite state belongs. First, we have to calculate all three reduced density matrices (see Eqs. (7.2), (7.3)). By determining if the reduced density matrices correspond to pure or mixed states we can decide if the state belongs to the classes (i), (ii), or (iii)3. If we get the result that the state is truly tripartite entangled (all reduced density matrices are mixed states), we can calculate the 3-tangle τ [24] to decide whether the state belongs to the GHZ or W class.

The discussed features of the diﬀerent equivalence classes of tripartite pure states are put together in Table 7.1.

Example 1. The representative of the GHZ class is the state |GHZi (7.13). In matrix notation in the standard product basis we have, according to

3 Equivalently we can calculate the ranks of the reduced density matrices: If all ranks are 1 we have class (i), if two ranks are 2 and one is 1 we have class (ii), and if all ranks are 2 we have class (iii). 7. Tripartite Systems 86

Eqs. (7.1), (7.6)

 1 1  2 0 0 0 0 0 0 2    0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0    GHZ  0 0 0 0 0 0 0 0  ρ := |GHZi hGHZ| =   . (7.17)  0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0  1 1 2 0 0 0 0 0 0 2 If we trace out only one subsystem, the resulting 2-qubit reduced density matrices are  1  2 0 0 0  0 0 0 0  ρGHZ = ρGHZ = ρGHZ =   . (7.18) AB BC AC  0 0 0 0  1 0 0 0 2 These are separable mixed states, since the above matrix is invariant under partial transposition (see Theoreom 3.6). So we notice that if tracing out one subsystem of the state |GHZi we obtain a separable state. The reduced density matrices (7.2), (7.3) for each part of the system are µ ¶ 1 GHZ GHZ GHZ 2 0 ρA = ρB = ρC = 1 . (7.19) 0 2 The reduced density matrices (7.19) all correspond to mixed states, since GHZ 2 Tr(ρA ) = 1/2 < 1 (or, equivalently, since the matrix is of rank 2), which is in agreement with the state being truly tripartite entangled.

Example 2. The representative of the W class is the state |Wi (7.16). In matrix notation (7.1), (7.6) this state becomes   0 0 0 0 0 0 0 0    0 1/3 1/3 0 1/3 0 0 0     0 1/3 1/3 0 1/3 0 0 0    W  0 0 0 0 0 0 0 0  ρ := |Wi hWi =   . (7.20)  0 1/3 1/3 0 1/3 0 0 0     0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 7. Tripartite Systems 87

By tracing out one subsystem we obtain the 2-qubit reduced density matrices   1/3 0 0 0  0 1/3 1/3 0  ρW = ρW = ρW =   . (7.21) AB BC AC  0 1/3 1/3 0  0 0 0 0

The partial transposition of the matrices√ (7.21) is no longer positive because one eigenvalue is negative (1/6(1 − 5)), and so the 2-qubit reduced density matrices (7.21) describe entangled mixed states - contrary to the GHZ case in the last example. All one particle reduced density matrices are mixed states, since we have µ ¶ 2/3 0 ρW = ρW = ρW = , (7.22) A B C 0 1/3 which again is in no disagreement with the properties of the W class states.

7.4 General States

7.4.1 Equivalence Classes of General Tripartite States The classiﬁcation of general tripartite states is a generalization of the pure state case (Sec. 7.3.2) and is introduced in Ref. [1]. In this classiﬁcation, general states can belong to several classes at the same time. Each class is actually a set of states with special properties (for a graphical illustration see Fig. 7.1):

(i) Class S is the set of separable states S which consists of all states that can be expressed like in Eq. (7.5).

(ii) Class B is the set of all states that can be written as a convex combination of pure product states and/or pure states that contain only bipartite entanglement.

(iii) Class W equals the set of all states that can be expressed as a convex combination of pure product states and/or pure states with bipartite entanglement and/or pure W class states.

(iv) Class GHZ is the set of all states. 7. Tripartite Systems 88

GHZ

W B S

Fig. 7.1: Illustration of the diﬀerent classes of general tripartite states

Remark. The classiﬁcation is realized under the viewpoint that all classes are convex and compact sets [1]. Any invertible local operation on a state ρ of class X \ Y (where X can be B, W or GHZ and Y is the ‘lower’ class of states, e.g., GHZ \ W or W \ B), that is (A ⊗ B ⊗ C)ρ(A ⊗ B ⊗ C)† (A ⊗ B ⊗ C is an ILO, see Theorem 7.8), does not map the state outside the set X \ Y 4. The class of a general state is invariant under LOCC [1] and can additionally ‘obtain’ only lower classes (e.g. for a state ρ ∈ GHZ and ρ ∈ GHZ \ W we have ρ∈ / B, but after a suitable LOCC the state can be transformed into the state σ for which σ ∈ GHZ and σ ∈ B).

7.4.2 Tripartite Witnesses Since all classes (i) - (iv) are convex and compact sets, as a consequence of the Hahn-Bahnach Theorem we can state the following theorem, in the same way as we did in connection with the entanglement witness Theorem 3.1 in Sec. 3.3.1: Theorem 7.2. A tripartite state σ is an element of GHZ \ X (where X can be S, B, or W) if and only if there exists a Hermitian operator Atri, such that ® σ, Atri = TrσAtri < 0 , ® ρ, Atri = TrρAtri ≥ 0 ∀ρ ∈ X. (7.23)

Example 1. The tripartite witness 2 Atri = 1 − |Wi hW| (7.24) W 3 4 This is because we express any state as a convex combination of pure states, and the class of each pure state is invariant under invertible local operations. 7. Tripartite Systems 89 determines that |Wi hW| ∈ GHZ \ B (where |Wi is deﬁned in Eq. (7.16)), since 1 TrAtri |Wi hW| = − < 0 (7.25) W 3 and tri TrAW ρB ≥ 0 ∀ρB ∈ B, (7.26) because Tr |Wi hW| ρB ≤ 2/3 (see Ref. [1]).

This work presents an overview of results achieved on the subject of detecting, classifying, and quantifying entanglement. Methods of detecting entanglement are divided into nonoperational and operational separability criteria, where the later provide a ‘recipe’ for the detection of entanglement but the former do not. These criteria give fundamental insight into the theory of entanglement; in particular a connection with a way of quantifying entanglement is discussed (between the concept of entanglement witnesses and the Hilbert-Schmidt measure). Furthermore two possibilities to classify entanglement and several methods to quantify entanglement are treated. For lower dimensional bipartite systems the characterization of entanglement is more or less complete, since there exist very useful operational methods to detect and quantify entanglement (i.e. the PPT and reduction criterion and the concurrence). For higher dimensional systems, however, there are no such simple methods, although for a lot of states entanglement is successfully detected, classified, and quantified. The characterization of entanglement for systems of more than two particles is in general difficult to perform. The situation is different to the two particle case because there exist a priori different ‘kinds’ of entanglement. For lower dimensions and few particles some operational methods to classify and quantify entanglement have been found, but not for the detection of multipartite entanglement. It is therefore of great interest to extend and deepen the understanding of entanglement beyond 2 particles and/or low dimensions, not only due to a sheer mathematical interest, but also for a successful future development of quantum information theory and applications. LIST OF FIGURES

3.1 Geometric illustration of a plane in Euclidean space and the ~ diﬀerent values of the scalar product for states above (bu), ~ ~ within (bp) and under (bd) the plane...... 18 3.2 Illustration of an optimal entanglement witness ...... 19 3.3 Plot of S1, S2 as functions of the parameter p and intersections with the entropies of the reduced density matrices Sred = 1 . . 24 3.4 Comparison of the information gained about the Werner state ρα with 3 diﬀerent separability criteria: 2 entropy inequalities and the CHSH inequality ...... 25 3.5 Comparison of the PPT criterion with other separability criteria for the 2-qubit Werner state ρα: The PPT criterion clearly distinguishes between separable and entangled states and gives a wider range of entanglement that the other criteria...... 28

4.1 Plot of the ﬁdelity g(F ) of the distilled state ρ0 ...... 35 4.2 Illustration of entanglement and distillability. Since all entangled 2-qubit states are distillable and NPT, we have a clear distinction in this case. For general states, however, there are entangled PPT states (bound entangled) and maybe bound entangled NPT states, which are those outside the “box” of the free entangled states. Note that this is not a geometric representation of sets of states...... 38 4.3 Plot of the function f(x1, x2 = 0, x3 = 0, y1, y2 = 0, y3 = 0). We can see that the global minimum f = 0 is not taken at a single point but for many diﬀerent values of x1 and y1. . . . . 41 4.4 Illustration of the various properties of the state ρβ (4.14). The question mark says that in this area we do not have enough information, we only know that for 0 ≤ β < 1 the state is NPT and therefore entangled...... 42

5.1 Plot of entanglement of formation EF as a function of concurrence C ...... 57 List of Figures 92

5.2 Plot of concurrence C(ρα) and entanglement of formation EF (ρα) of the Werner state ρα for values of α where ρα is entangled . 58 5.3 Plot of the relative entropy of entanglement DRE(ρα) of the Werner state ρα for values of α where ρα is entangled . . . . . 62 5.4 Plot of the Hilbert-Schmidt measure DHS(ρα) of the Werner state ρα for values of α where ρα is entangled ...... 63 5.5 Comparison of the concurrence C(ρα), the entanglement of formation EF(ρα), the relative entropy of entanglement DRE(ρα) and the ‘normalized’ Hilbert-Schmidt measure DHSN(ρα) of the Werner state ρα for values of α where ρα is entangled . . . 65

6.1 Illustration of Eqs. (6.7) and (6.8): The scalar product hρl −

ρ1, ρ1 − ρ2i is negative because the projection (ρl − ρ1)k onto ρ1 − ρ2 points in the opposite direction to ρ1 − ρ2. On the other side, hρr − ρ1, ρ1 − ρ2i is positive for states ρr, because

then the projection (ρr − ρ1)k points in the same direction as ρ1 − ρ2...... 67 6.2 Illustration of the Bertlmann-Narnhofer-Thirring Theorem . . 70 6.3 Illustration why C˜ cannot be an entanglement witness ifρ ˜ is not the nearest separable state. The hatched area is the one were the condition hρ, C˜i ≥ 0 ∀ρ ∈ S is violated...... 71

7.1 Illustration of the diﬀerent classes of general tripartite states . 88 LIST OF TABLES

7.1 Features of the diﬀerent equivalent classes: Possible transformations via LOCC into other classes, the von Neumann entropies of the reduced density matrices SA, SB, SC , and the 3-tangle τ...... 85 BIBLIOGRAPHY

[1] A. Ac´ın, D. Bruß, M. Lewenstein, and A. Sanpera, Classification of mixed three-qubit states, Phys. Rev. Lett. 87 (2001), 040401. [2] Arvind, K. S. Mallesh, and N. Mukunda, A generalized Pancharatnam geometric phase formula for three-level quantum systems, Math. Gen. 30 (1997), 2417. [3] J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1 (1964), 195. [4] , Speakable and unspeakable in quantum mechanics, Cambridge University Press, 1987. [5] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70 (1993), 1895. [6] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Concentrating partial entanglement by local operations, Phys. Rev. A 53 (1996), 2046. [7] , Purification of noisy entanglement and faithful teleportation via noisy channels, Phys. Rev. A 76 (1996), 722. [8] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A 54 (1996), 3824. [9] C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, and A. V. Thap- liyal, Exact and asymptotic measures of multipartite pure state entanglement, quant-ph/9908073. [10] R. A. Bertlmann and B. C. Hiesmayr, Bell inequalities for entangled kaons and their unitary time evolution, Phys. Rev. A 63 (2001), 062112. [11] R. A. Bertlmann, B. C. Hiesmayr, K. Durstberger, and P. Krammer, Entanglement witnesses for qubits and qutrits, quant-ph/0508043. Bibliography 95

[12] R. A. Bertlmann, H. Narnhofer, and W. Thirring, Geometric picture of entanglement and Bell inequalities, Phys. Rev. A 66 (2002), 032319.

[13] R. A. Bertlmann and A. Zeilinger (eds.), Quantum [un]speakables, from Bell to quantum information, Springer, Berlin Heidelberg New York, 2002.

[14] D. Bohm, Quantum theory, Prentice-Hall, Englewood Cliﬀs, NJ, 1951.

[15] D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, Ex- perimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 80 (1998), 1121.

[16] D. Bouwmeester, A. Ekert, and A. Zeilinger (eds.), The physics of quantum information: quantum cryptography, quantum teleportation, quantum computation, Springer, Berlin, Heidelberg, New York, 2000.

[17] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Experimental quantum teleportation, Nature 390 (1997), 575.

[18] G. Brassard, R. Cleve, and A. Tapp, Cost of exactly simulating quantum entanglement with classical communication, Phys. Rev. Lett. 83 (1999), 1874.

[19] D. Bruß, Characterizing entanglement, J. Math. Phys. 43 (2002), 4237.

[20] C. M. Caves and G. J. Milburn, Qutrit entanglement, Optics Commu- nications 179 (2000), 439.

[21] M. D. Choi, Positive semideﬁnite biquadratic forms, Lin. Algebra Appl. 12 (1975), 95.

[22] J. F. Clauser and M. A. Horne, Experimental consequences of objective local theories, Phys. Rev. D 10 (1974), 526.

[23] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23 (1969), 880.

[24] V. Coﬀman, J. Kundu, and W. K. Wootters, Distributed entanglement, Phys. Rev. A 61 (2000), 052306. Bibliography 96

[25] O. Cohen and T. A. Brun, Distillation of Greenberger-Horne-Zeilinger states by selective information manipulation, Phys. Rev. Lett 84 (2000), 5908.

[26] J. Dehaene, F. Verstraete, and B. De Moor, On the geometry of entangled states, J. Mod. Opt. 49 (2002), 1277.

[27] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. San- pera, Quantum privacy ampliﬁcation and the security of quantum cryptography over noisy channels, Phys. Rev. Lett. 77 (1996), 2818.

[28] D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and A. V. Thapliyal, Evidence for bound entangled states with negative partial transpose, Phys. Rev. A 61 (2000), 062312.

[29] W. D¨ur,J. I. Cirac, M. Lewenstein, and D. Bruß, Distillability and partial transposition in bipartite systems, Phys. Rev. A 61 (2000), 062313.

[30] W. D¨ur,G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62 (2000), 062314.

[31] A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 47 (1935), 777.

[32] A. K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett 67 (1991), 661.

[33] T. Gao, F. L. Yan, and Z. X. Wang, Deterministic secure direct communication using GHZ states and swapping quantum entanglement, J. of Phys. A: Mathematical and General 38 (2005), 5761.

[34] A. Garg and N. D. Mermin, Farkas’s lemma and the nature of reality: statistical implications of quantum correlations, Foundations of Physics 14 (1984), 1.

[35] N. Gisin, Bell’s inequality holds for all non-product states, Phys. Lett. A 154 (1991), 201.

[36] D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Bell’s Theorem without inequalities, Am. J. of Phys. 58 (1990), 1131.

[37] D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going beyond Bell’s Theorem, Bell’s Theorem, quantum theory and conceptions of the uni- verse (M. Kafatos, ed.), Kluwer Academic Publishers, 1989, p. 73. Bibliography 97

[38] O. Guehne, P. Hyllus, D. Bruß, M. Lewenstein A. Ekert, C. Macchi- avello, and A. Sanpera, Experimental detection of entanglement via witness operators and local measurements, J. Mod. Opt. 50 (2003), 1079.

[39] M. Horodecki and P. Horodecki, Reduction criterion of separability and limits for a class of distillation protocols, Phys. Rev. A 59 (1999), 4206.

[40] M. Horodecki, P. Horodecki, and R. Horodecki, Violating Bell inequality by mixed spin-1/2 states: necessary and suﬃcient condition, Phys. Lett. A 200 (1995), 340.

[41] , Quantum α-entropy inequalities: independent condition for local realism?, Phys. Lett. A 210 (1996), 377.

[42] , Separability of mixed states: necessary and suﬃcient conditions, Phys. Lett. A 223 (1996), 1.

[43] , Inseparable two spin-1/2 density matrices can be distilled to a singlet form, Phys. Rev. Lett. 78 (1997), 574.

[44] , Mixed-state entanglement and distillation: Is there a “bound” entanglement in nature?, Phys. Rev. Lett. 80 (1998), 5239.

[45] , Mixed-state entanglement and quantum communication, Quan- tum Information (G. Alber et al., ed.), Springer Tracts in Modern Physics, vol. 173, Springer Verlag Berlin, 2001, p. 151.

[46] P. Horodecki, J. A. Smolin, B. M. Terhal, and A. V. Thapliyal, Rank two bipartite bound entangled states do not exist, quant-ph/9910122.

[47] P. Hyllus, O. Guehne, D. Bruss, and M. Lewenstein, Relations between entanglement witnesses and Bell inequalities, quant-ph/0504079.

[48] L. M. Ioannou, B. C. Travaglione, D. Cheung, and A. Ekert, Improved algorithm for quantum separability and entanglement detection, Phys. Rev. A 70 (2004), 060303.

[49] M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, Optimization of entanglement witnesses, Phys. Rev. A 62 (2000), 052310.

[50] G. Lindblad, Expectations and entropy inequalities for ﬁnite quantum systems, Comm. Math. Phys. 39 (1974), 111.

[51] , Completely positive maps and entropy inequalities, Comm. Math. Phys. 40 (1975), 147. Bibliography 98

[52] N. D. Mermin, Quantum mysteries revisited, Am. J. of Phys. 58 (1990), 731.

[53] H. Narnhofer, private communication.

[54] T. Ogawa, A. Sasaki, M. Iwamota, and H. Yamamoto, Quantum se- cret sharing schemes and reversibility of quantum operations, quant- ph/0505001.

[55] M. Ozawa, Entanglement measures and the Hilbert-Schmidt distance, Phys. Lett. A 268 (2000), 158.

[56] A. Peres, Quantum theory: concepts and methods, Kluwer Academic Publishers, 1993.

[57] , Separability criterion for density matrices, Phys. Rev. Lett. 77 (1996), 1413.

[58] , All the Bell inequalities, Foundations of Physics 29 (1999), 589.

[59] I. Pitowski, Correlation polytopes: their geometry and complexity, Math- ematical Programming 50 (1991), 395.

[60] , George Boole’s ‘conditions of possible experience’ and the quantum puzzle, Brit. J. Phil. Sci. 45 (1994), 95.

[61] A. O. Pittenger and M. H. Rubin, Geometry of entanglement witnesses and local detection of entanglement, Phys. Rev. A 67 (2003), 012327.

[62] S. Popescu, Bell’s inequalities versus teleportation: what is nonlocality?, Phys. Rev. Lett. 72 (1994), 797.

[63] , Bell’s inequalities and density matrices: revealing “hidden” non- locality, Phys. Rev. Lett. 74 (1995), 2619.

[64] E. M. Rains, Bound on distillable entanglement, Phys. Rev. A 60 (1999), 179.

[65] M. Reed and B. Simon, Methods of modern mathematical physics I: functional analysis, Academic Press, New York and London, 1972.

[66] R. T. Rockafellar, Convex analysis, Princeton University Press, 1970.

[67] E. Schmidt, Zur Theorie der linearen und nichtlinearen Integralgle- ichungen. I. Teil: Entwicklung willk¨urlicherFunktionen nach Systemen vorgeschriebener, Math. Ann. 63 (1907), 433. Bibliography 99

[68] E. Schr¨odinger, Die gegenw¨artigeSituation in der Quantenmechanik, Naturwissenschaften 23 (1935), 807, 823, 844.

[69] E. Størmer, Positive linear maps of operator algebras, Acta Math. 110 (1963), 233.

[70] B. M. Terhal, Bell inequalities and the separability criterion, Phys. Lett. A 271 (2000), 319.

[71] , Detecting quantum entanglement, Theoretical Computer Sci- ence 287 (2002), 313.

[72] V. Vedral and M. B. Plenio, Entanglement measures and puriﬁcation procedures, Phys. Rev. A 57 (1998), 1619.

[73] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Quantifying entanglement, Phys. Rev. Lett. 78 (1997), 2275.

[74] G. Vidal and J. I. Cirac, Irreversibility in asymptotic manipulations of entanglement, Phys. Rev. Lett. 86 (2001), 5803.

[75] G. Vidal, W. D¨ur,and J. I. Cirac, Reversible combination of inequivalent kinds of multipartite entanglement, Phys. Rev. Lett. 85 (2000), 658.

[76] R. F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, Phys. Rev. A 40 (1989), 4277.

[77] R. F. Werner and M. M. Wolf, Bell’s inequalities for states with positive partial transpose, Phys. Rev. A 61 (2000), 062102.

[78] C. Witte and M. Trucks, A new entanglement measure induced by the Hilbert-Schmidt norm, Phys. Lett. A 257 (1999), 14.

[79] W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80 (1998), 2245.

[80] S. L. Woronowicz, Positive maps of low dimensional matrix algebras, Rep. Math. Phys. 10 (1976), 165.

[81] K. Zyczkowski,˙ Volume of the set of separable states II, Phys. Rev. A 60 (1998), 3496.

[82] K. Zyczkowski,˙ P. Horodecki, A. Sanpera, and M. Lewenstein, Volume of the set of separable states, Phys. Rev. A 58 (1998), 883.