Quantum Entanglement: Detection, Classification, and ......ested in pure states only, either the vector representation jˆi or the operator representation ‰pure = jˆihˆj can be

Quantum Entanglement: Detection,Classification, and Quantification

Diplomarbeitzur Erlangung des akademischen Grades

,,Magister der Naturwissenschaften’’an der

Universität Wien

eingereicht vonPhilipp Krammer

betreut vonAo. Univ. Prof. Dr. Reinhold A. Bertlmann

Wien, Oktober 2005

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2. Basic Mathematical Description . . . . . . . . . . . . . . . . 62.1 Spaces, Operators and States in a Finite Dimensional Hilbert

Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Bipartite Systems . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Qutrits . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Positive and Completely Positive Maps . . . . . . . . . . . . . 14

3. Detection of Entanglement . . . . . . . . . . . . . . . . . . . 163.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Pure States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 General States . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.1 Nonoperational Separability Criteria . . . . . . . . . . 173.3.2 Operational Separability Criteria . . . . . . . . . . . . 20

4. Classification of Entanglement . . . . . . . . . . . . . . . . 324.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Free and Bound Entanglement . . . . . . . . . . . . . . . . . . 32

4.2.1 Distillation of Entangled States . . . . . . . . . . . . . 324.2.2 Bound Entanglement . . . . . . . . . . . . . . . . . . . 36

4.3 Locality vs. Non-locality . . . . . . . . . . . . . . . . . . . . . 424.3.1 EPR and Bell Inequalities . . . . . . . . . . . . . . . . 424.3.2 General Bell Inequality . . . . . . . . . . . . . . . . . . 444.3.3 Bell Inequalities and the Entanglement Witness Theorem 49

5. Quantification of Entanglement . . . . . . . . . . . . . . . . 525.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2 Pure States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.3 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 . . . . . . . . . . . . . . . 585.3.4 Distance Measures . . . . . . . . . . . . . . . . . . . . 595.3.5 Comparison of Different Entanglement Measures for

the 2-Qubit Werner State . . . . . . . . . . . . . . . . 64

6. Hilbert-Schmidt Measure and Entanglement Witness . . 666.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.2 Geometrical Considerations about the Hilbert-Schmidt Distance 666.3 The Bertlmann-Narnhofer-Thirring Theorem . . . . . . . . . . 686.4 How to Check a Guess of the Nearest Separable State . . . . . 706.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.5.1 Isotropic Qubit States . . . . . . . . . . . . . . . . . . 726.5.2 Isotropic Qutrit States . . . . . . . . . . . . . . . . . . 746.5.3 Isotropic States in Higher Dimensions . . . . . . . . . . 76

7. Tripartite Systems . . . . . . . . . . . . . . . . . . . . . . . . . 797.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.3 Pure States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.3.1 Detection of Entangled Pure States . . . . . . . . . . . 817.3.2 Equivalence Classes of Pure Tripartite States . . . . . . 81

7.4 General States . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.4.1 Equivalence Classes of General Tripartite States . . . . 877.4.2 Tripartite Witnesses . . . . . . . . . . . . . . . . . . . 88

8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

1. INTRODUCTION

What is Quantum Entanglement? If we look up the word ‘entanglement’ ina dictionary, we find something like ‘state of being involved in complicatedcircumstances’, the term also denotes an affair between two people. Thus inquantum mechanics we could describe quantum entanglement literally as a‘complicated affair’ between two or more particles.

The first one to introduce the term was Erwin Schrödinger in Ref. [68].Since this article was published in German, ‘entanglement’ is a later transla-tion of the word ‘Verschränkung’. Schrödinger does not refer to a mathemat-ical definition of entanglement. He introduces entanglement as a correlationof possible measurement outcomes and states the following:

Maximal knowledge of a whole system does not necessarily includeknowledge of all of its parts, even if these are totally divided fromeach other and do not influence each other at the present time.

Note that ‘system’ is always a generalized expression for some physical real-ization; in this context a system of two or more particles is meant.

Nowadays the definition of entanglement is a mathematical one and rathersimple (see Chapter 2) – however, the phenomenological description of en-tanglement is still difficult. Since J.S. Bell introduced his ‘Bell Inequality’ [3]it has become clear that the correlations related to quantum entanglementcan be stronger than merely classical correlations. Classical correlations arethose that are explainable by a local realistic theory, and it was propagatedby Einstein, Podolsky, and Rosen [31] that quantum mechanics should alsobe a local realistic theory (see Sec. 4.3). The mysteriousness inherent toquantum entanglement mainly comes from the fact that often in cannot beexplained with a classical deterministic model [3, 4] and so underlines the‘new physics’ that comes with quantum mechanics and distinguishes it fromclassical physics.

Why do we need quantum entanglement? What at first seemed to be a morephilosophical investigation became of practical use in recent years. With thedevelopment of quantum information theory a new ‘quantum’ way of informa-tion 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 andcryptography (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 hasto be encoded into less physical material. To be able to keep pace with thetechnological demand, quantum information theory could serve as the futureconcept of information processing and communication devices. It is thereforenot only of philosophical but also of practical use to deepen and extend thedescription of quantum entanglement.

Aim and Structure of this Work. The aim of this work is to provide abasic mathematical overview of quantum entanglement which includes thefundamental aspects of detecting, classifying, and quantifying entanglement.Several examples should give insight of the explicit application of the giventheory. There is no emphasis on detailed proofs. Nevertheless some proofsthat 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 workis concerned with bipartite systems, these are systems that consist of twoparts (i.e. particles in experimental application).

The work is organized as follows: In Chapter 2 we start with basic math-ematical concepts. Next, in Chapter 3, we address the problem of detectingentanglement; in Chapter 4 entanglement is classified according to certainproperties, and in Chapter 5 we discuss several methods to quantify entan-glement. In Chapter 6 we combine the concept of detecting and quantifyingentanglement. Finally, in Chapter 7, we briefly take a look on tripartitesystems.

2. BASIC MATHEMATICAL DESCRIPTION

2.1 Spaces, Operators and States in a Finite DimensionalHilbert Space

Operators act on the Hilbert space H of a quantum mechanical system, theymake up a Hilbert Space themselves, called Hilbert-Schmidt space A. We areonly interested in finite dimensional Hilbert spaces, so that in fact A can beregarded as a space of matrices, taking into account that in finite dimensionsoperators can be written in matrix form. A scalar product defined on A is(A,B ∈ A)

〈A,B〉 = TrA†B , (2.1)with the corresponding Hilbert-Schmidt norm

‖A‖ =√〈A,A〉 A ∈ A . (2.2)

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

Aij = 〈ei|A |ej〉 , (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 ordensity matrix) if 1

Trρ = 1, ρ ≥ 0 , (2.4)where ρ ≥ 0 means that ρ is a positive operator (more precise: positivesemidefinite), 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 vec-tors |ψ〉 in Hilbert Space. This concept is widened with the introduction ofmixed states, so that in general states are viewed as operators. If one is inter-ested in pure states only, either the vector representation |ψ〉 or the operatorrepresentation ρpure = |ψ〉〈ψ| 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 theconvention of quantum communication, the two parties are usually referredto as ‘Alice’ and ‘Bob’. For bipartite systems the Hilbert space is denoted asHd1A ⊗Hd2B , where d1 is the dimension of Alice’s subspace and d2 is Bob’s, orjust 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 oftenconsider the Hilbert Space H2 ⊗H2, states on this space are called 2-qubitstates.

Matrix Notation. In general, we can write a state ρ as a matrix accordingto Eq. (2.3). However, often we have to use a product basis, to guarantee thatcertain calculations etc. make sense. In this case for the matrix notation ofa state ρ on Hd1A ⊗Hd2B we have

ρmµ,nν = 〈em ⊗ fµ| ρ |en ⊗ fν〉 . (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 exampleneeded to calculate the reduced density matrices of a state ρ. These areobtained if Alice neglects Bob’s system, or vice versa, which mathematicallymeans 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 cannotgive a matrix of negative trace, since projectors in matrix notation need to have positivediagonal entries.

3 This is because all eigenvalues have to be smaller than 1.


system. The notation is

ρA = TrBρ ,

ρB = TrAρ , (2.8)

where ρA denotes Alices reduced density matrix and ρB Bob’s. The matrixelements of the reduced density matrices are

(ρA)mn =

d2∑

β=1

ρmβ,nβ ,

(ρB)µν =

d1∑a=1

ρaµ,aν . (2.9)

Definition of Entangled Pure States. A bipartite pure state is called ‘en-tangled’ if it cannot be written as a single product of vectors which describestates of the subsystems, i.e.

|ψprod〉 = |ψA〉 ⊗ |ψB〉 . (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 canbe written as a convex combination of product states, i.e. [76]

ρ =∑

i

pi ρiA ⊗ ρiB, 0 ≤ pi ≤ 1,

∑i

pi = 1 . (2.11)

All separable states are the elements of the set of separable states S. If astate 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 ex-actly (2.11) is the definition of separability (as being the counterpart of entan-glement). When it was introduced by Werner in Ref. [76] he gave a plausiblephysical reasoning: Werner differentiated between ‘uncorrelated’ states and‘classically correlated’ states (which both were denoted later as separablestates).

An uncorrelated state is a product state that can be written as

ρ = ρA ⊗ ρB , (2.12)


because then expectation values of joint measurements (denoted by operatorsA for Alice and B for Bob) on such a state factorize:

〈A⊗B〉 = TrρA⊗B = Tr (ρA ⊗ ρB) (A⊗B) = TrρAA TrρBB . (2.13)

Here the classical rule of multiplying probabilities occurs, and this corre-sponds to the fact that the measurements by Alice and Bob are independentof each other.

For the classically correlated states one can think of the following physicalpreparation devices: Alice and Bob each have a device with a switch thatcan be set in different positions i = 1, ..., n, n > 1. For each setting of theswitch the devices prepare states ρAi and ρ

Bi . Before the measurement, a

random number between 1 and n is drawn, and the switches of the devicesare set according to this number. Furthermore, each number i occurs withprobability pi. Now the expectation value of a measurement A⊗B will be aweighed sum of factorized expectation values:

〈A⊗B〉 =n∑

i=1

piTrρAi ATrρ

Bi B

=n∑

i=1

piTr(ρAi ⊗ ρBi

)(A⊗B)

=: TrρA⊗B . (2.14)

Here we defined ρ like in Eq. (2.11). With this definition of ρ we can writethe expectation value as one obtained from a single state, and this state iscalled classically correlated. We say ‘classically’ because the preparation ofthis state is done merely classical, and ‘correlated’ because the expectationvalue 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 corre-lated states, since here n ≥ 1, so the uncorrelated states are referred to aswell if n = 1.

Fraction. The fraction or fidelity of a state ρ with respect to a maximallyentangled pure state |ψmax〉 is given by

Fψmax := 〈ψmax| ρ |ψmax〉 (2.15)

Eq. (2.15) is nothing but the probability that the resulting state of a projec-tive measurement (in a basis where |ψmax〉 is one basis vector) is |ψmax〉. Sothe range of possible values of Fψmax(ρ) is 0 ≤ Fψmax(ρ) ≤ 1.


Isotropic States. We define an isotropic state ρ(d)α on a Hilbert SpaceHd⊗Hd

as (see Refs. [39, 45]):

ρ(d)α = α∣∣φd+

〉 〈φd+

∣∣ + 1− αd2

1⊗ 1, α ∈ R, − 1d2 − 1 ≤ α ≤ 1 . (2.16)

Here d is the dimension of the Hilbert space Hd⊗Hd, the range of α is deter-mined by the positivity of the state. The state

∣∣φd+〉

is maximally entangledand given by

∣∣φd+〉

=1√d

d−1∑i=0

|i〉 ⊗ |i〉 , (2.17)

where {|i〉} 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)

(U ⊗ U∗)ρ(d)α (U ⊗ U∗)† = ρα . (2.18)

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

− 1d2 − 1 ≤ α ≤

1d + 1

⇒ ρ(d)α separable ,1

d + 1< α ≤ 1 ⇒ ρ(d)α entangled .

(2.19)

Instead of the parameter α in Eq. (2.16) we can also define an equivalentisotropic state ρF with the fraction F (2.15) as the parameter. In case of|ψmax〉 =

∣∣φd+〉

(2.17) we write shortly Fφ+ := F . According to Eq. (2.15) weget

F =〈φd+

∣∣ ρ(d)α∣∣φd+

〉=

1 + α(d2 − 1)d2

, (2.20)

or

α =d2F − 1d2 − 1 . (2.21)

Inserting Eq. (2.21) into the definition (2.16) we get the equivalent form ofan isotropic state

ρ(d)F =

d2

d2 − 1((

F − 1d2

) ∣∣φd+〉 〈

φd+∣∣ + (1− F ) 1

d2

)(2.22)

4 The entangled property of the isotropic state is prooved by using the reduction crite-rion (see Theorem 3.8) in Sec. 3.3.2. It is shown in Ref. [39] that for the remaining valuesof the parameter α the state can be written as a mixture of product states and thus isseparable (see Eq. (2.11)).


2.2.1 Qubits

Single Qubits. A qubit state ω, acting on H2, can be decomposed in termsof Pauli matrices (we use the convention to sum over same indices):

ω =1

2

(1+ niσi

), ni ∈ R,

∑i

n2i = |~n| ≤ 1 . (2.23)

Note that for |~n|2 < 1 the state is mixed (corresponding to Trω2 ≤ 1) whereasfor |~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

ρ =

ρ11,11 ρ11,12 ρ11,21 ρ11,22ρ12,11 ρ12,12 ρ12,21 ρ12,22ρ21,11 ρ21,12 ρ21,21 ρ21,22ρ22,11 ρ22,12 ρ22,21 ρ22,22

. (2.24)

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

{|0〉 ⊗ |0〉 , |0〉 ⊗ |1〉 , |1〉 ⊗ |0〉 , |1〉 ⊗ |1〉} , (2.25)

which has the properties〈i|j〉 = δij . (2.26)

Alternatively, we can write any 2-qubit density matrix in a basis of the4× 4 matrices composed of the identity matrix and the Pauli matrices,

ρ =1

4

(1⊗ 1+ aiσi ⊗ 1+ bi1⊗ σi + cijσi ⊗ σj

), ai, bi, cij ∈ R . (2.27)

A product state ρA ⊗ ρB has the form

ρA ⊗ ρB = 14 (1⊗ 1+ niσi ⊗ 1+ mi1⊗ σi + nimjσi ⊗ σj) ,ni,mi ∈ R, |~n| ≤ 1, |~m| ≤ 1 . (2.28)

Any separable state (2.11) can be written as the convex combination of ex-pressions (2.28),

ρsep =∑

k pk14

(1⊗ 1+ nki σi ⊗ 1+ mki 1⊗ σi + nki mkj σi ⊗ σj

),

nki ,mki ∈ R,

∣∣∣ ~nk∣∣∣ ≤ 1,

∣∣∣ ~mk∣∣∣ ≤ 1 . (2.29)


Bell Basis. A basis in H2 ⊗H2 is the Bell basis, which consists of 4 ortho-normal maximally entangled pure states:

|ψ−〉 = 1√2

(|0〉 ⊗ |1〉 − |1〉 ⊗ |0〉) (2.30)

|ψ+〉 = 1√2

(|0〉 ⊗ |1〉+ |1〉 ⊗ |0〉) (2.31)

|φ−〉 = 1√2

(|0〉 ⊗ |0〉 − |1〉 ⊗ |1〉) (2.32)

|φ+〉 = 1√2

(|0〉 ⊗ |0〉+ |1〉 ⊗ |1〉) . (2.33)

Isotropic Qubit State. We can write a 2-qubit isotropic state ρ(2)F (2.22) as

a mixture of the Bell states (2.30) - (2.33):

ρ(2)F =: ρF = F |φ+〉〈φ+| +

1− F3

|ψ−〉〈ψ−|+ 1− F3

|ψ+〉〈ψ+|+

+1− F

3|φ−〉〈φ−| , 0 ≤ F ≤ 1 . (2.34)

Werner State. A state we will often use in examples is the 2-qubit Wernerstate (introduced for general dimensions in [76] and for 2-qubits in this formin [62])

ρα = α |ψ−〉〈ψ−|+ 1− α4

1⊗ 1, −13≤ α ≤ 1 . (2.35)

Note that the interval for α follows from the necessity that Trρ = 1. Thematrix notation of ρα in the standard basis (2.25) is, according to Eq. (2.24):

ρα =

1−α4

0 0 00 1+α

4−α2

00 −α

21+α

40

0 0 0 1−α4

. (2.36)

2.2.2 Qutrits

Single Qutrits. The description of qutrits is very similar to the one forqubits. A qutrit state ω on H3 can be expressed in the matrix basis {1, λ1,λ2, . . . , λ8} with an appropriate set of coefficients {ni}

ω =1

3

(1+

√3 ni λ

i)

, ni ∈ R ,∑

i

n2i = |~n|2 ≤ 1 . (2.37)


The factor√

3 is included for a proper normalization, i.e. Trω2 ≤ 1 (see alsoRefs. [2, 20]). The matrices λi (i = 1, ..., 8) are the eight Gell-Mann matrices

λ1 =

0 1 01 0 00 0 0

, λ2 =

0 −i 0i 0 00 0 0

, λ3 =

1 0 00 −1 00 0 0

,

λ4 =

0 0 10 0 01 0 0

, λ5 =

0 0 −i0 0 0i 0 0

, λ6 =

0 0 00 0 10 1 0

,

λ7 =

0 0 00 0 −i0 i 0

, λ8 = 1√

3

1 0 00 1 00 0 −2

, (2.38)

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 existsets {ni} for which the matrix is not a state, as can be seen in the followingexample [53]:

Example. Let us consider a set of coefficients {ni} where all coefficientsvanish except n8. According to Eq. (2.37) the only possible values for thiscoefficient are n8 = +1 or n8 = −1. If we have n8 = +1, then we get for amatrix A+1 formed like in Eq. (2.37)

A+1 =1

3

(1+

√3)

=

23

0 00 2

30

0 0 −13

. (2.39)

Although we have TrA+1=1, A+1 is not a state because one eigenvalue, i.e.−1/3, is negative.

On the other hand, if n8 = −1, we find

A−1 =1

3

(1−

√3)

=

0 0 00 0 00 0 1

, (2.40)

which clearly is a state since TrA+1=1 and A+1 ≥ 0, we can write A−1 = ωto maintain the notation of Eq. (2.37).


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

ρ =

ρ11,11 ρ11,12 ρ11,13 ρ11,21 ρ11,22 ρ11,23 ρ11,31 ρ11,32 ρ11,33ρ12,11 ρ12,12 ρ12,13 ρ12,21 ρ12,22 ρ12,23 ρ12,31 ρ12,32 ρ12,33ρ13,11 ρ13,12 ρ13,13 ρ13,21 ρ13,22 ρ13,23 ρ13,31 ρ13,32 ρ13,33ρ21,11 ρ21,12 ρ21,13 ρ21,21 ρ21,22 ρ21,23 ρ21,31 ρ21,32 ρ21,33ρ22,11 ρ22,12 ρ22,13 ρ22,21 ρ22,22 ρ22,23 ρ22,31 ρ22,32 ρ22,33ρ23,11 ρ23,12 ρ23,13 ρ23,21 ρ23,22 ρ23,23 ρ23,31 ρ23,32 ρ23,33ρ31,11 ρ31,12 ρ31,13 ρ31,21 ρ31,22 ρ31,23 ρ31,31 ρ31,32 ρ31,33ρ32,11 ρ32,12 ρ32,13 ρ32,21 ρ32,22 ρ32,23 ρ32,31 ρ32,32 ρ32,33ρ33,11 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 =|0〉, e2 = f2 = |1〉, e3 = f3 = |2〉)

{ |0〉 ⊗ |0〉 , |0〉 ⊗ |1〉 , |0〉 ⊗ |2〉 , |1〉 ⊗ |0〉 , |1〉 ⊗ |1〉 ,|1〉 ⊗ |2〉 , |2〉 ⊗ |0〉 , |2〉 ⊗ |1〉 , |2〉 ⊗ |2〉} . (2.42)

The basis (2.42) has the properties (2.26).A 2-qutrit state can also be represented in a basis of 9 × 9 matrices

consisting of the unit matrix 1 and the eight Gell-Mann matrices λi,

ρ =1

9

(1⊗ 1 + ai λi ⊗ 1 + bi 1⊗ λi + cij λi ⊗ λj

), ai, bi, cij ∈ R .

(2.43)By the same argumentation as for qubits any separable 2-qutrit state is aconvex combination of product states,

ρsep =∑

k

pk1

9

(1⊗ 1 +

√3 nki λ

i ⊗ 1 +√

3 mki 1⊗ λi + 3 nki mkj λi ⊗ λj)

.

(2.44)

2.3 Positive and Completely Positive Maps

A linear mapΛ : A1 → A2 (2.45)

maps operators from a space A1 into a space A2. Λ is called positive if itmaps positive operators into positive operators,

Λ(A) ≥ 0 ∀A ≥ 0 . (2.46)


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 thematrix 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 agiven quantum mechanical state is entangled or not. We will see that forpure 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 andnumber of particles) that always gives a result whether the state is entangledor not. Nevertheless there are some relatively simple methods for states onlower dimensional Hilbert spaces [57, 42, 39, 45].

We have to distinguish between two ‘classes’ of methods of detectingentanglement: Nonoperational and operational separability criteria. We calla criterion ‘nonoperational’ if there exists no ‘recipe’ to perform the criterionon a given state, and ‘operational’ if such a recipe indeed exists. Apartfrom that, separability criteria can be necessary or necessary and sufficientconditions for separability. A necessary condition for separability has to befulfilled by every separable state. So if a state does not fulfill the condition, ithas 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 byseparable 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 theresults for pure states, in Sec. 3.3 we consider general states (pure and mixedstates) - in particular we investigate nonoperational separability criteria inSec. 3.3.1, whereas in Sec. 3.3.2 operational criteria are discussed. We willsee that for the 2-qubit case H2⊗H2 (and for H2⊗H3 orH3⊗H2) there existoperational separability criteria that are necessary and sufficient conditionsfor separability.

3. Detection of Entanglement 17

3.2 Pure States

We can check easily if a pure state |ψ〉 is entangled by looking at the reduceddensity matrices of |ψ〉〈ψ|: According to Eq. (2.10) the state is a productstate if and only if the reduced density matrices are pure states. 1

Example. Let us consider the pure state |ψ−〉, where |ψ−〉 is the singletstate (2.30). When written as a density matrix in the standard productbasis (2.25) we get (see (2.24))

|ψ−〉〈ψ−| =

0 0 0 00 1/2 −1/2 00 −1/2 1/2 00 0 0 0

. (3.1)

Now we can calculate the reduced density matrices, according to Eqs. (2.8)and (2.9),

ρA = ρB =

(1/2 00 1/2

). (3.2)

We see that the above matrix is a mixed state, since (according to Eq. (2.6))Trρ2A = Trρ

2B < 1. So we conclude that |ψ−〉 is entangled.

3.3 General States

If a state ρ is a mixed state (2.6) then the results of Sec. 3.2 are not valid.The following considerations are valid for mixed and pure states.

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 ofthe 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 aHermitian operator A ∈ A, called entanglement witness, such that

〈ρent, A〉 = TrAρent < 0 ,〈ρ,A〉 = TrAρ ≥ 0 ∀ρ ∈ S . (3.3)

1 A similar method uses the Schmidt decomposition [67] of a pure state |ψ〉 (for detailssee, e.g., Ref. [45]).


Fig. 3.1: Geometric illustration of a plane in Euclidean space and the differentvalues 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-BanachTheorem of functional analysis; this is done in Ref. [42]. Here we wantto illustrate how the theorem can be derived with help of the geometricalrepresentation 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 thereexists a hyper-plane that separates b from the set A.

First, let us consider the following geometric consideration: In Euclideanspace a plane is defined by its orthogonal vector ~a. The plane separatesvectors for which their scalar product with ~a is negative from vectors withpositive scalar product, vectors in the plane have, of course, a vanishingscalar product with ~a (see Fig. 3.1).

This can be compared with our situation: A scalar function 〈ρ,A〉 = 0 de-fines a hyperplane in the set of all states, and this plane separates ‘up’ statesρu for which 〈ρu, A〉 < 0 from ‘down’ states ρd with 〈ρd, A〉 > 0. States ρpwith 〈ρp, A〉 = 0 are inside the hyperplane. According to the Hahn-BanachTheorem 3.2, we conclude that due to the convexity of the set of separablestates, there always exists a plane that separates an entangled state from theset of separable states.

An entanglement witness is ‘optimal’, i.e. Aopt, if apart from Eqs. (3.1)there exists a separable state ρ̃ ∈ S for which

〈ρ̃, Aopt〉 = 0 . (3.4)

It is optimal in the sense that it defines a tangent plane to the set of separablestates S and is called tangent functional for that reason [12]. It detects moreentangled states than non optimal entanglement witnesses, see Fig. 3.2.


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 theoremcan be derived:

Theorem 3.3 (PMT). A bipartite state ρ is separable if and only if

(1⊗ Λ)ρ ≥ 0 ∀ positive maps Λ . (3.5)The fact that we have (1 ⊗ Λ)ρ ≥ 0 for a separable state ρ can be seen

easily [57]: Applying (1⊗ Λ) to a separable state (2.11) gives

(1⊗ Λ)ρ =n∑

i=1

piρAi ⊗ Λ(ρBi ), (3.6)

and since Λ is positive, Λ(ρBi ) is as well, and so (I ⊗ Λ)ρ is positive. InRef. [42] the PMT is proved in the other direction (that a state ρ has to beseparable if (1⊗ Λ)ρ ≥ 0 ∀ positive maps Λ).

To put it another way, Theorem 3.3 says that a state ρent is entangled ifand 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 com-pletely positive maps help to detect entangled states.

Example. An example for a not completely positive map is the transpositionT . To see this, it is enough to show that

(1⊗ T ) |φ+〉〈φ+| < 0 , (3.8)


where |φ+〉 is defined in Eq. (2.33). Written in matrix notation (2.24) in thestandard product basis (2.25) we have:

|φ+〉〈φ+| =

1/2 0 0 1/20 0 0 00 0 0 0

1/2 0 0 1/2

. (3.9)

We can check the positivity of the state by calculating the eigenvalues: Theseare {1, 0, 0, 0}, all are positive, as expected.

Now what happens if we apply 1 ⊗ T? We know that the transpositionof a 2 × 2 matrix (Aij) is simply done by interchanging the indices of theelements: T ((Aij)) =: (A

Tij) = (Aj i). So 1 ⊗ T means that only Bob’s part

is subjected to transposition, we speak of partial transposition. Only theGreek indices of the matrix elements (2.7) are interchanged:

(1⊗ T )(ρmµ,nν) =: (ρTBmµ,nν) = (ρmν,nµ) . (3.10)

Applying (3.10) on Eq. (2.24) we obtain (1⊗ T ) |ψ+〉〈ψ+|:

(|ψ+〉〈ψ+|)TB =

1/2 0 0 00 0 1/2 00 1/2 0 00 0 0 1/2

. (3.11)

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 forinequalities that can be derived out of the assumption of a local realistictheory, and is violated by states that do not admit such a theory. SpecialBIs are often named differently, for example ‘CHSH inequality’. BIs arefamous for showing that for many entangled states it is not possible to applya local realistic description of measurement processes. For a more detaileddiscussion and references see Sec. 4.3.

Apart from that, BIs can serve as necessary - but not sufficient - separa-bility conditions: Every separable state has to satisfy a BI [76]. So if a stateviolates a BI, it must be entangled - but if it fulfills it, we cannot be sure.


The CHSH Inequality as a Seperability Criterion. The CHSH inequality wasintroduced 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 satisfythe inequality

〈ρ, 21−B〉 ≥ 0, B = ~a · ~σ ⊗ (~b +~b′) · ~σ + ~a′ · ~σ ⊗ (~b−~b′) · ~σ , (3.12)

where ~a,~a′,~b,~b′ are any unit vectors in R3; ~σ is the vector out of the threePauli matrices, ~σ = (σx, σy, σz).

If for a given state the inequality (3.12) is not fulfilled, then the state isentangled for sure. If it is fulfilled, then we cannot be sure. What at firstdoes not look ‘user friendly’ is the fact that in order to check if a given state ρviolates the inequality (3.12), we have to check many or even all measurement

directions ~a,~a′,~b,~b′. Of course we could also minimize over all directions, butin Ref. [40] a theorem is proved that allows to check a violation quite faster:

Theorem 3.5. A 2-qubit state violates the CHSH inequality (3.12) for some

operator B (some set of measurement directions ~a,~a′,~b,~b′) if and only if

M(ρ) > 1 . (3.13)

Here M(ρ) is the sum of the two greater eigenvalues of a matrix Uρ. Thematrix Uρ can be constructed in the following way: First we calculate thematrix elements of a matrix Tρ, (Tρ)

nm = Trρσn ⊗ σm (n,m = 1, 2, 3, σ1corresponds to σx, etc.). Then Uρ = T

Tρ Tρ.

Example. We want to examine if the Werner state (2.35) violates the CHSHinequality (3.12), and if yes, for what interval of the parameter α. The matrixnotation (2.36) can be expressed in a basis of Pauli matrices (see Eq. (2.27)),

ρα =1

4(1− α~σ ⊗ ~σ) , −1

3≤ p ≤ 1 , (3.14)

where we defined ~σ⊗ ~σ := σx⊗ σx + σy ⊗ σy + σz ⊗ σz. Written in this way,the matrix elements (Tρ)

nm can easily be calculated. When taking the trace,we remember that

TrA⊗B = TrATrB . (3.15)Since Trσn = 0 ∀n = x, y or z, only the diagonal terms (Tρ)nn do not vanish,since here Tr(σn ⊗ σn)(σn ⊗ σn) = 4. These are

(Tρ)nn =

−α4· 4 = −α . (3.16)


So we have

T =

−α 0 00 −α 00 0 −α

, U =

α2 0 00 α2 00 0 α2

. (3.17)

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

M(ρα) = 2α2 . (3.18)

According to Theorem 3.5, ρα violates the CHSH inequality (3.12) if

α >1√2

, (3.19)

so we conclude that all Werner states with α > 1√2

are entangled for sure.

Entropy Inequalities

Other necessary separability criteria are inequalities that compare certainquantum 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 (seeEqs. (2.8) and (2.9)). The inequalities originated from an observation bySchrödinger [68] that an entangled state provides more information aboutthe whole system than about the subsystems. If we associate entropy withthe absence of information, then the inequalities (3.20) state the opposite,which is assumed to be a property of separable states. Indeed, for certainquantum 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)S2(ρ) = − log Trρ2 , (3.23)

where R(ρ) is the rank of the matrix ρ, i.e. the number of nonvanishingeigenvalues. The logarithm can be taken to any base, since for differentbases, the logarithm functions differ only in some constant which cancels outin the inequality.


Example. As an example we want to check the inequalities for the Wernerstate ρα (2.35). To do this, we first consider the matrix notation (2.36) andcalculate the reduced density matrices. We get

(ρα)A = (ρα)B =

(12

00 1

2

). (3.24)

S0. First we calculate the S0 entropies (3.21). The rank of the reduceddensity matrix is 2, since it has two nonvanishing eigenvalues (can beseen directly from the matrix (3.24), since it is diagonal). In order todetermine the rank of ρα we need to calculate the eigenvalues of ρα.These are

λ1 = λ2 = λ3 =1− α

4, λ4 =

1 + 3α

4. (3.25)

If α 6= 1, all eigenvalues are greater than zero and therefore do notvanish. 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 cannotsay anything if or for what α the state is entangled.

If, however, α = 1, then only λ4 = 1, the other eigenvalues are 0. Inthis 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 isthe special case in which the Werner state equals |ψ−〉〈ψ−|, we can sayfor sure that the state is entangled.

S1. The ’von Neumann entropy’ S1 (3.22) is the most common quantumentropy used for many purposes. First we need to remember that func-tions acting on a matrix are defined by acting on the elements of thediagonalized matrix, that is, acting on the eigenvalues. When takingthe trace, we can always write a state in diagonal matrix form, sincethe trace operation is independent of the choice of basis. Therefore

−Trρ log ρ = −∑

i

λi log λi , (3.28)

where the λis are the eigenvalues of the state ρ. Using Eq. (3.28) weget for the reduced density matrices

S1(ρA) = S1(ρB) = −2 · 12

log1

2= − log 1

2= log 2 . (3.29)


S 1

S

a

0,7476

S 2

red

31 @ 0,5774

0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

Fig. 3.3: Plot of S1, S2 as functions of the parameter p and intersections with theentropies 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

S1(ρα) = −3(

1− α4

)log2

1− α4

− 1 + 3α4

log21 + 3α

4. (3.31)

The entropy inequalities (3.20) are satisfied if S1(ρα) ≥ 1. Since wecannot solve the equation S1(ρα) ≥ 1 analytically, we plot the functionS1(ρα) in dependence of α (see Fig. 3.3) and calculate the intersectionwith the entropy of both reduced density matrices numerically. Weobtain a violation of the inequalities (3.20) for α > 0, 7476, which is aweaker condition than the CHSH inequality, since that gave a violationfor α > 1√

2= 0, 7071. So the entropy inequalities with the S1 or von

Neumann entropy do not give a greater range of the parameter α wherewe can know for sure that the state is entangled.

S2. To calculate the S2 entropy (3.23) we use

S2(ρ) = − log (Trρ2) = − log∑

i

λ2i , (3.32)


?

entangled

0 0,5 1

S

S

CHSH

1

2

a

entangled

entangled

?

?

Fig. 3.4: Comparison of the information gained about the Werner state ρα with3 different separability criteria: 2 entropy inequalities and the CHSHinequality

and obtain for the reduced density matrices (where it is useful again touse log2)

S2(ρA) = S2(ρB) = − log2(

1

4+

1

4

)= − log2

1

2= log2 2 = 1 , (3.33)

and for the whole state we get

S2(ρα) = − log2(

3

(1− p

4

)2+

(1 + 3p

4

)2). (3.34)

Now we can analytically solve the inequality S2(ρp) < 1 and find thatfor α > 1√

3the entropy inequalities (3.20) are violated. Hence for this

value of α the state is entangled for sure (see Fig. 3.3). This is a strongercondition than the CHSH inequality, since 1√

3< 1√

2and so we got a

larger range of the parameter with certain entanglement. In Ref. [41]it is shown that for all 2-qubit states the S2 entropy inequalities arealways stronger than the CHSH inequality.

The gained information about the entanglement of the Werner state ρα isillustrated in Figure 3.4. (To be precise, in all the figures of course thepossible values of α could be extended to the value −1/3, for reasons ofsimplicity this is neglected there.)

The Positive Partial Transpose (PPT) Criterion

The PPT Criterion is very useful for 2-qubit systems, since it is an operationalcriterion and a necessary and sufficient condition for separability. It was


recognized as a necessary separability criterion in Ref. [57] and extended toa necessary and sufficient one for 2 qubits in Ref. [42].

Theorem 3.6 (PPT Criterion). A state ρ acting on H2 ⊗ H2, H3 ⊗ H2or H2 ⊗ H3 is separable if and only if its partial transposition is a positiveoperator,

ρTB = (1⊗ T )ρ ≥ 0 . (3.35)For states acting on higher dimensional Hilbert spaces, the criterion is onlynecessary for separability. We call any state ρ for which Eq. (3.35) is satisfieda ‘PPT state’.

Proof. We have already seen in section 3.3.1 that the transposition is apositive, but not completely positive map. In Eq. (3.6) we have seen that forany positive map Λ the operation (1⊗ Λ)ρ on a separable ρ gives a positiveoperator. So of course for Λ = T this has to be true as well. But so far onlya necessary condition for separability has been gained. This fact was alreadyapprehended by Peres [57].

To prove that the criterion is also a sufficient one for H2 ⊗H2, H3 ⊗H2or 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 spacesH2 ⊗H2, H3 ⊗H2 or H2 ⊗H3 can be decomposed in the following way:

Λ = ΛCP1 + ΛCP2 ◦ T . (3.36)

Here ΛCP1 and ΛCP2 are completely positive maps.

Now let us suppose we have a state for which (1 ⊗ T )ρ ≥ 0, and wewant to show that this fact is sufficient for separability, which means thatthe state has to be separable for sure. Since ΛCP1 and Λ

CP2 are completely

positive maps the following statement has to be true:

(1⊗ ΛCP1 )ρ + (1⊗ ΛCP2 )(1⊗ T )ρ ≥ 0 (3.37)or

(1⊗ ΛCP1 )ρ + (1⊗ ΛCP2 ◦ 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 adecomposition (3.36) where the steps (3.37) and (3.38) can be done. ThePMT Theorem is a necessary and sufficient condition for separability and sothe proof is completed.


Example. We want to investigate the Werner state again. The partial trans-position of the matrix (2.36) is, according to Eq. (3.10),

ρα =

1−α4

0 0 −α2

0 1+α4

0 00 0 1+α

40

−α2

0 0 1−α4

. (3.40)

The eigenvalues of this matrix are

λ1 = λ2 = λ3 =1 + α

4, λ4 =

1− 3α4

. (3.41)

The first three eigenvalues are positive for all possible parameters α. λ4 canbe negative, and we get, applying the PPT Criterion (Theorem 3.6):

−13≤ α ≤ 1

3⇒ ρα is separable ,

1

3< α ≤ 1 ⇒ ρα is entangled . (3.42)

It is interesting that the PPT Criterion gives a remarkable wider range ofentanglement of the Werner state than the other necessary separability condi-tions discussed in the last paragraphs did. This becomes particularly obviouswhen looking at a graphical comparison of different separability criteria (seeFig. 3.5).

The Reduction Criterion

Another separability criterion whose properties are similar to the PPT cri-terion (Theorem 3.6) is the reduction criterion [39]:

Theorem 3.8 (Reduction Criterion). A state ρ acting on H2⊗H2, H3⊗H2or 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 onlynecessary 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.


separablePPT entangled

a

?

entangled

0 0,5 1

S

S

CHSH

1

2

a

entangled

entangled

?

?

Fig. 3.5: Comparison of the PPT criterion with other separability criteria for the2-qubit Werner state ρα: The PPT criterion clearly distinguishes betweenseparable and entangled states and gives a wider range of entanglementthat the other criteria.

Proof. According to the PMT Theorem (3.3) we know that for a positivemap 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 isequivalent to the PPT criterion (3.6) for H2⊗H2, H2⊗H3 or H3⊗H2 andthus 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 lookat Eq. (3.39), we see that the only way it can be violated is a violation ofthe PPT criterion. So the reduction criterion is not stronger than the PPTcriterion (it does not detect more entangled states).

2 Proof of positivity: If we write Λ(M) in its diagonal form Λ(M)d, for a positive Mwe have (λi are the eigenvalues of M , Md is the diagonalized M) Λ(M)d =

∑i λi1−Md.

The diagonal elements of this matrix are the eigenvalues µj of Λ(M), µj =∑

i λi − λj =∑i 6=j λi ≥ 0, and so Λ(M) ≥ 0.


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):

(ρα)A =

(12

00 1

2

)=

1

21 . (3.46)

And furthermore we obtain

(ρα)A ⊗ 1 = 121⊗ 1 . (3.47)

If we want to apply the reduction criterion (Theorem 3.8), we calculate thediagonal matrix ((ρα)A ⊗ 1 − ρα)d, because then the eigenvalues are thediagonal elements. We find with the help of Eq. (3.47)

((ρα)A ⊗ 1− ρα)d = ((ρα)A ⊗ 1)d − (ρα)d = 121⊗ 1− (ρα)d . (3.48)

We conclude from Eq. (3.25) that the diagonalized Werner state is

(ρα)d =

1−α4

0 0 00 1−α

40 0

0 0 1−α4

00 0 0 1+3α

4

. (3.49)

So Eq. (3.48) becomes

((ρα)A ⊗ 1− ρα)d =

1+α4

0 0 00 1+α

40 0

0 0 1+α4

00 0 0 1−3α

4

. (3.50)

The eigenvalue 1−3p4

can be negative for some range of the parameter α, sowe obtain

−13≤ α ≤ 1

3⇒ ρα is separable ,

1

3< α ≤ 1 ⇒ ρα is entangled , (3.51)

which is exactly the same result as Eq. (3.42) in connection with the PPTcriterion.


Example 2. The following example illustrates that for states on Hilbertspaces of more general dimensions, the reduction criterion (Theorem 3.8)can be more useful than the PPT criterion. The state of interest is theisotropic state ρ

(d)α (2.16) of any dimension d ≥ 2. We first calculate the

reduced density matrix

(ρ(d)α )A = TrBρ(d)α = αTrB

∣∣φd+〉 〈

φd+∣∣ + 1− α

d2TrB1⊗ 1 , (3.52)

and because the reduced density matrix of the maximally entangled purestate

∣∣φd+〉

has to be the maximally mixed state 1d1 of the subsystem, we

obtain

(ρ(d)α )A = TrBρ(d)α =

α

d1+

1− αd

1 =1

d1 . (3.53)

The term of interest for the reduction criterion is

(ρ(d)α )A ⊗ 1− ρ(d)α =1

d1⊗ 1− α

∣∣φd+〉 〈

φd+∣∣− 1− α

d21⊗ 1 . (3.54)

Like in the first example we can diagonalize the whole term (3.54),

((ρ(d)α )A ⊗ 1− ρ(d)α

)d

=α + d− 1

d21⊗ 1− α (

∣∣φd+〉 〈

φd+∣∣)

d. (3.55)

Since∣∣φd+

〉 〈φd+

∣∣ is a pure state, the diagonal matrix always has one elementequal to 1 and all others equal to 0. So with help of Eq. (3.55) we find theeigenvalues

λ1 =α(1− d2) + d− 1

d2, λ2, . . . , λd =

d− 1 + αd2

(3.56)

of (ρ(d)α )A ⊗ 1 − ρ(d)α . The eigenvalues λ2, . . . , λd are positive for all possible

values of α and d ≥ 2. The eigenvalue λ1 is, however, negative for somevalues of α and we have

1

d + 1< α ≤ 1 ⇒ ρ(d)α is entangled . (3.57)

In Ref. [39] it is shown that for the other possible values of α the state canalways be written as a mixture of product states, and so

− 1d2 − 1 ≤ α ≤

1

d + 1⇒ ρ(d)α is separable . (3.58)

Finally, we want to formulate Eqs. (3.57) and (3.58) with the fraction F in-stead of α, since we know that the notations (2.16) and (2.22) are equivalent.


We insert Eq. (2.21) in Eqs. (3.57) and (3.58) and find

1

d< F ≤ 1 ⇒ ρ(d)F is entangled ,

0 ≤ F ≤ 1d

⇒ ρ(d)F is separable . (3.59)

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, clas-sify 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 ob-tain a maximal entangled pure state out of a mixed entangled state via localoperations and classical communication (LOCC)). The distillation of mixedentangled states was introduced in Ref. [7], for further application of thesubject see, e.g., Refs. [8, 27, 45]. Distillable entangled states are called freeentangled and non-distillable entangled states are called bound entangled [44].

The chapter is organized as follows: The concept of distillation and theclassification connected with it is discussed in Sec. 4.2. In Sec. 4.3 we inves-tigate LHV models under general viewpoints, that is, Bell’s original idea isextended 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 quan-tum communication, e.g., teleportation. Thus Alice produces 2-qubit singletstates |ψ−〉 (2.30) and sends one particle from each pair to Bob. But thechannel she uses for her transmission is noisy, so when Bob receives his par-ticle, Alice and Bob share no pure singlet state |ψ−〉 any longer, but somemixed state ρ.

Can they, by any means, obtain the singlet states again? The answeris yes [7], for some mixed states ρ, Alice and Bob can do local operationsand classical communication (LOCC) to recover from a given number ofthe same mixed states ρ a smaller number of (nearly) maximally entangled

4. Classification of Entanglement 33

singlets |ψ−〉. Note the word ‘nearly’ in the last sentence. It means that witha finite number n of ‘input’ states ρ, we can distill a smaller number k (withsome probability pk) of states ρdist out of them that have a higher fidelityFψ−(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 statesρdist. So we can get ‘output’ states ρout with an arbitrarily high fidelityFψ−(ρout) by applying the same protocol again and again.

However, for some protocols, (e.g., the BBPSSW protocol [7]) in the limitof infinitely many input states ρ,1 the distillation rate Rdist(ρ) of distilledoutput states per input state (asymptotic distillation rate) tends to zero.Nevertheless there are distillation protocols [7, 8] for which Rdist(ρ) does nottend to zero, but to some positive constant c ∈ R,

Rdist(ρ) = limn→∞

k

n= c . (4.1)

The maximal possible distillation rate that can be achieved out of input statesρ and with any distillation protocol is called entanglement of distillation [8]

Edist(ρ) = maxLOCC

Rdist(ρ) (4.2)

and is used as an entanglement measure (see Chapter 5).

The BBPSSW Distillation Protocol

The first distillation protocol was introduced in Ref. [7] by Bennett, Brassard,Popescu, Schumacher, Smolin and Wootters, and is thus called BBPSSWprotocol. It works for all entangled 2-qubit states ρ for which a maximallyentangled state |ψmax〉 exists such that2

Fψmax(ρ) > 1/2 , (4.3)

where Fψmax(ρ) is the fraction given in Eq. (2.15). Note that if a state ρhas the property (4.3) then it cannot have a fraction higher than 1/2 withrespect to any other pure state. The protocol itself consists of the followingsteps:

1 That means we can apply the protocol infinitely many times, since we have an infinitesource of input pairs. So Fψ−(ρout) → 1.

2 The BBPSSW protocol is suitable for general states that satisfy the mentioned prop-erties. There also exist ‘distillation’ (more precise: concentration) protocols for pure statesonly [6] and it can be shown that all entangled pure states are distillable.


1. First, the state ρ is subjected to a suitable local unitary transformationUA ⊗ UB that transforms it into a state ρ1 with a fraction Fφ+ =: F >1/2, where |φ+〉 is the state defined in Eq. (2.33) (i.e. the maximallyentangled state (2.17) with d = 2). Such a transformation is alwayspossible [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 complexconjugate (Alice performs a random U , then tells Bob, who performsU∗). This transforms the state into a isotropic state ρF (2.34) [45]:

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, eachpair is in the state ρF . This means that Alice holds two particles, andBob as well. Each of them now applies a so-called XOR-operation toher / his particles. A XOR-operation is defined as

UXOR |a〉 ⊗ |b〉 = |a〉 ⊗ |(a + b)mod 2〉 , (4.6)where a, b = 0 or 1 and xmod 2 means that if x ≥ 2, we have to subtract2 from x so many times until we have x < 2 (thus in our case we have(a + b)mod 2 = 0 if a + b = 2). Here |a〉 is called ‘source’, |b〉 is called‘target’. We obtain the state ρ̃ that is a state of two pairs:

ρF ⊗ ρF → ρ̃ = UXOR(ρF ⊗ ρF )U †XOR (4.7)

4. In the next step Alice and Bob measure the spin of the target pair alongthe z-axis. If their outcomes are parallel (both measure |0〉 or bothmeasure |1〉), then the source pair is kept. We calculate the resultingstate of the source pair via performing a projection according to themeasurement and tracing out the target pair,

ρ̃ → ρ′ := Trtarget( (

1⊗ P‖)ρ̃

(1⊗ P‖

)

Tr(1⊗ P‖

)ρ̃

(1⊗ P‖

))

, (4.8)

where P‖ = |00〉〈00|+|11〉〈11|. The factor Tr(1⊗ P‖

)ρ̃

(1⊗ P‖

)gives

the probability that Alice and Bob measure parallel spins and is neededfor the normalization of the state (Trρ′ = 1).


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

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

F ′(F ) := 〈φ+| ρ′ |φ+〉 =F 2 + 1

9(1− F )2

F 2 + 23F (1− F ) + 5

9(1− F )2 . (4.9)

Let us take a look at a plot of the functions g(F ) := F ′(F ) and f(F ) := F inFig. 4.1: Only for F > 1/2 we always have g(F ) > f(F ). So only if we startwith a state ρ for which F > 1/2, we can increase the fidelity by iteratingthe process 1 - 4.

What about the number k of output states ρout after l iterations of theprotocol 1 - 4? According to the protocol, we get

k =np

2 l, (4.10)

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 productof the probabilities to get parallel spins after measurement). If we want toreach a fidelity F ′ = 1, we have to iterate the process infinitely often. Thatalso means we need an infinite supply of input pairs, and the probability ptends 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 withFψmax > 1/2 (see Refs. [7, 8]).


Distillable Entangled States

Entangled 2-qubit States. In Ref. [43] it is shown that with a special LOCCoperation 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). Sowe 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 aprojective operation P ⊗ P to an isotropic state ρ(d)α (2.16), where

P = |0〉〈0|+ |1〉〈1| . (4.12)

We can also say that Alice and Bob measure the state of their particles, andthey only keep their pair if they get |0〉 or |1〉. The resulting state is a 2-qubit isotropic state ρ

(2)α , where we normalized the outcome of the operation

according to Trρ(2)α = 1. If ρ

(d)α is entangled, that is, we have 1d+1 < α ≤ 1,

then for the resulting state ρ(2)α we get 13 < α ≤ 1, so the 2-qubit isotropic

state is entangled too. This state is distillable, since all entangled 2-qubitstates are distillable. If we use the equivalent form ρ

(2)F (2.34) of the 2-qubit

isotropic state, then, according to Eq. (3.59), we have 12

< F ≤ 1, and so theresulting state can be distilled with the BBPSSW protocol without any priorfiltering. 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 byapplying a suitable filtering operation on a state that violates the reductioncriterion (Theorem 3.8), we obtain (with a certain probability) a state thathas a fraction F > 1/d, and this state can then be transformed via a random

U ⊗ U∗-transformation (4.5) into an entangled isotropic state ρ(d)F (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 entan-gled state that is not distillable is called bound entangled, whereas distillable


entangled states are called free entangled [44]. In particular we have thefollowing theorem:

Theorem 4.4. A PPT state (i.e. a state that remains positive under partialtransposition) cannot be distilled.

Theorem 4.4 can be proved in different ways. One way [44] uses yetanother theorem from which Theorem 4.4 can be derived. Here we want tosketch a proof from Ref. [45]: This proof is done in two steps, first, it isshown that any LOCC on PPT states result in PPT states [44]. Second, itis shown that for a PPT state ρPPT we always have

F (ρPPT ) ≤ 1d

, (4.13)

(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 quantuminformation, 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 purestate out of the bound entangled state again.

Another consequence is the following: One can prove [46] that any boundentangled state has to satisfy the S0 entropy inequality (3.20), (3.21). So thisinequality is also a necessary condition not only for separability, but also forbound entanglement. To prove that a bound entangled state has to satisfythe S0 entropy one can show that [46] any state violating the inequality alsohas to violate the reduction criterion, and according to Theorem 4.3 we knowthat such a state is distillable.

Do there exist bound entangled NPT states?

We have already learned in the last section that all entangled PPT statesare not distillable (bound entangled). Now the question arises if there existentangled states that are not positive under partial transposition (NPT), butnevertheless are not distillable. There have not been any rigorously conclusiveresults yet, but there is a strong implication that bound entangled NPT statesexist [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 formin Ref. [45] and based on matrices of Ref. [69]):

ρβ =2

7

∣∣φ3+〉 〈

φ3+∣∣ + β

7σ+

5− β7

σ− , 0 ≤ β ≤ 5 , (4.14)


PPT states NPT states

general states

separable states free entangled states

separable states free entangled states

bound entangled states

2-qubit states

Fig. 4.2: Illustration of entanglement and distillability. Since all entangled 2-qubitstates are distillable and NPT, we have a clear distinction in this case. Forgeneral states, however, there are entangled PPT states (bound entan-gled) and maybe bound entangled NPT states, which are those outsidethe “box” of the free entangled states. Note that this is not a geometricrepresentation of sets of states.


where, according to Eq. (2.17)

∣∣φ3+〉

=1√3

(|0〉 ⊗ |0〉+ |1〉 ⊗ |1〉+ |2〉 ⊗ |2〉) , (4.15)

and

σ+ =13(|01〉〈01|+ |12〉〈12|+ |20〉〈20|) , (4.16)

σ− = 13 (|10〉〈10|+ |21〉〈21|+ |02〉〈02|) . (4.17)If we write the state (4.14) in matrix notation (2.41) in the standard basis

(2.42) we obtain

ρβ =

221

0 0 0 221

0 0 0 221

0 5−β21

0 0 0 0 0 0 0

0 0 β21

0 0 0 0 0 0

0 0 0 β21

0 0 0 0 0221

0 0 0 221

0 0 0 221

0 0 0 0 0 5−β21

0 0 0

0 0 0 0 0 0 5−β21

0 0

0 0 0 0 0 0 0 β21

0221

0 0 0 221

0 0 0 221

. (4.18)

A check of the eigenvalues of the matrix (4.18) gives the result that ρβ ≥ 0for 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 transposedstate ρTBβ . We find

λ1 = λ2 = λ3 =2

21

λ4 = λ5 = λ6 =1

42

(5−

√41− 20β + 4β2

)

λ7 = λ8 = λ9 =1

42

(5 +

√41− 20β + 4β2

). (4.19)

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 necessaryfor separability, from Eq. (4.20) we know that ρβ is entangled for sure if


0 ≤ β < 1 or 4 < β ≤ 5 ; but for 1 ≤ β ≤ 4 the state is PPT and wecannot be certain if the state is separable or entangled. If we want to findout if somewhere within this range of β the state is entangled, we have to useanother 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 thusbound entangled according to Theorem 4.4.

Remark. A positive map Λ for which (1 ⊗ Λ)ρ < 0 and ρTB ≥ 0 cannotbe decomposable like Eq. (3.36), because we argued in the proof of the PPTcriterion (Theorem 3.6) in Sec. 3.3.2 that if a map Λ is decomposable andρTB ≥ 0, then this fact is equivalent to (1⊗Λ)ρ ≥ 0, which is a contradictionto the premises.

Clearly the difficulty lies in finding a suitable positive map Λ. The followingmap3 turns out to be useful:

Λ

a11 a12 a13a21 a22 a23a31 a32 a33

=

a11 + a22 −a12 −a13−a21 a22 + a33 −a23−a31 −a32 a33 + a11

. (4.21)

Proof that Λ (4.21) is positive. In Ref. [21] it is argued that a map Λ ispositive if the corresponding biquadratic form

f(x, y) := yT ·Λ (x · xT )·y, x =

x1x2x3

, y =

y1y2y3

, xi, yi ∈ R (4.22)

is positive for all x, y. Inserting our Λ from Eq. (4.21) we obtain

f(x, y) = x21y21 + x

22y

22 + x

23y

23 − 2x2x3y2y3 − 2x1x3y1y3 − 2x1x2y1y2 +

+ x23y22 + x

22y

21 + x

21y

23 . (4.23)

We can search for minima of this function and find that a global minimumis f = 0. So f(x, y) ≥ 0 ∀x, y and we proved that Λ (4.21) is positive. InFigure 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). Themap 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 inRef. [45]. The map presented and proved to be positive in Ref. [21] is, however, slightlydifferent to the map of Ref. [45] and to the map (4.21) (the map of Ref. [21] would notgive any evidence for bound entanglement either).


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

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

(1⊗ Λ) ρβ =

7−β21

0 0 0 − 221

0 0 0 − 221

0 521

0 0 0 0 0 0 0

0 0 2+β21

0 0 0 0 0 0

0 0 0 2+β21

0 0 0 0 0

− 221

0 0 0 7−β21

0 0 0 − 221

0 0 0 0 0 521

0 0 00 0 0 0 0 0 5

210 0

0 0 0 0 0 0 0 2+β21

0

− 221

0 0 0 − 221

0 0 0 7−β21

. (4.24)

The eigenvalues of the above matrix are

λ1 = λ2 = λ3 =5

21

λ4 =3− β

21

λ5 = λ6 =9− β

21

λ7 = λ8 = λ9 =2 + β

21. (4.25)


0 2 3 4 5 b

? separable bound

entangled

free

entangled

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

All eigenvalues are positive (within the allowed region of β), except for λ4we 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 boundentangled, for

3 < β ≤ 4 . (4.27)In Ref. [45] it is shown that for 2 ≤ β ≤ 3 the state ρβ is separable and for4 < β ≤ 5 the state is free entangled (because it can be projected onto anentangled 2-qubit state, and thus is distillable, see Theorem 4.1.) A graphicalillustration 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 theirown interpretation of quantum mechanics. They came to the conclusion thatquantum mechanics is incomplete; that there have to be intrinsic properties ofquantum mechanical objects which determine the outcome of measurements.In order to illustrate their viewpoint they stated a gedankenexperiment, inwhich a source emits two entangled particles in opposite direction. Let ushere consider Bohm’s variant of the experiment [14] where the source emitstwo spin 1/2 particles in a singlet state |ψ−〉 (2.30). Note that the vector|0〉 denotes “spin up” and |1〉 stands for “spin down”. EPR considered fourrequirements which they considered necessary to be fulfilled by any physicaltheory:


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

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

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

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

Translating the above requirements to our ‘gedanken’ experiment we canconclude the following: Once we measure the spin of one particle, we instan-taneously know what the outcome of the measurement of the other particlewill be. If we consider that the second measurement is performed immedi-ately after the first, that is, information could not have been transmittedfrom one particle to the other viewing the speed of light as the maximumpossible speed, then, due to locality (ii) the particles cannot influence oneanother. Thus, according to reality (iii) there has to be an ‘element of reality’corresponding to measurement outcomes that should be included in quantumtheory.

Now for a long time the question if quantum theory could be completedwith 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 topairs of particles, which predetermine the outcome of spin measurements inarbitrary directions. He considered two spin 1/2 particles (which we wouldcall a 2-qubit state in quantum information) in a pure state and set up the fa-mous Bell inequality (BI) which every 2-qubit state should satisfy if it admitshis local hidden variable theory (LHV). The BI involves expectation valuesof spin measurements. To his own surprise, he found that for some spin mea-surement directions the singlet state |ψ−〉 (2.30) violates the BI. That meansthat for this state local hidden variables cannot be assigned to the particlestelling them how to behave in measurements. So we cannot help acceptingsome kind of non-locality of quantum mechanics, there is some ‘spooky actionat distance’ that makes the particle which is measured after the other behavein the anti-correlated way. There is no need to believe in some faster-than-light information exchange, but for sure quantum correlations are strongerthan classical correlations in a barely comprehensible way.


There have been many variations and extensions of Bell’s inequality for-mulated until now. An example is the CHSH inequality (3.12) [23], alreadymentioned in Sec. 3.3.2. If an entangled state does not violate a specific kindof BI, it is not at all sure that it does not violate some other kind. There havebeen many efforts to give a more generalized formulation of Bell inequalities.In Ref. [76] Werner showed that some bipartite entangled states, the Wernerstates (for 2-qubits see Eq. (2.35)), do not violate an inequality derived byassuming general projective measurements of Alice and Bob. In this case onecan definitely use a LHV model to describe the process. Nevertheless, if onedoes not restrict the measurements to projective ones but to the most gen-eral 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 possiblekind of BI, or, equivalently, if it does not violate the most general expressionof a BI (which we will call general Bell inequality) even after subjection toany LOCC. If any BI is violated, then a state is called ‘non-local’. Thequestion arises whether non-locality is a necessary feature of all entangledstates, or if there exist ‘local’ entangled states for which there exist LHVsaccording to general measurements. It is in particular useful to determine ifan entangled state is local, since in quantum information a state admittinga LHV theory is not useful; such a state could be replaced by classical bitsaccording to the LHV model [18].

It is known that any pure entangled state violates a BI (e.g. the CHSHinequality after applying a particular LOCC to it, see Ref. [35]). For mixedstates the situation is not clear (yet). All distillable (see Sec. 4.2) entangledstates violate a Bell inequality, since they can be transformed into pure en-tangled 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 towardsolving the problem we want to state a most general formulation of Bellinequalities in the following section.

4.3.2 General Bell Inequality

This formulation of a general Bell inequality follows mostly Ref. [70] as wellas 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


asAlice: MA1 ,M

A2 , . . . ,M

An Bob: M

B1 ,M

B2 , . . . ,M

Bm . (4.28)

They are general because the outcomes of each measurement are describedby operators

MAi : EAi,1, E

Ai,2, . . . , E

Ai,p(i) ;

MBj : EBj,1, E

Bj,2, . . . , E

Bj,q(i) ; (4.29)

here p(i) is the number of possible outcomes of the i-th measurement forAlice, and equivalently for Bob. The operators (4.29) correspond to POVMmeasurements (see, e.g., [56]) and satisfy the condition

p(i)∑

k=1

EAi,k = 1, EAi,k ≥ 0 (4.30)

(and equivalently for Bob) but are not necessarily orthonormal like in thecase of projectors Qi,k, which always satisfy Qi,kQi,r = δkr.

We can calculate the following probabilities for a (bipartite) state ρ:

PAi,k = Tr(EAi,k ⊗ 1

)ρ (4.31)

PBj,l = Tr(1⊗ EBi,k

)ρ (4.32)

PA,Bi,k;j,l = Tr(EAi,k ⊗ EBi,k

)ρ . (4.33)

Eq. (4.31) gives the probability that Alice measures in the i-th measurementthe k-th outcome, with Eq. (4.32) we obtain the probability that Bob mea-sures in the j-th measurement the l-th outcome, and if we want to calculatethe probability that Alice measures the k-th and Bob the l-th outcome ina 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 ‘probabilityvector’ of a state ρ corresponding to measurements (4.28),

~Pρ =(

~PA,B, ~PA, ~PB)

, (4.34)

where ~PA,B, ~PA and ~PB contain all probabilities PA,Bi,k;j,l, PAi,k and P

Bj,l ((4.31)

- (4.33)) corresponding to the various combinations of measurements / out-comes for joint measurements, Alice’s measurements and Bob’s measure-ments.

Hidden variables ‘instruct’ the system which outcome a certain measure-ment should give. That is, a specific hidden variable λi defines an instruction


vector with entries 0 or 1, which give the probabilities that in a certain mea-surement a particular outcome is realized. There are instruction vectors forthe measurements of Alice, Bob, or both, denoted as ~BAλi ,

~BBλi ,~BA,Bλi . For

example if Alice has 2 possible measurements with 2 outcomes each, theinstruction vector ~BAλi defined by one hidden variable λi would be, e.g.,

~BAλi = (0, 1; 1, 0) , (4.35)

where the first two entries give the probabilities for the realization of out-comes 1 and 2 of the first measurement; here it is determined that withcertainty outcome 2 is realized; and the last two entries describe the sec-ond measurement equivalently. Of course Bob’s instruction vector ~BBλi hasa similar form, according to the number of his possible measurements andoutcomes.

It is important that the instruction vector for measurements of both Aliceand Bob takes the assumption of locality into account. That means it isassumed that the measurements are independent from each other and wecan write

~BA,Bλi =~BAλi ⊗ ~BBλi . (4.36)

The ‘total’ instruction vector ~Bλ is given by

~Bλi =(

~BA,Bλ ,~BAλ , ~B

Bλ

). (4.37)

For example, if Alice has 2 possible measurements with 2 outcomes each andBob 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 measure-ments 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 iswritten as a convex combination of instruction vectors [59, 60],

~PLHV =∑

i

qi ~Bλi , qi ≥ 0,∑

i

qi = 1 . (4.39)

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 differentfor different possible measurements (4.28) of Alice and Bob. Now we canformulate the following theorem:


Theorem 4.5. A bipartite state ρ can be described by a LHV theory withrespect 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 thereexists 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 ofLLHV (M). What about the entangled states? There exists a useful Lemma,called the Minkowski-Farkas Lemma (see, e.g., Ref. [66]), that gives a condi-tion for a vector not being an element of a convex cone, and which appliedto our case is of the following form:

Lemma 4.1. The probability vector ~Pρ̃ of a state ρ̃ is not an element of

LLHV (M) if and only if there exists a ‘Farkas vector’ ~F , such that

~F · ~Pρ̃ < 0 and ~F · ~Bλi ≥ 0 ∀λi . (4.41)

In general the Farkas vector ~F can have any real components; however, inRef. [58] it is shown that it suffices to consider integers only. From Lemma 4.1and Theorem 4.5 we can induce a general Bell inequality. If we have a Farkasvector ~F for which ~F · ~Bλi ≥ 0 ∀λi, we can also say that

∑i

qi ~F · ~Bλi ≥ 0 , ∀qi ≥ 0,∑

i

qi = 1 , (4.42)

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).


Since the probability vector ~PρLHV of a state ρLHV that can be describedby 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 state ρ̃violates the general Bell inequality (4.44) if

~F · ~Pρ̃ < 0 , (4.46)

and, according to the Minkowski-Farkas Lemma 4.1 and Theorem 4.5, sucha state can in general not be described by a LHV theory.

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

MA1 = ~a · ~σ, MA2 = ~a′ · ~σ ,MB1 =

~b · ~σ, MB2 = ~b′ · ~σ . (4.47)

We only have to consider probabilities to measure the outcome +1 (in suitableunits), 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′ with outcomes +1)

(PLHV )A,Ba,+1;b′,+1 =: P

LHVab′ , (4.48)

and similar for the single probabilities. The CH inequality is of the form

PLHVa − PLHVab + PLHVb′ − PLHVa′b′ + PLHVa′b − PLHVab′ ≥ 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 thatthe 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 .


Remark. If a state violates the CH inequality (4.49) (or any other Bellinequality) it can in general not be described by a LHV theory and is thereforecalled ‘non-local’ (and is of course entangled). If it does not violate theinequality we cannot definitely say that the state can be described by a LHVtheory, not even with respect to the regarded measurement ensemble, sincewe only checked one possible Farkas vector and not all possible Farkas vectors.Furthermore, we should note that in the literature often the expression ‘localstate’ is connected with a particular ensemble of measurements - it is meantthat for a particular ensemble of measurements we can apply a LHV theory.Nevertheless it might be the case that with other measurements a non-localityof the state is revealed. To be accurate only states that satisfy all general Bellinequalities (with all possible Farkas vectors and all possible measurementensembles), 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, butin 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 anentanglement witness for this state. But we cannot, in general, construct aviolation of a general Bell inequality out of a given entanglement witness foran entangled state.

Construction of an Entanglement Witness out of a Violation of the GeneralBell 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 ofthe Farkas vector similar to the probability vector (4.34),

~F =(

~FA,B, ~FA, ~FB)

, ~FA,B =(FA,Bi,k;j,l

), ~FA =

(FAi,k

), ~FB =

(FBj,l

),

(4.51)and define an operator

A :=∑

i,k,j,l

FA,Bi,k;j,lEAi,k ⊗ EBj,l +

∑

i,k

FAi,kEAi,k ⊗ 1+

∑

j,l

FBj,l1⊗ EBj,l , (4.52)

where the operators EAi,k are those introduced in Eq. (4.29). With Eqs. (4.31)- (4.33) we calculate for any state ρ̃

~F · ~Pρ̃ = TrAρ̃ . (4.53)


The violation of the general Bell inequality (4.46) clearly corresponds toTrAρent < 0. Since we already know that any separable state ρ can bedescribed by a LHV theory we have ~Pρ = ~PρLHV and therefore, according toEq. (4.45)

~F · ~Pρ = TrAρ ≥ 0 ∀ separable states ρ . (4.54)Thus we have

〈ρent, A〉 = TrAρent < 0 ,〈ρ,A〉 = TrAρ ≥ 0 ∀ρ ∈ S , (4.55)

which is exact

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Quantum Entanglement: Detection, Classification, and ......ested in pure states only, either the vector representation jˆi or the operator representation ‰pure = jˆihˆj can be