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Quantum Entanglement: Detection,Classification, and Quantification

Diplomarbeitzur Erlangung des akademischen Grades

,,Magister der Naturwissenschaften’’an der

Universitat 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 Schrodinger in Ref. [68].Since this article was published in German, ‘entanglement’ is a later transla-tion of the word ‘Verschrankung’. Schrodinger 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 asHd1

A ⊗Hd2B , 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 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 Hd1

A ⊗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 ⊗ ρi

B, 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 ρA

i and ρBi . 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 ATrρB

i B

=n∑

i=1

piTr(ρA

i ⊗ 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− α

d21⊗ 1, α ∈ R, − 1

d2 − 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 entangled

and 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 ,

1d + 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) we

get

F =⟨φd

+

∣∣ ρ(d)α

∣∣φd+

⟩=

1 + α(d2 − 1)

d2, (2.20)

or

α =d2F − 1

d2 − 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 − 1

d2

) ∣∣φd+

⟩ ⟨φd

+

∣∣ + (1− F )1d2

)(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+ niσ

i), 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 ρ11,22

ρ12,11 ρ12,12 ρ12,21 ρ12,22

ρ21,11 ρ21,12 ρ21,21 ρ21,22

ρ22,11 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+ aiσ

i ⊗ 1+ bi1⊗ σi + cijσi ⊗ σj

), ai, bi, cij ∈ R . (2.27)

A product state ρA ⊗ ρB has the form

ρA ⊗ ρB = 14(1⊗ 1+ niσ

i ⊗ 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+ nk

i σi ⊗ 1+ mk

i 1⊗ σi + nki m

kj σ

i ⊗ σj),

nki ,m

ki ∈ 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− F

3|ψ−〉〈ψ−|+ 1− F

3|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− α

41⊗ 1, −1

3≤ 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 ρ11,33

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

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 =|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 λ

i ⊗ 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 nk

i λi ⊗ 1 +√

3 mki 1⊗ λi + 3 nk

i 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))Trρ2

A = Trρ2B < 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 ρp

with 〈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 ρ ∈ S for which

〈ρ, 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 seeneasily [57]: Applying (1⊗ Λ) to a separable state (2.11) gives

(1⊗ Λ)ρ =n∑

i=1

piρAi ⊗ Λ(ρB

i ), (3.6)

and since Λ is positive, Λ(ρBi ) 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 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)) =: (AT

ij) = (Aj i). So 1 ⊗ T means that only Bob’s partis 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, σ1

corresponds 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 =

−α

4· 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 bySchrodinger [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 · 1

2log

1

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− α

4

)log2

1− α

4− 1 + 3α

4log2

1 + 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 ⊗ H2

or 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 ⊗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 spacesH2 ⊗H2, H3 ⊗H2 or H2 ⊗H3 can be decomposed in the following way:

Λ = ΛCP1 + ΛCP

2 ◦ T . (3.36)

Here ΛCP1 and ΛCP

2 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 ΛCP

1 and ΛCP2 are completely

positive maps the following statement has to be true:

(1⊗ ΛCP1 )ρ + (1⊗ ΛCP

2 )(1⊗ T )ρ ≥ 0 (3.37)

or(1⊗ ΛCP

1 )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− 3α

4. (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):

−1

3≤ 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⊗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 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 =1

21⊗ 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 =1

21⊗ 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

−1

3≤ 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 1

d1 of the subsystem, we

obtain

(ρ(d)α )A = TrBρ(d)

α =α

d1+

1− α

d1 =

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 + α

d2(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

− 1

d2 − 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 ≤ 1

d⇒ 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 thestate, 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 ρ that is a state of two pairs:

ρF ⊗ ρF → ρ = 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,

ρ → ρ′ := Trtarget

( (1⊗ P‖

)ρ

(1⊗ P‖

)

Tr(1⊗ P‖

)ρ

(1⊗ P‖

))

, (4.8)

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

)ρ

(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 1

d+1< α ≤ 1,

then for the resulting state ρ(2)α we get 1

3< 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 ) ≤ 1

d, (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− β

7σ− , 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 ρTB

β . 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 a13

a21 a22 a23

a31 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 =

x1

x2

x3

, y =

y1

y2

y3

, xi, yi ∈ R (4.22)

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

f(x, y) = x21y

21 + x2

2y22 + x2

3y23 − 2x2x3y2y3 − 2x1x3y1y3 − 2x1x2y1y2 +

+ x23y

22 + x2

2y21 + x2

1y23 . (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 λ4

we haveλ4 < 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: MA

1 ,MA2 , . . . ,MA

n Bob: MB1 ,MB

2 , . . . ,MBm . (4.28)

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

MAi : EA

i,1, EAi,2, . . . , EA

i,p(i) ;

MBj : EB

j,1, EBj,2, . . . , EB

j,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, EA

i,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

(EA

i,k ⊗ 1)ρ (4.31)

PBj,l = Tr

(1⊗ EB

i,k

)ρ (4.32)

PA,Bi,k;j,l = Tr

(EA

i,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, PA

i,k and PBj,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 ~BA

λi, ~BB

λi, ~BA,B

λi. For

example if Alice has 2 possible measurements with 2 outcomes each, theinstruction vector ~BA

λidefined 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 ~BB

λihas

a 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⊗ ~BB

λi. (4.36)

The ‘total’ instruction vector ~Bλ is given by

~Bλi=

(~BA,B

λ , ~BAλ , ~BB

λ

). (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 ~Bλiand 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 ~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, 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 ~PρLHVof 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 state ρviolates 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, 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, MA

2 = ~a′ · ~σ ,

MB1 = ~b · ~σ, MB

2 = ~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 =: PLHV

ab′ , (4.48)

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

PLHVa − PLHV

ab + PLHVb′ − PLHV

a′b′ + PLHVa′b − PLHV

ab′ ≥ 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,B

i,k;j,l

), ~FA =

(FA

i,k

), ~FB =

(FB

j,l

),

(4.51)and define an operator

A :=∑

i,k,j,l

FA,Bi,k;j,lE

Ai,k ⊗ EB

j,l +∑

i,k

FAi,kE

Ai,k ⊗ 1+

∑

j,l

FBj,l1⊗ EB

j,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 ρ

~F · ~Pρ = TrAρ . (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 exactly the entanglement witness Theorem 3.1.

Example. We want to construct an entanglement witness out of a violationof the CH inequality (4.49). The operators (4.29) corresponding to the out-come +1 of the possible measurements (4.47) are projectors, for example theprojector for a measurement along ~a is

EAa,+1 = EB

a,+1 =: Q+1 =1

2(1+ ~a · ~σ) , (4.56)

and equivalently for the other measurement directions. We obtain the entan-glement witness [70] with help of Eq. (4.52), where we sum over the measure-

ment directions ~a, ~a′,~b, ~b′ (we do not have to sum over all possible outcomes,

since terms for the outcome −1 do not matter because the component of ~Fis 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 allstates violating the CH inequality (4.49), but also for all states that violatethe CHSH inequality.

Can We Construct a General Bell Inequality out of an EntanglementWitness?

In the last section we have seen that starting with a violation of a general Bellinequality (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 anentanglement witness (and a violation of the inequality for the state ρent thatis detected by the entanglement witness)? In general we cannot do this. Thisis due to the following: If we have an entanglement witness A for a particular


entangled state ρent, we know that for all separable states ρ we have TrAρ ≥ 0or ~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 vectorsthat do not correspond to quantum mechanical probability vectors. So thecondition ~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 Bellinequality from an entanglement witness if we do not allow LHV theoryvectors 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 everyentangled 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 classifyit according to certain properties of the entangled state. In quantum infor-mation 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 arebetter suited than others. So it is of interest to somehow quantify or measurethe entanglement of states. We will see that for pure states we have a usefulmeasure that can be calculated for all states on a finite dimensional Hilbertspace Hd1⊗Hd2 - the entropy measure1 [8, 6] - it is the von Neumann entropyof the reduced density matrices. Unfortunately, the von Neumann entropyturns out to be a bad entanglement measure for mixed states. Here we haveto 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 systemsthere exist algebraic ‘recipe’ methods for calculating a measure. Addition-ally, often it is not clear which entanglement measure is better suited thanothers, and still we do not know which properties are more and which areless important to be satisfied by a measure.

The chapter is organized as follows: In Sec. 5.2 we start with quantify-ing the entanglement of pure states and introduce the entropy measure, inSec. 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 = |ψ〉〈ψ| a good and convenient entanglement measureis 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 introducedin Eq. (3.22). It is useful to take the logarithm in the definition of the vonNeumann 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, |ψ〉 =|ψA〉 ⊗ |ψB〉 (this is independent of the choice of logarithm basis), and 1 isachieved for maximally entangled states (since for those states the reduceddensity matrices are maximally mixed states).

There are some characteristic properties of EvN that fulfill natural expec-tations of an entanglement measure [8, 6]:

(i) EvN is additive:EvN(ρ⊗n) = 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)

(iii) After LOCC2 the expectation value of EvN cannot have increased:

∑i

piEvN(ρi) ≤ EvN(ρ) , (5.5)

where the states ρi are the residual states after LOCC that occur withprobability pi.

We know that there are protocols in which from a large number of less en-tangled 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 onestate a state with a higher value of entropy measure. Considering the otherway 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 ofinfinitely many input states) rate of output states ρout per input state ρin is[6]

limn→∞

mout

nin

=EvN(ρin)

EvN(ρout). (5.6)

2 See also footnote to Eq. (5.28)


Here nin is the number of input states and mout is the number of outputstates. If we consider the case in which the output states are non maximallyentangled and the input states are singlets, we have (since EvN(|ψ−〉〈ψ−|) =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 entropymeasure 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 entropymeasure (see Sec. 5.2),

EF (ρpure) = EvN(ρpure) . (5.8)

(ii) The entanglement of formation of an ensemble of pure states ε :={pi, ρ

ipure

}, where pi are the probabilities for the states ρi

pure to occurin the ensemble, is

EF (ε) =∑

i

piEF (ρipure) =

∑i

piEvN(ρipure) . (5.9)

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

EF (ρ) = minε

EF (ε) (5.10)

Practical Justification of the Definition

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

3 Remember that a mixed state ρ can be written as ρ =∑

i piρipure, but this decompo-

sition is not unique.


cost) [8]4. Why is this the case? We have already seen in the last section thatin the case of pure states Alice and Bob need EvN(ρpure) singlets to create anarbitrary pure state ρpure. So in order to prepare an ensemble ε :=

{pi, ρ

ipure

}(see (ii) in the definition above) they need

∑i piEvN(ρi

pure) = EF (ε) singlets.Obviously the minimum number of singlets needed to create the state ρ isthe minimum of the entanglement of formation of all possible ensembles ε,which is EF (ρ), according to (iii).

However, one might object the following to the above reasoning: What ifAlice and Bob prepare a state ρ′ with a cost EF (ρ′) and then apply LOCCto increase the entanglement of formation so that they obtain a state ρ forwhich

EF (ρ) > EF (ρ′) ? (5.11)

No, they cannot5, because we can proof that the expected entanglement offormation is not increasing under LOCC. The proof is done in Ref. [8] anduses the assumption that any LOCC can be separated into the following basicoperations (here, e.g., by Alice, but equivalently by Bob):

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

(b) Alice performs a unitary transformation.

(c) Alice performs a projective measurement.

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

Example. We can demonstrate that the entanglement of formation (5.9) ofdifferent ensembles realizing the same mixed state ρ is in general different:Consider the maximally mixed 2-qubit state 1

21. This state can be prepared

as an equal mixture of four orthogonal product states, so the entanglement offormation of such an ensemble is 0. On the other hand, we could also preparethe state as an equal mixture of the four Bell states, where the entanglementof formation for this ensemble is 1 (all Bell states have entropy measure 1and all appear with probability 1/4).

4 This statement involves the conviction that any LOCC operation can be separated intoseveral basic operations (a) - (d), mentioned later on. Although intuitively and physicallyevident, there is no strict mathematical proof that this can be done. Therefore someauthors 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 toconsider expectation values.


5.3.2 Concurrence and Calculating the Entanglement of Formation for2 Qubits

The definition of entanglement of formation for mixed states, Eq. (5.10), isclear and rather simple. Nevertheless the calculation is in general hard todo; it is often difficult to determine all possible ensembles of pure states thatrealize a given mixed state.

For 2-qubit states, however, there exists an operational method to calcu-late the entanglement of formation [79]. This is done via introducing anotherfunction of a 2-qubit density matrix ρ, the so-called concurrence. We will seethat the concurrence itself can be used as an entanglement measure.

First, we have to start with some definitions. The spin flip operation ona pure state |ψ〉 is given by

|ψ〉 →∣∣∣ψ

⟩= σy |ψ∗〉 , (5.12)

where ‘*’ denotes complex conjugation. The operation is called spin flip,because it ‘flips’ the spin of the |0〉 and |1〉 states:

σy |0〉 = i |1〉 , σy |1〉 = −i |0〉 . (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 con-

currence are of importance (introduced in Refs. [8, 79]):

Theorem 5.1. The entanglement of formation of a 2-qubit state ρ is a func-tion of the concurrence C,

EF (ρ) = EF (C(ρ)) = H

(1 +

√1− C2

2

). (5.15)

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 |ψ〉 is

C(|ψ〉) =∣∣∣⟨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 indecreasing order.


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]. InFig. 5.1 we can see a plot of the entanglement of formation EF as a functionof the concurrence C: As C increases from 0 to 1, EF increases monotonicallyfrom 0 to 1 - so we can consider C as an entanglement measure itself. If weare only interested in the comparison of the ‘amount’ of entanglement ofdifferent states, the concurrence does its job as an entanglement measure asgood as the entanglement of formation; and we do not need to perform theextra calculation of Theorem 5.1.

Example. We want to determine the entanglement of formation and concur-rence for the Werner state ρα (2.35) of 2 qubits. To obtain the concurrenceC according to Eq. (5.18) we first have to calculate the spin flipped Wernerstate ρα using Eq. (5.14). Using the matrix notation (2.36) (in the standardbasis, as needed) of the Werner state and

σy ⊗ σy =

0 0 0 −10 0 1 00 1 0 0−1 0 0 0

(5.19)

we getρα = ρα . (5.20)

Now we need to calculate the square roots of the eigenvalues of ρ · ρ. In ourcase we have to calculate the square roots of the eigenvalues of ρ2, whichare nothing but the eigenvalues λi of the Werner state (3.25) themselves.We calculate (where we ordered the eigenvalues in decreasing order, we have


0.4 0.5 0.6 0.7 0.8 0.9

0.2

0.4

0.6

0.8

1

E (r )F a

C (r )a

a

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

λ4 ≥ λ3 = λ2 = λ1 )

µ1 − µ2 − µ3 − µ4 = λ4 − λ3 − λ2 − λ1 =3α− 1

2, (5.21)

and so for the concurrence we have, according to Eq. (5.18),

C(ρα) = 0 for −1/3 ≤ α ≤ 1/3 ,

C(ρα) =3α− 1

2for 1/3 < α ≤ 1 . (5.22)

We use Theorem 5.1 and obtain the entanglement of formation

EF (ρα) =1

4

((√3 + 6p− 9p2 − 2

)log2

1

4

(2−

√3 + 6p− 9p2

)−

−(2−

√3 + 6p− 9p2

)log2

1

4

(2 +

√3 + 6p− 9p2

))(5.23)

Comparing the expressions (5.22) and (5.23) we can see that the concurrencehas a much simpler form which makes it more convenient to work with. Theresults are in perfect agreement with the results of section 3.3.2, where wefound 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) forthe entangled Werner state (Fig. 5.2) reveals that the entanglement of for-mation 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. Itis given by [8]

Edist(ρ) = maxLOCC

Rdist(ρ) , (5.24)


and is the maximal possible rate of distilled output singlet states per inputstate ρ (in the limit of infinitely many input states) that can be achievedby a distillation protocol. As an entanglement measure the value of Edist(ρ)varies between 0 and 1 which allows a comparison with the previous in-troduced measures concurrence and entanglement of formation. However,Edist(ρ) is ‘nonoperational‘, it is in general not known what the ‘best’ distil-lation 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 anyentanglement measure E 6:

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

(ii) The entanglement measure should be invariant under local unitary op-erations:

E(ρ) = E((UA ⊗ UB) ρ (UA ⊗ UB)†

). (5.26)

(iii) (a) The entanglement measure should not increase under any com-plete positive map (which corresponds to a general local physicaloperation and classical communication that can be performed):

E

(∑i

ViρV †i

)≤ E(ρ) , (5.27)

where Vi is of the form Vi,A ⊗ Vi,B and∑

i ViV†i = 1.

(b) The expectation value of the entanglement measure after selectiveoperations7 should not increase:

∑i

Tr(ViρiV

†i

)E

ViρiV

†i

Tr(ViρiV

†i

) ≤ E(ρ) . (5.28)

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 a certain operation, forexample a measurement that occurs with a certain probability Tr

(ViρiV

†i

).


Remark. Note that the properties (5.25) - (5.28) are similar to the prop-erties (5.3) - (5.5) in connection with the entropy measure, but they do notinclude the property of additivity (5.3). In principle we would not have tomention property (ii) (Eq. (5.26)), since it follows from Eq. (5.27), never-theless it is useful to emphasize that the entanglement measure should notchange 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 state ρ as the minimal distance (5.29)of a state ρ to the set S of all separable states [73],

D(ρ) := minρ∈S

d(ρ, ρ) . (5.30)

In Refs. [73, 72] various sufficient conditions for distance functions (5.29) aregiven that guarantee, if fulfilled, that the distance measure (5.30) has theproperties (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) := minρ∈S

dRE(ρ, ρ) . (5.32)

It is shown in Refs. [73, 72] that the relative entropy of entanglement indeedhas the properties (5.25) - (5.28). In general, to compute Eq. (5.31) for anytwo states ρ1, ρ2 we use (similar to the calculations for the entropy inequalitiesin Sec. 3.3.2)

Tr (ρ2 (log2 ρ2 − log2 ρ1)) =∑

i

(λi (log2 λi − log2 µi)) , (5.33)

where µi are the eigenvalues of the state ρ1 and λi those of ρ2.


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

α , so that we have

DRE(ρentα ) = dRE(ρ0,α, ρα) (5.34)

In Ref. [73] it is determined that ρ0,α is independent of the parameter α andis 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),

µ1 = µ2 = µ3 =1

6, µ4 =

1

2. (5.36)

We obtain

DRE(ρentα ) = dRE(ρ0, ρα) =

∑i

(λi (log2 λi − log2 µi))

= 3

(1− α

4

(log2

1− α

4+ log2 6

))+

+1 + 3α

4

(log2

1 + 3α

4+ 1

)

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

where we defined a := 1+3α4

, 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 thenorm (2.2),

dHS(ρ1, ρ2) = ‖ρ1 − ρ2‖ , (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, applythe results of Ref. [73] and again translate it into our notation with the parameter α.


Fig. 5.3: Plot of the relative entropy of entanglement DRE(ρα) of the Werner stateρα 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 state ρ, is [78]

D(ρ) = minρ∈S

dHS(ρ, ρ) = minρ∈S

‖ρ− ρ‖ . (5.40)

Although the Hilbert-Schmidt measure is much more convenient to handle incalculations than the relative entropy of entanglement, and despite of otheradvantages (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 measureto 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). InRef. [78] it is conjectured that Eq. (5.41) is indeed fulfilled in the case ofthe Hilbert-Schmidt distance (5.39). Nevertheless, in Ref. [55] it is shownthat the proof presented in Ref. [78] does not hold, and a counterexample ispresented. However, there has not been any indication yet that there existstates 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 determinethe Hilbert-Schmidt measure. In Refs. [78, 12] the nearest separable state (forwhich Eq. (5.40) minimizes) to an entangled Werner state ρent

α (see Eq. (3.42))


Fig. 5.4: Plot of the Hilbert-Schmidt measure DHS(ρα) of the Werner state ρα forvalues 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

DHS(ρentα ) =

∥∥ρ0 − ρentα

∥∥ . (5.43)

The explicit calculation can be done in different ways. One way is to use thenotation (2.27), like it is done (for the isotropic state) in Sec. 6.5. Here wewant to use the diagonalized form of ρ0 − ρent

α , because then we have

DHS(ρentα ) =


∥∥ =

√∑i

(µi − λi)2 , (5.44)

where µi are the eigenvalues of ρ0, see Eq. (5.36), and λi are the eigenvaluesof ρα, see Eq. (3.25). Inserting the values of Eqs. (5.36) and (3.25) we find

DHS(ρentα ) =

√3

2

(α− 1

3

). (5.45)

For a plot of the Hilbert-Schmidt measure for ρentα 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 procedureis nearly the same, we only have changed signs on the left-hand side of Eq. (6.34).


5.3.5 Comparison of Different Entanglement Measures for the 2-QubitWerner State

In the previous sections we calculated various entanglement measures for the2-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 entan-glement (5.37) and the Hilbert-Schmidt measure (5.45) for this particularstate.

First, we have to recognize that all of this entanglement measures varybetween 0 and 1, with the exception of the Hilbert-Schmidt measure. Inorder to be able to properly compare the Hilbert-Schmidt measure with theother measures, we have to ‘normalize’ Eq. (5.45); namely, we obtain thenormalized Hilbert-Schmidt measure by dividing DHS(ρ

entα ) by the value of

DHS(ρent1 ) and obtain

DHSN(ρentα ) =

3

2

(α− 1

3

)(5.46)

Remarkably this is exactly the same value as we obtained for the concurrenceC(ρent

α ) in Eq. (5.22), so that we have

DHSN(ρentα ) = C(ρent

α ) . (5.47)

In Fig. 5.5 the entanglement measures are graphically compared.


0.4 0.5 0.6 0.7 0.8 0.9

0.2

0.4

0.6

0.8

1

a

E (rE (E ( )E (E ( ) )

C (r )a)(HSN arD

D (r ( ( )D (D ( arr ) )

=

Fig. 5.5: Comparison of the concurrence C(ρα), the entanglement of formationEF(ρα), the relative entropy of entanglement DRE(ρα) and the ‘normal-ized’ Hilbert-Schmidt measure DHSN(ρα) of the Werner state ρα for valuesof 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-Schmidtmeasure of entanglement (5.40) and the concept of entanglement witnesses(see Theorem 3.1), it follows Refs. [12, 11]. This is of importance in manyaspects, e.g., it becomes evident that the procedures of detecting and measur-ing entanglement are closely connected to each other. Furthermore, as it isoften difficult to construct an entanglement witness, a method for construct-ing an entanglement witness is given. The chapter is organized as follows: InSec. 6.2 we start with geometrical considerations comparing Euclidean spacegeometry with Hilbert-Schmidt space geometry. Next, in Sec. 6.3 we derivea useful theorem related to the connection of the Hilbert-Schmidt measureto a so-called ‘generalized Bell inequality’1. Then we state a lemma concern-ing the detection of the nearest separable state to a given entangled state inSec. 6.4, and finally, in Sec. 6.5 we discuss examples.

6.2 Geometrical Considerations about the Hilbert-SchmidtDistance

We can write the Hilbert-Schmidt distance (5.39) of any two states ρ1, ρ2 ∈ Aas

dHS(ρ1, ρ2) = ‖ρ1 − ρ2‖ =

⟨ρ1 − ρ2,

ρ1 − ρ2

‖ρ1 − ρ2‖⟩

=⟨ρ1 − ρ2, C

⟩, (6.1)

where we define the operator

C :=ρ1 − ρ2

‖ρ1 − ρ2‖ . (6.2)

1 Mind that the ‘generalized Bell inequality’ introduced in this context is different fromthe general Bell inequality of Theorem 4.6. A more detailed discussion of the subject isgiven later on.

6. Hilbert-Schmidt Measure and Entanglement Witness 67

Fig. 6.1: Illustration of Eqs. (6.7) and (6.8): The scalar product 〈ρl − ρ1, ρ1− ρ2〉is negative because the projection (ρl − ρ1)‖ onto ρ1 − ρ2 points in theopposite direction to ρ1 − ρ2. On the other side, 〈ρr − ρ1, ρ1 − ρ2〉 ispositive for states ρr, because then the projection (ρr − ρ1)‖ points inthe same direction as ρ1 − ρ2.

Instead of C we may also choose C := C + c1 (c ∈ C) because we have

dHS(ρ1, ρ2) =⟨ρ1 − ρ2, C

⟩=

⟨ρ1 − ρ2, C

⟩+ 〈ρ1 − ρ2, c1〉 = 〈ρ1 − ρ2, C〉 ,

(6.3)where we used 〈ρ1 − ρ2,1〉 = Trρ1 − Trρ2 = 0 . For convenience2 we fix c to

c = −〈ρ1, ρ1 − ρ2〉‖ρ1 − ρ2‖ , (6.4)

and obtain

C =ρ1 − ρ2 − 〈ρ1, ρ1 − ρ2〉1

‖ρ1 − ρ2‖ . (6.5)

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

‖ρ1 − ρ2‖ 〈ρp − ρ1, ρ1 − ρ2〉 = 0 . (6.6)

For all states on one side of the plane, let us call them ‘left-hand’ states ρl,we have

1

‖ρ1 − ρ2‖ 〈ρl − ρ1, ρ1 − ρ2〉 < 0 , (6.7)

whereas the states on the other side, the ‘right-hand’ states ρr are given by

1

‖ρ1 − ρ2‖ 〈ρr − ρ1, ρ1 − ρ2〉 > 0 . (6.8)

2 This value of c will turn out to be useful.


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

〈ρ , C〉 =

⟨ρ ,

ρ1 − ρ2

‖ρ1 − ρ2‖⟩− 〈ρ1, ρ1 − ρ2〉

‖ρ1 − ρ2‖ 〈ρ ,1〉

=1

‖ρ1 − ρ2‖ 〈ρ− ρ1, ρ1 − ρ2〉 . (6.9)

Then the plane P is determined by

〈ρp , C〉 = 0 , (6.10)

and the ‘left-hand’ and ‘right-hand’ states satisfy the inequalities

〈ρl , C〉 < 0 and 〈ρr , C〉 > 0 . (6.11)

6.3 The Bertlmann-Narnhofer-Thirring Theorem

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

〈ρ,A〉 ≥ 0 ∀ρ ∈ S , (6.12)

a generalized Bell inequality. In this context ‘generalized’ means that it de-tects entanglement in the mathematical sense and not non-locality. Thus itdoes not serve as a criterion to determine if a state admits a LHV theory likethe 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 inequalitycan detect non-locality like usual Bell inequalities, but has a general form(arbitrary measurements, etc.; see, e.g., Refs. [70, 10]). As mentioned, thisgeneral Bell inequality (from Sec. 4.3.2) does not necessarily detect entangle-ment. Nevertheless, every entangled state violates the inequality (6.12) foran appropriate entanglement witness A.

We can re-write Eq. (3.3) as

〈ρ,A〉 − 〈ρent, A〉 ≥ 0 ∀ρ ∈ S . (6.13)

The maximal violation of the GBI is defined by

B(ρent) = maxA, ‖A−a1‖≤1

(minρ∈S

〈ρ,A〉 − 〈ρent, A〉)

, (6.14)


where the maximum is taken over all possible entanglement witnesses A,suitably normed3.

Interestingly, one can find connections between the Hilbert-Schmidt mea-sure and the concept of entanglement witnesses. In particular, there existsthe following equivalence stated in the Bertlmann-Narnhofer-Thirring Theo-rem [12]:

Theorem 6.1. The Hilbert-Schmidt measure of an entangled state equals themaximal 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 normis continuous and the set S is compact

minρ∈S

‖ρ− ρent‖ = ‖ρ0 − ρent‖ . (6.16)

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

dHS(ρ0, ρent) = D(ρent) = 〈ρ0, C〉 − 〈ρent, C〉 . (6.17)

In Eq. (6.17) the operator C has to be an optimal entanglement witnessfor the following reason: The state ρ0 lies on the boundary of the set of allseparable states S and the hyperplane defined by 〈ρp , C〉 = 0 is orthogonal toρ0−ρent. Because ρ0 is the nearest separable state to ρent the plane has to betangent 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

Aopt = C =ρ0 − ρent − 〈ρ0, ρ0 − ρent〉1

‖ρ0 − ρent‖ , (6.18)

which we use to rewrite the Hilbert-Schmidt measure (6.17)

D(ρent) = 〈ρ0, Aopt〉 − 〈ρent, Aopt〉 . (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) isnot an entanglement witness.


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

Since we havemax

A(−〈ρent, A〉) = −〈ρent, Aopt〉 , (6.20)

where A is restricted by ‖A− a1‖ ≤ 1 and 〈ρ0, Aopt〉 = 0 , we obtain

D(ρent) = 〈ρ0, Aopt〉 − 〈ρent, Aopt〉 = maxA, ‖A−a1‖≤1

(〈ρ0, A〉 − 〈ρent, A〉)

= maxA, ‖A−a1‖≤1

(minρ∈S

〈ρ, A〉 − 〈ρent, A〉)

= B(ρent) , (6.21)

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 tocalculate 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 thedistance (for specific procedures, see, e.g., Refs. [26, 82, 81]). However, wecan 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 uscall ρ the guess of the nearest separable state. From previous considerations(Eqs. (6.5), (6.6) and (6.10)) we know that the operator

C =ρ− ρent − 〈ρ, ρ− ρent〉1

‖ρ− ρent‖ (6.22)


Fig. 6.3: Illustration why C cannot be an entanglement witness if ρ is not thenearest separable state. The hatched area is the one were the condition〈ρ, C〉 ≥ 0 ∀ρ ∈ S is violated.

defines a hyperplane which is orthogonal to ρ− ρent and includes ρ. Now westate the following lemma:

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

Proof. We already know from Sec. 6.3 that if ρ is the nearest separable statethen the operator C is an entanglement witness. So we need to prove theopposite: If C is an entanglement witness the state ρ has to be the nearestseparable state ρ0. We prove it indirectly. If ρ is not the nearest separablestate then ‖ρent − ρ‖ does not give the minimal distance to S; the planedefined by 〈ρp, C〉 = 0 is not tangent to S and thus the existence of ‘left-hand’ separable states ρsep satisfying 〈ρsep, C〉 < 0 follows. That means Ccannot be an entanglement witness (inequalities (3.3) are not fulfilled), seeFig. 6.3.

Remark. Of course it is in general not easy to apply Lemma 6.1 to determinethe nearest separable state, since in general it is difficult to check whether theoperator 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 methodsto 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 isthe operator for which the GBI is maximally violated.


6.5 Examples

6.5.1 Isotropic Qubit States

In Ref. [12] the 2-qubit Werner state has been studied5 – here we considerthe isotropic state in 2 dimensions (acting on H2 ⊗ H2, it is obtained ford = 2 in Eq. (2.16))

ρα = α∣∣φ2

+

⟩ ⟨φ2

+

∣∣ +1− α

41 , −1

3≤ α ≤ 1 , (6.23)

where∣∣φ2

+

⟩:= |φ+〉 (2.33). In matrix notation in the standard product basis

(2.25) we get

ρα =

1 + α4 0 0 α

20 1− α

4 0 0

0 0 1− α4 0

α2 0 0 1 + α

4

, (6.24)

whereas in terms of the Pauli matrices basis (2.27) the state can be expressedby

ρα =1

4(1+ α Σ) , (6.25)

with the definition

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

We know that ρα is (recall Eq. (2.19))

for − 1

3≤ α ≤ 1

3separable , for

1

3< α ≤ 1 entangled . (6.27)

To compute the Hilbert-Schmidt measure (5.40) for an entangled isotropicstate ρent

α we need to calculate D(ρentα ) = minρ∈S ‖ρ− ρent

α ‖ , that is, we needto find the nearest separable state ρ0 to the entangled state in order to obtainD(ρent

α ) = ‖ρ0 − ρentα ‖ . From the separability condition (6.27) we see that the

state with α = 1/3 lies on the boundary between separable and entangledisotropic states. Thus our guess for all isotropic entangled qubit states is(and we call it ρ):

ρ = ρ1/3 =1

4

(1 +

1

3Σ

). (6.28)

5 We also calculated the Hilbert-Schmidt measure for the 2-qubit Werner state inSec. 5.3.4.


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

ρ− ρentα =

1

4

(1

3− α

)Σ with

∥∥ρ− ρentα

∥∥ =

√3

2

(α− 1

3

), (6.29)

(note that ‖Σ‖ = 2√

3) and

⟨ρ, ρ− ρent

α

⟩= Tr ρ(ρ− ρent

α ) =1

4

(1

3− α

). (6.30)

Then the operator C is explicitly given by

C =ρ− ρent

α − 〈ρ, ρ− ρentα 〉1

‖ρ− ρentα ‖ =

1

2√

3(1− Σ) . (6.31)

We need to examine that C is an entanglement witness, i.e., we check theinequalities (3.3). For the entangled state (where α > 1/3) we get

⟨ρent

α , C⟩

= Tr ρentα C = −

√3

2

(α− 1

3

)< 0 . (6.32)

So the first condition is satisfied. The second one, the positivity of 〈ρ, C〉 forall separable states ρ, can be seen in the following way: With notation (2.29)for ρsep the scalar product is

⟨ρsep, C

⟩=

∑

k

pk1

2√

3

(1− nk

xmkx + nk

ymky − nk

zmkz

),

√(nk

x)2 +

(nk

y

)2+ (nk

z)2 =:

∣∣~nk∣∣ ≤ 1,

∣∣~mk∣∣ ≤ 1 . (6.33)

We have to show that

−nkxm

kx + nk

ymky − nk

zmkz ≥ −1 , (6.34)

then the right-hand side of Eq. (6.33) remains always positive. (The convexsum 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,

−nkxm

kx + nk

ymky − nk

zmkz ≥ −nk

xmkx − nk

ymky − nk

zmkz = −~nk · ~mk ≥ −1 ,

(6.36)


which completes the proof that 〈ρ, C〉 ≥ 0 ∀ρ ∈ S. So C represents anentanglement witness

Aopt = C =1

2√

3(1− Σ) , (6.37)

and our guess for the nearest separable state was correct, ρ = ρ0 .

The Hilbert-Schmidt measure for the entangled isotropic state is deter-mined by Eq. (6.29),

D(ρentα ) =


∥∥ =

√3

2

(α− 1

3

). (6.38)

It only remains to check the Bertlmann-Narnhofer-Thirring Theorem 6.1.Thus we calculate the maximal violation B(ρent

α ) (6.14) of the GBI. Themaximum is attained for the optimal entanglement witness Aopt and theminimum for the nearest separable state ρ0 . Then Eq. (6.32) determines thevalue of B(ρent

α ) (recall that 〈ρ0, Aopt〉 = 0)

B(ρentα ) = − ⟨

ρentα , Aopt

⟩=

√3

2

(α− 1

3

). (6.39)

So, indeed D(ρentα ) = B(ρent

α ) , the Hilbert-Schmidt measure equals the max-imal violation of the GBI.

6.5.2 Isotropic Qutrit States

Eq. (2.16) defines the isotropic qutrit state for d = 3, i.e.

ρα = α∣∣φ3

+

⟩ ⟨φ3

+

∣∣ +1− α

91 , −1

8≤ α ≤ 1 , (6.40)

where ∣∣φ3+

⟩=

1√3

(|0〉 ⊗ |0〉+ |1〉 ⊗ |1〉+ |2〉 ⊗ |2〉

). (6.41)

In matrix notation in the standard product basis (2.42) we have

ρα =

1+2α9

0 0 0 α3

0 0 0 α3

0 1−α9

0 0 0 0 0 0 00 0 1−α

90 0 0 0 0 0

0 0 0 1−α9

0 0 0 0 0α3

0 0 0 1+2α9

0 0 0 α3

0 0 0 0 0 1−α9

0 0 00 0 0 0 0 0 1−α

90 0

0 0 0 0 0 0 0 1−α9

0α3

0 0 0 α3

0 0 0 1+2α9

. (6.42)


In the Gell-Mann matrices representation (2.43) the state ρα can be expressedas (see also Ref. [20] 6)

ρα =1

9

(1+

3α

2Λ

), (6.43)

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

−18 ≤ α ≤ 1

4 ⇒ ρα separable ,

14 < α ≤ 1 ⇒ ρα entangled .

(6.45)

By the same argument as in the qubit case we guess the nearest separablestate to the state (6.43),

ρ = ρ1/4 =1

9

(1+

3

8Λ

). (6.46)

Again, to check our guess we examine if the operator C (6.22) is an entan-glement witness. We need the following expressions

ρ− ρentα =

1

6

(1

4− α

)Λ with

∥∥ρ− ρentα

∥∥ =2√

2

3

(α− 1

4

), (6.47)

(where ‖Λ‖ = 4√

2) and

⟨ρ, ρ− ρent

α

⟩= Tr ρ(ρ− ρent

α ) =2

9

(1

4− α

). (6.48)

Then C (6.22) is explicitly given by

C =1

3√

2

(1− 3

4Λ

). (6.49)

Now let us check the entanglement witness conditions (3.3) for C:

⟨ρent

α , C⟩

= Tr ρentα C = −2

√2

3

(α− 1

4

)< 0 . (6.50)

6 In Ref. [20] the projectors |i〉〈j| (i, j = 0, 1, 2) are expressed as linear combinations ofthe Gell-Mann matrices to obtain this form of the isotropic state ρα.


So the first condition is satisfied since α > 1/4; for the second one we obtain(with help of Eq. (2.44))

⟨ρsep, C

⟩=

∑

k

pk1

3√

2

(1 − nk

1mk1 + nk

2mk2 − nk

3mk3 − nk

4mk4 + nk

5mk5

− nk6m

k6 + nk

7mk7 − nk

8mk8

),∣∣~nk

∣∣ ≤ 1,∣∣~mk

∣∣ ≤ 1 . (6.51)

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

−nk1m

k1 + nk

2mk2 − nk

3mk3 − nk

4mk4 + nk

5mk5 −

− nk6m

k6 + nk

7mk7 − nk

8mk8

≥ −~nk · ~mk ≥ −1 , (6.52)

so that 〈ρsep, C〉 ≥ 0 . Indeed, C represents an entanglement witness and weidentify

Aopt = C =1

3√

2

(1− 3

4Λ

)and ρ = ρ0 . (6.53)

With Eq. (6.47) the Hilbert-Schmidt measure is

D(ρentα ) =


∥∥ =2√

2

3

(α− 1

4

), (6.54)

and by the same argumentation as for qubits the maximal violation B(ρentα )

(6.14) of the GBI is determined by Eq. (6.50)


ρentα , Aopt

⟩=

2√

2

3

(α− 1

4

). (6.55)

So again, D(ρentα ) = B(ρent

α ) , we see that the Bertlmann-Narnhofer-ThirringTheorem 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 qutritresults to arbitrary dimensions. A general state on Hd can be written in a

matrix basis{1, γ1, . . . , γd2−1

}as

ω =1

d

(1+

√d(d− 1)

2ni γ

i

),

∑i

n2i =: |~n|2 ≤ 1 . (6.56)


We have included the factor√

d(d−1)2

for the correct normalization (Trω2 ≤ 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 separablestates is a straight forward extension to Eqs. (2.29) and (2.44)

ρsep =∑

k

pk1

d2

(1⊗ 1 +

√d(d− 1)

2nk

i γi ⊗ 1

+ ,

√d(d− 1)

2mk

i 1⊗ γi +d(d− 1)

2nk

i mkj γi ⊗ γj

). (6.58)

We express a d-dimensional isotropic state (on Hd ⊗Hd) – a generalizationof the isotropic qubit state (6.25) and qutrit state (6.43) – as

ρα =1

d2

(1 +

d

2α Γ

), − 1

d2 − 1≤ α ≤ 1 , (6.59)

where we define

Γ :=d2−1∑i=1

ci γi ⊗ γi , ci = ±1 . (6.60)

The factor d2

in Eq. (6.59) is due to normalization. The splitting of ρα intoentangled 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 thebeginning, 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 forall 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 densitymatrices according to the standard basis notation.

In this way a generalization of our previous results is possible and canbe 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 asbefore, we find the following expressions for the nearest separable state ρ0,the Hilbert-Schmidt measure D(ρent

α ) and the optimal entanglement witnessAopt :

ρ0 = ρ 1d+1

=1

d2

(1 +

d

2(d + 1)Γ

), (6.61)


D(ρentα ) =


∥∥ =

√d2 − 1

d

(α − 1

d + 1

), (6.62)

Aopt =d− 1

d√

d2 − 1

(1 − d

2(d− 1)Γ

). (6.63)

The maximal violation of the GBI gives


ρentα , Aopt

⟩=

√d2 − 1

d

(α − 1

d + 1

), (6.64)

thus we see that again D(ρentα ) = B(ρent

α ) and Theorem (6.1) is satisfied.

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

7. TRIPARTITE SYSTEMS

7.1 Introduction

So long we have only been viewing bipartite systems. Foundations of quan-tum information and much of its theory nowadays is related to bipartite sys-tems where much has been achieved. Nevertheless, tripartite or multipartitesystems 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: WhereBell’s Theorem says that quantum mechanics contradicts LHV theories viaexpectation values, the GHZ theorem predicts a contradiction that can inprinciple be verified with only one experiment with spin measurements inthe same direction (Bell’s Theorem does not exhibit a contradiction for mea-surements in the same direction). Of course, tri- and multipartite systemsare 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 entan-glement is in this case defined as a straight forward extension of the bipartitecase. It is essentially different that for tripartite states there exists a a pri-ori classification of entanglement (apart from the question of distillabilityand admission of a LHV theory, discussed in Chapter 4). First steps towardthis revelation were done in Refs. [9, 75]; in Ref. [30] an exact description of3-qubit entanglement for pure states was introduced.

The chapter is organized as follows: First, in Sec. 7.2 we give a basicmathematical description of tripartite systems. In Sec. 7.3 we examine aclassification of pure entangled tripartite states, whereas in Sec. 7.4 we discussgeneral (pure or mixed) states. In this context we introduce the concept oftripartite 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 HilbertspacesHdi of dimension di, so that the whole system is described by a Hilbert

7. Tripartite Systems 80

space Hd1A ⊗Hd2

B ⊗Hd3C . 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= 〈emA

⊗ fmB⊗ gmC

| ρ |enA⊗ fnB

⊗ gnC〉 . (7.1)

Here {ei}, {fi} and {gi} are bases of the Hilbert spaces Hd1A , Hd2

B and Hd3C .

Reduced Density Matrices. The reduced density matrices are obtained bytracing out subsystems. If one subsystem is traced out, the resulting densitymatrix is a bipartite one. By again tracing out another subsystem we obtain aone 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:

(ρA)mA nA=

d3∑c=1

d2∑

b=1

ρmA b c , nA b c ,

(ρB)mB nB=

d3∑c=1

d1∑a=1

ρa mB c , a nB c ,

(ρC)mC nC=

d2∑

b=1

d1∑a=1

ρa b mC , a b nC. (7.3)

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

|ψprod〉 = |ψA〉 ⊗ |ψB〉 ⊗ |ψC〉 . (7.4)

Such a state that is not entangled is called ‘product’ state.


General Definition of Entanglement. In a quite similar way as we definedthe separability of general (pure or mixed) bipartite states we can define theseparability (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.

ρ =∑

i

pi ρiA ⊗ ρi

B ⊗ ρiC , 0 ≤ pi ≤ 1,

∑i

pi = 1 . (7.5)

All separable states are the elements of the set of separable states S. If astate 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

ρ =

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

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 = |0〉, e2 =f2 = g2 = |1〉 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 densitymatrices (7.2), (7.3): The state is a product state if all reduced densitymatrices 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 |ψ〉 can be converted into the state |φ〉 with some

non vanishing probability if a local operator A⊗B ⊗ C exists such that

|ψ〉 → |φ〉 = A⊗B ⊗ C |ψ〉 . (7.7)

According to the following theorem all pure tripartite states can be dividedinto different equivalence classes:


Theorem 7.1. Two pure tripartite states |ψ〉, |φ〉 belong to the same equiv-alence class if there exists a local operator A⊗ B ⊗ C that relates the statesin the following way:

|ψ〉 → |φ〉 = A⊗B ⊗ C |ψ〉 and

|φ〉 → |ψ〉 = A−1 ⊗B−1 ⊗ C−1 |ψ〉 . (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 trans-forms one state into another with some nonzero probability can be expressedwith an operator A⊗B⊗C. Furthermore, if and only if there exists an ILOrelating two pure states, then the states can be obtained from each otherwith some nonzero probability via LOCC. So in general we can say that allstates that can be obtained from each other via LOCC (with some nonzeroprobability) 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 be-ginning of this section that a tripartite pure state is a product state ifall of its reduced density matrices are pure states. By local unitary op-erations any product state |ψprod〉 = |ψA〉⊗|ψB〉⊗|ψC〉 can be obtainedfrom the state

|prod〉 = |0〉 ⊗ |0〉 ⊗ |0〉 (7.9)

and the other way round, thus there always exists an ILO that relatestwo product states in the way of Theorem 7.8, so that all product statesbelong to the same equivalence class. We say that the state |prod〉 (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 theother two are mixed states, then we speak of tripartite states withbipartite entanglement. This means that only two parts of the systemare entangled with each other, and these form a product state with theremaining part. Any such pure state can be reversibly transformed bya suitable unitary transformation into the state

|ψbip〉 = |0〉 ⊗ (α |0〉 ⊗ |0〉+ β |1〉 ⊗ |1〉) α, β ∈ R, α2 + β2 = 1 .(7.10)


The above state can be obtained with a suitable ILO from the state(and the other way round)

|bip〉 =1√2|0〉 ⊗ (|0〉 ⊗ |0〉+ |1〉 ⊗ |1〉) , (7.11)

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 tothe same equivalence class.

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

(a) GHZ Class. All states of this equivalence class can be written asa sum of only two product terms. Such states can be obtainedfrom or transformed into the state (with a suitable local unitarytransformation)

|ψGHZ〉 = γ(α |0〉 ⊗ |0〉 ⊗ |0〉+ βeiφ |φA〉 ⊗ |φB〉 ⊗ |φC〉

)

α, β ∈ R, α2 + β2 = 1, 0 ≤ φ < 2π , (7.12)

where γ is a normalization factor and the states |φA〉, |φB〉, and|φC〉 are arbitrary superpositions of the states |0〉 and |1〉. Thereexists an ILO [30] relating the state |ψGHZ〉 with the representativestate

|GHZ〉 =1√2

(|0〉 ⊗ |0〉 ⊗ |0〉+ |1〉 ⊗ |1〉 ⊗ |1〉) (7.13)

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

|ψW〉 = a |0〉 ⊗ |0〉 ⊗ |1〉+ b |0〉 ⊗ |1〉 ⊗ |0〉+

+ c |1〉 ⊗ |0〉 ⊗ |0〉+ d |0〉 ⊗ |0〉 ⊗ |0〉 (7.14)

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

|ψW〉 = a |0〉 ⊗ |0〉 ⊗ |1〉+ b |0〉 ⊗ |1〉 ⊗ |0〉+

+ (c |1〉+ d |0〉)⊗ |0〉 ⊗ |0〉 (7.15)


The representative of this class is the state

|W〉 =1√3

(|0〉 ⊗ |0〉 ⊗ |1〉+ |0〉 ⊗ |1〉 ⊗ |0〉+ |1〉 ⊗ |0〉 ⊗ |0〉)(7.16)

since it is related to the state |ψW〉 (7.14) with a suitable ILO [30].

How do we know that there do not exist any ILOs relating states of differentequivalent classes (i) - (iii)? First, it can be seen [30] that any possible ILOcannot change the rank of the reduced density matrices1, which at once im-plies that class (i) and (ii) are inequivalent. Additionally, any ILO conservesthe number of product terms needed to express a state2. Therefore the GHZand W class are inequivalent.

Possible State Transformations under General LOCC. The equivalenceclasses (i) - (iii) are invariant under reversible LOCC (i.e. under ILOs).But what about non invertible local operators? It is shown in Ref. [30] thatif an operator A ⊗ B ⊗ C is invertible, then at least one of the operatorsA, B, and C must have rank 1. So for states belonging to the GHZ or Wclass, 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 anyLOCC that transforms a GHZ class state into a W class state - but thereare non invertible operations that transform a GHZ class or W class stateinto a A-BC, B-CA, C-AB or A-B-C class state. Furthermore, noninvertibleoperations can transform a A-BC, B-CA, or C-AB class state into a A-B-Cclass state (see Tab. 7.1).

Entanglement Measures for Pure Tripartite States. What entanglement mea-sures can be used to quantify the entanglement of tripartite states? Onemeasure is the entropy measure, introduced in Sec. 5.2. Since this measure isdefined for bipartite states, we can interpret a tripartite state as a bipartiteone with, e.g., ‘A’ being one part of the system and ‘B,C’ being the otherpart. In this manner we can calculate the von Neumann entropy (3.22) ofthe reduced density matrix ρA and similarly for the other reduced densitymatrices. 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 vanishingeigenvalues. So the reduced density matrices can have rank 2, which corresponds to amixed 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 inde-pendent since the operator has to be invertible (A−1 etc. exists).


Class LOCC into other classes SA SB SC τ

A-B-C - 0 0 0 0A-BC A-B-C 0 > 0 > 0 0C-AB A-B-C > 0 > 0 0 0B-AC A-B-C > 0 0 > 0 0GHZ A-B-C, A-BC, B-CA, C-AB > 0 > 0 > 0 > 0W A-B-C, A-BC, B-CA, C-AB > 0 > 0 > 0 0

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

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

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

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

Example 1. The representative of the GHZ class is the state |GHZ〉 (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 ranksare 1 we have class (i), if two ranks are 2 and one is 1 we have class (ii), and if all ranksare 2 we have class (iii).


Eqs. (7.1), (7.6)

ρGHZ := |GHZ〉〈GHZ| =

12

0 0 0 0 0 0 12

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 012

0 0 0 0 0 0 12

. (7.17)

If we trace out only one subsystem, the resulting 2-qubit reduced densitymatrices are

ρGHZAB = ρGHZ

BC = ρGHZAC =

12

0 0 00 0 0 00 0 0 00 0 0 1

2

. (7.18)

These are separable mixed states, since the above matrix is invariant underpartial transposition (see Theoreom 3.6). So we notice that if tracing outone subsystem of the state |GHZ〉 we obtain a separable state. The reduceddensity matrices (7.2), (7.3) for each part of the system are

ρGHZA = ρGHZ

B = ρGHZC =

(12

00 1

2

). (7.19)

The reduced density matrices (7.19) all correspond to mixed states, sinceTr(ρGHZ

A )2 = 1/2 < 1 (or, equivalently, since the matrix is of rank 2), whichis in agreement with the state being truly tripartite entangled.

Example 2. The representative of the W class is the state |W〉 (7.16). Inmatrix notation (7.1), (7.6) this state becomes

ρW := |W〉〈W〉 =

0 0 0 0 0 0 0 00 1/3 1/3 0 1/3 0 0 00 1/3 1/3 0 1/3 0 0 00 0 0 0 0 0 0 00 1/3 1/3 0 1/3 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

. (7.20)


By tracing out one subsystem we obtain the 2-qubit reduced density matrices

ρWAB = ρW

BC = ρWAC =

1/3 0 0 00 1/3 1/3 00 1/3 1/3 00 0 0 0

. (7.21)

The partial transposition of the matrices (7.21) is no longer positive becauseone eigenvalue is negative (1/6(1−√5)), and so the 2-qubit reduced densitymatrices (7.21) describe entangled mixed states - contrary to the GHZ casein the last example.

All one particle reduced density matrices are mixed states, since we have

ρWA = ρW

B = ρWC =

(2/3 00 1/3

), (7.22)

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 purestate 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 isactually a set of states with special properties (for a graphical illustrationsee Fig. 7.1):

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

(ii) Class B is the set of all states that can be written as a convex com-bination of pure product states and/or pure states that contain onlybipartite entanglement.

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

(iv) Class GHZ is the set of all states.


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 classesare convex and compact sets [1]. Any invertible local operation on a stateρ of class X \ Y (where X can be B, W or GHZ and Y is the ‘lower’ classof 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 outsidethe set X \ Y 4. The class of a general state is invariant under LOCC [1] andcan additionally ‘obtain’ only lower classes (e.g. for a state ρ ∈ GHZ andρ ∈ GHZ \W we have ρ /∈ B, but after a suitable LOCC the state can betransformed 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 ofthe Hahn-Bahnach Theorem we can state the following theorem, in the sameway as we did in connection with the entanglement witness Theorem 3.1 inSec. 3.3.1:

Theorem 7.2. A tripartite state σ is an element of GHZ \X (where X canbe 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

AtriW =

2

31− |W〉〈W| (7.24)

4 This is because we express any state as a convex combination of pure states, and theclass of each pure state is invariant under invertible local operations.

8. CONCLUSION

This work presents an overview of results achieved on the subject of de-tecting, classifying, and quantifying entanglement. Methods of detectingentanglement are divided into nonoperational and operational separabilitycriteria, where the later provide a ‘recipe’ for the detection of entanglementbut the former do not. These criteria give fundamental insight into the the-ory of entanglement; in particular a connection with a way of quantifyingentanglement is discussed (between the concept of entanglement witnessesand the Hilbert-Schmidt measure). Furthermore two possibilities to classifyentanglement and several methods to quantify entanglement are treated.

For lower dimensional bipartite systems the characterization of entan-glement is more or less complete, since there exist very useful operationalmethods to detect and quantify entanglement (i.e. the PPT and reductioncriterion and the concurrence). For higher dimensional systems, however,there are no such simple methods, although for a lot of states entanglementis successfully detected, classified, and quantified.

The characterization of entanglement for systems of more than two par-ticles is in general difficult to perform. The situation is different to the twoparticle case because there exist a priori different ‘kinds’ of entanglement.For lower dimensions and few particles some operational methods to classifyand quantify entanglement have been found, but not for the detection ofmultipartite entanglement.

It is therefore of great interest to extend and deepen the understandingof entanglement beyond 2 particles and/or low dimensions, not only due toa sheer mathematical interest, but also for a successful future developmentof quantum information theory and applications.

LIST OF FIGURES

3.1 Geometric illustration of a plane in Euclidean space and thedifferent values of the scalar product for states above (~bu),

within (~bp) and under (~bd) the plane. . . . . . . . . . . . . . . 183.2 Illustration of an optimal entanglement witness . . . . . . . . 193.3 Plot of S1, S2 as functions of the parameter p and intersections

with the entropies of the reduced density matrices Sred = 1 . . 243.4 Comparison of the information gained about the Werner state

ρα with 3 different separability criteria: 2 entropy inequalitiesand the CHSH inequality . . . . . . . . . . . . . . . . . . . . . 25

3.5 Comparison of the PPT criterion with other separability crite-ria for the 2-qubit Werner state ρα: The PPT criterion clearlydistinguishes between separable and entangled states and givesa wider range of entanglement that the other criteria. . . . . . 28

4.1 Plot of the fidelity g(F ) of the distilled state ρ′ . . . . . . . . . 354.2 Illustration of entanglement and distillability. Since all entan-

gled 2-qubit states are distillable and NPT, we have a cleardistinction in this case. For general states, however, there areentangled PPT states (bound entangled) and maybe boundentangled NPT states, which are those outside the “box” ofthe free entangled states. Note that this is not a geometricrepresentation 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 asingle point but for many different values of x1 and y1. . . . . 41

4.4 Illustration of the various properties of the state ρβ (4.14). Thequestion mark says that in this area we do not have enoughinformation, we only know that for 0 ≤ β < 1 the state isNPT and therefore entangled. . . . . . . . . . . . . . . . . . . 42

5.1 Plot of entanglement of formation EF as a function of concur-rence 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 theWerner state ρα for values of α where ρα is entangled . . . . . 62

5.4 Plot of the Hilbert-Schmidt measure DHS(ρα) of the Wernerstate ρα for values of α where ρα is entangled . . . . . . . . . 63

5.5 Comparison of the concurrence C(ρα), the entanglement of for-mation EF(ρα), the relative entropy of entanglement DRE(ρα)and the ‘normalized’ Hilbert-Schmidt measure DHSN(ρα) ofthe 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 〈ρl −ρ1, ρ1 − ρ2〉 is negative because the projection (ρl − ρ1)‖ ontoρ1 − ρ2 points in the opposite direction to ρ1 − ρ2. On theother side, 〈ρr − ρ1, ρ1 − ρ2〉 is positive for states ρr, becausethen the projection (ρr − ρ1)‖ points in the same direction asρ1 − ρ2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2 Illustration of the Bertlmann-Narnhofer-Thirring Theorem . . 706.3 Illustration why C cannot be an entanglement witness if ρ is

not the nearest separable state. The hatched area is the onewere the condition 〈ρ, C〉 ≥ 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 transfor-mations via LOCC into other classes, the von Neumann en-tropies of the reduced density matrices SA, SB, SC , and the3-tangle τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

BIBLIOGRAPHY

[1] A. Acın, D. Bruß, M. Lewenstein, and A. Sanpera, Classification ofmixed three-qubit states, Phys. Rev. Lett. 87 (2001), 040401.

[2] Arvind, K. S. Mallesh, and N. Mukunda, A generalized Pancharatnamgeometric 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, CambridgeUniversity Press, 1987.

[5] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K.Wootters, Teleporting an unknown quantum state via dual classical andEinstein-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 oper-ations, Phys. Rev. A 53 (1996), 2046.

[7] , Purification of noisy entanglement and faithful teleportation vianoisy 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. A54 (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 entan-glement, quant-ph/9908073.

[10] R. A. Bertlmann and B. C. Hiesmayr, Bell inequalities for entangledkaons 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 ofentanglement and Bell inequalities, Phys. Rev. A 66 (2002), 032319.

[13] R. A. Bertlmann and A. Zeilinger (eds.), Quantum [un]speakables, fromBell 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 viadual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett.80 (1998), 1121.

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

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

[18] G. Brassard, R. Cleve, and A. Tapp, Cost of exactly simulating quantumentanglement 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 objectivelocal theories, Phys. Rev. D 10 (1974), 526.

[23] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposedexperiment 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-Zeilingerstates by selective information manipulation, Phys. Rev. Lett 84 (2000),5908.

[26] J. Dehaene, F. Verstraete, and B. De Moor, On the geometry of entangledstates, 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 cryp-tography 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 partialtranspose, Phys. Rev. A 61 (2000), 062312.

[29] W. Dur, J. I. Cirac, M. Lewenstein, and D. Bruß, Distillability and par-tial transposition in bipartite systems, Phys. Rev. A 61 (2000), 062313.

[30] W. Dur, G. Vidal, and J. I. Cirac, Three qubits can be entangled in twoinequivalent ways, Phys. Rev. A 62 (2000), 062314.

[31] A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical de-scription 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 com-munication 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 Physics14 (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’sTheorem without inequalities, Am. J. of Phys. 58 (1990), 1131.

[37] D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going beyond Bell’sTheorem, 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 wit-ness operators and local measurements, J. Mod. Opt. 50 (2003), 1079.

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

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

[41] , Quantum α-entropy inequalities: independent condition for lo-cal 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 asinglet 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 ModernPhysics, vol. 173, Springer Verlag Berlin, 2001, p. 151.

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

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

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

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

[50] G. Lindblad, Expectations and entropy inequalities for finite quantumsystems, 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 AcademicPublishers, 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 quan-tum puzzle, Brit. J. Phil. Sci. 45 (1994), 95.

[61] A. O. Pittenger and M. H. Rubin, Geometry of entanglement witnessesand 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 willkurlicher Funktionen nach Systemenvorgeschriebener, Math. Ann. 63 (1907), 433.

Bibliography 99

[68] E. Schrodinger, Die gegenwartige Situation 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 purificationprocedures, Phys. Rev. A 57 (1998), 1619.

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

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

[75] G. Vidal, W. Dur, and J. I. Cirac, Reversible combination of inequivalentkinds of multipartite entanglement, Phys. Rev. Lett. 85 (2000), 658.

[76] R. F. Werner, Quantum states with Einstein-Podolsky-Rosen correla-tions 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 positivepartial transpose, Phys. Rev. A 61 (2000), 062102.

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

[79] W. K. Wootters, Entanglement of formation of an arbitrary state of twoqubits, 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. A60 (1998), 3496.

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

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