8.1 Introduction

Quantum theory, formalized in the first few decades of the twentieth century, contains elements that are radically different from the classical description of Nature. An important aspect in these fundamental differences is the existence of quantum correlations in the quantum formalism. In the classical description of Nature, if a system is formed by different subsystems, complete knowledge of the whole system implies that the sum of the information of the subsystems makes up the complete information for the whole system. This is no longer true in the quantum formalism. In the quantum world, there exist states of composite systems for which we might have the complete knowledge, while our knowledge about the subsystems might be completely random. In technical terms, one can have pure quantum states of a two‐party system, whose local states are completely mixed. One may reach some paradoxical conclusions if one applies a classical description to states that have characteristic quantum signatures.

During the last two decades, it has been realized that these fundamentally nonclassical states, also denoted as “entangled states,” can provide us with something else than paradoxes. They may be used to perform tasks that cannot be achieved with classical states. As benchmarks of this turning point in our view of such nonclassical states, one might mention the spectacular discoveries of (entanglement‐based) quantum cryptography (1991) (1), quantum dense coding (1992) (2), and quantum teleportation (1993) (3).

In this chapter, we consider both bipartite and multipartite composite systems. We define formally what entangled states are, present some important criteria to discriminate entangled states from separable ones, and show their classification according to their capability to perform some precisely defined tasks. Our knowledge in the subject of entanglement is still far from complete, although significant progress has been made in the recent years and very active research is currently underway. We will consider multipartite quantum states (states of more than two parties) in Section 8.9, until then, we consider only bipartite quantum states.

8.2 Bipartite Pure States: Schmidt Decomposition

Consider a bipartite system in a shared pure state. The two parties in possession of the system are traditionally denoted as Alice (A) and Bob (B), who can be located in distant regions. Let Alice's physical system be described by the Hilbert space and that of Bob by . Then the joint physical system of Alice and Bob is described by the tensor product Hilbert space .

An example of an entangled state is the well‐known singlet state , where and are two orthonormal states. Operationally, product states correspond to those states that can be locally prepared by Alice and Bob at two separate locations. Entangled states can, however, be prepared only after the particles of Alice and Bob have interacted either directly or by means of an ancillary system. The second option is necessary due to the existence of the phenomenon of entanglement swapping (4). A very useful representation, only valid for pure bipartite states, is the, so‐called, Schmidt representation.

The positive numbers are known as the Schmidt coefficients and the vectors as the Schmidt vectors of . Note that product pure states correspond to those states, whose Schmidt decomposition has one and only one Schmidt coefficient. If the decomposition has more than one Schmidt coefficients, the state is entangled. Note that the squares of the Schmidt coefficients of a pure bipartite state are the eigenvalues of both the reduced density matrices ( ) and ( ) of . The last fact gives us an easy method to find the Schmidt coefficients and the Schmidt vectors.

8.3 Bipartite Mixed States: Separable and Entangled States

As discussed in the last section, the question whether a given pure bipartite state is separable or entangled is straightforward. One has just to check if the reduced density matrices are pure. This condition is equivalent to the fact that a bipartite pure state has a single Schmidt coefficient. The determination of separability for mixed states is much harder, and currently lacks a complete answer, even in composite systems of dimension as low as .

To reach a formal definition of separable and entangled states, consider the following preparation procedure of a bipartite quantum state between Alice and Bob. Suppose that Alice prepares her physical system in the state and Bob prepares his physical system in the state . Then, the combined state of their joint physical system is given by

8.2

We now assume that they can communicate over a classical channel (a phone line, for example). Then, whenever Alice prepares the state ( ), which she does with probability , she communicates that to Bob, and correspondingly Bob prepares his system in the state ( ). Of course, and . The state that they prepare is then

8.3

The important point to note here is that the state displayed in Eq. 8.3 is the most general state that Alice and Bob will be able to prepare by local quantum operations and classical communication (LOCC) (5). In LOCC protocols, two parties Alice and Bob perform local quantum operations separately in their respective Hilbert spaces, and they are allowed to communicate classical information about the results of their local operations. Let us make the definition somewhat more formal.

Local Operations and Classical Communication ( LOCC ). Suppose Alice and Bob share a quantum state defined on the Hilbert space . Alice performs a quantum operation on her local Hilbert space , using a complete set of complete general quantum operations , satisfying , and sends her measurement result to Bob via a classical channel. Depending on the measurement result of Alice, Bob operates a complete set of general quantum operations , satisfying on his part belonging to the Hilbert space . This joint operation along with the classical communication is called one‐way LOCC. Furthermore, Bob can send his result to Alice, and she can choose another set of local operations , satisfying , according to Bob's outcome. They can continue this process as long as required, and the entire operation is termed as LOCC, or two‐way LOCC. The operators and are the identity operators on and , respectively.

Entangled states cannot be prepared by two parties if only LOCC is allowed between them. To prepare such states, the physical systems must be brought together to interact.⁰¹

The question whether a given bipartite state is separable or not turns out to be quite complicated. Among the difficulties, we note that for an arbitrary state , there is no stringent bound on the value of in Eq. 8.3, which is only limited by the Caratheodory theorem to be with (see (6,7)). Although the general answer to the separability problem still eludes us, there has been significant progress in recent years, and we will review some such directions in the following sections.

8.4 Operational Entanglement Criteria

In this section, we introduce some operational entanglement criteria for bipartite quantum states. In particular, we discuss the partial transposition criterion (8,9), the majorization criterion (10), the cross‐norm or realignment criterion (11–13), and the covariance matrix criterion (14,15). There exist several other criteria (see, e.g., Refs. (16–21)), which will not be discussed here. However, note that, up to now, a necessary and sufficient criterion for detecting entanglement of an arbitrary given mixed state is still lacking.

8.4.1 Partial Transposition

A similar definition exists for the partial transposition of with respect to Bob's subsystem. Note that . Although the partial transposition depends upon the choice of the basis in which is written, its eigenvalues are basis independent. We say that a state has positive partial transposition (PPT), whenever , that is, the eigenvalues of are nonnegative. Otherwise, the state is said to be nonpositive under‐partial transposition (NPT).

The partial transposition criterion, for detecting entanglement is simple: Given a bipartite state , find the eigenvalues of any of its partial transpositions. A negative eigenvalue immediately implies that the state is entangled. Examples of states for which the partial transposition has negative eigenvalues include the singlet state.

The partial transposition criterion allows to detect in a straightforward manner all entangled states that are NPT states. This is a huge class of states. However, it turns out that there exist PPT states, which are not separable, as pointed out in Ref. (6) (see also (22)). Moreover, the set of PPT entangled states is not a set of measure zero (23). It is, therefore, important to have further independent criteria of entanglement detection, which permits to detect entangled PPT states. It is worth mentioning here that entangled PPT states form the only known examples of the “bound entangled states” (see Refs. ( 22,24) for details). Bound entangled states of bipartite quantum states are the states that cannot be distilled, that is, converted into singlet states under LOCC (25,26), with other entangled states being distillable. We will talk about distillation of quantum states later in this chapter in a bit more detail. Although as yet not found, it is conjectured that there also exist NPT bound entangled states (24). Note also that both separable and PPT states form convex sets. Figure 8.1 depicts the structure of the state space with respect to the partial transposition criteria and distillability.

Theorem 8.2 is a necessary condition of separability in any arbitrary dimension. However, for some special cases, the partial transposition criterion is both a necessary and a sufficient condition for separability:

As mentioned above, PPT bound entangled states exist. However, as Theorem 8.3 shows, they can exist only in dimensions higher than and .

8.4.3 Cross‐Norm or Matrix Realignment

The cross‐norm or matrix realignment criterion ( 11– 13) provides another way to delineate separable and entangled states, and more importantly, can successfully detect various PPT entangled states. There are various ways to formulate this criterion. Here, we present a formulation given in Ref. (13) as Corollary 18.

A density matrix on a Hilbert space , where and are the dimensions of the Hilbert spaces and respectively, can be written as

8.11

We have used the same notations as in Eq. 8.4, except that we have added zeros to the tensor , so that the indices run until the dimensions of the Hilbert spaces. and are complete sets of orthonormal Hermitian operators on the Hilbert spaces and respectively, with and . Without loss of generality, we assume that . After singular value decomposition of the matrix , we have

8.12

where and are and dimensional unitary matrices respectively, and is a dimensional diagonal matrix. Denoting the th column vector of and by and , the above expression becomes

8.13

where are the diagonal elements of . So, we have the matrix elements of as

8.14

If and are the matrix representations of and in and basis respectively, then using Eqs. 8.11 and 8.14, we obtain

8.15

Equation. 8.15 can be interpreted as the Schmidt decomposition of the density matrix in operator space, where the singular values are real and nonnegative. The cross‐norm or realignment criterion of separability is given by the following theorem:

8.4.4 Covariance Matrix

There exist several other operational criteria in the literature to detect whether a quantum state is separable or entangled ( 16– 21). We conclude this section by briefly illustrating one such separability criteria, known as the covariance matrix criterion ( 14, 15), which provides a general framework to link and understand several existing criteria including the cross‐norm or realignment criterion. Like the cross‐norm or realignment criterion, this method can identify entangled state for which the partial transposition criterion fails. Before delving into the theory of the covariance matrix criterion, let us first discus the definition and properties of the covariant matrices.

8.4.4.1 Definition and Properties

The complete set of orthonormal observables has to satisfy the Hilbert–Schmidt orthonormality condition . One example for such a set of observables for the case of single qubits in terms of the Pauli matrices, can be given by

8.19

In general, for the ‐dimensional case, one can consider the following matrices to form the complete set of orthonormal observables:

8.20

8.21

8.22

Let us now focus on the situation in which the Hilbert space is a tensor product of Hilbert spaces of two subsystems and with dimensions and , respectively. We can consider the complete set of orthonormal observables in as and in as , and construct a set of observables as . Although this set is not complete, it can be utilized to define a very useful form of covariant matrices, known as the block covariant matrices. The block covariant matrix for a given bipartite state and orthonormal observables is defined as follows.

Similarly, we can define the symmetric version of the block covariance matrix by replacing and with their symmetrized counterparts, while keeping unchanged. Clearly, if is a product state, then its block covariant matrix reduces to the block diagonal form, , as become zero .

If is a pure state on a ‐dimensional Hilbert space, then the corresponding covariance matrix satisfies the following properties:

1. The rank of is equal to .
2. The nonzero eigenvalues of are equal to 1, hence .
3. .

The corresponding symmetric covariance matrix satisfies the following:

1. The rank of is equal to .
2. The nonzero eigenvalues of are equal to 1/2, hence .

For mixed state on a ‐dimensional Hilbert space, we have , and the same for . The covariance matrix (symmetric and nonsymmetric) also satisfies the concavity property, that is, if is a convex combination of states , then

8.24

8.4.4.2 Covariance Matrix Criterion separability"?> for Separability

Clearly, it is not evident from Theorem 8.7 that the covariance matrix criterion leads to an efficient and physically plausible operational indication for separability. The main problem is to identify possible and , as this requires an optimization over all pure state decompositions of . Therefore, we now focus on the cases where the above criterion can be used efficiently to identify entangled states, by stating several corollaries of the above theorem.

Now we will give another operational entanglement criterion based on the Schmidt decomposition on operator space, then try to relate covariance matrix criterion with the cross‐norm or realignment criterion. A general bipartite quantum state on a Hilbert space , where and are the dimensions of the Hilbert spaces and , respectively, can be written as

8.30

where are real quantities, and and are complete sets of orthonormal Hermitian operators on the Hilbert spaces and , respectively. As we have seen earlier, in Eq. 8.30, can be written in the Schmidt decomposed‐like form (in operator space) via the singular value decomposition as

8.31

where singular values are real and nonnegative, and we have assumed that .

Now using the relations and we can have,

8.33

8.34

Using inequalities 8.32 8.33 8.34, we get

8.35

This is the cross‐norm or realignment criterion of separability mentioned earlier, which we get as a corollary of the covariance matrix criterion.

There are several other corollaries of the covariance matrix criterion, which enable one to efficiently detect entangled states in several cases. Moreover, the covariance matrix criterion can be improved by using local filtering operation (30). See Ref. (15) for details.

8.5 Non‐operational Entanglement Criteria

In this section, we discuss three further entanglement criteria. We show how the Hahn–Banach theorem can be used to obtain “entanglement witnesses.” We also introduce the notion of positive maps and present the entanglement criterion based on it. And finally, we present the range criterion of separability. All three criteria are “nonoperational,” in the sense that they are not state‐independent. Nevertheless, they provide important insights into the structure of the set of entangled states. Moreover, the concept of entanglement witnesses can be used to detect entanglement experimentally, by performing only a few local measurements, assuming some prior knowledge of the density matrix (31,32).

8.5.1 Technical Preface

The following lemma and observation will be useful for later purposes.

8.5.1.1 Entanglement Witnesses

8 Entanglement Witness from the Hahn–Banach Theorem

Central to the concept of entanglement witnesses is the Hahn–Banach theorem, , which we will present here limited to our situation and without proof (see, e.g., (33) for a proof of the more general theorem).

The statement of the theorem is illustrated in Figure 8.2. The figure motivates the introduction of a new coordinate system located within the hyperplane (supplemented by an orthogonal vector , which is chosen such that it points away from ). Using this coordinate system, every state can be characterized by its distance from the plane, by projecting onto the chosen orthonormal vector and using the trace as scalar product, that is, . This measure is either positive, zero, or negative. We now suppose that is the convex compact set of all separable states. According to our choice of basis in Figure 8.2, every separable state has a positive distance, while there are some entangled states with a negative distance. More formally, this can be phrased as follows.

Using these definitions, we can restate the consequences of the Hahn–Banach theorem in several ways:

From a theoretical point of view, the theorem is quite powerful. However, it does not give any insight of how to construct for a given state , the appropriate witness operator.

8 Examples

For a decomposable witness

8.41

8.42

for all separable states .

This argumentation shows that is a suitable witness also. Let us now consider the simplest case of . We can use

8.45

to write the density matrix

8.46

One can quickly verify that indeed fulfills the witness requirements. Using

8.47

we can rewrite the witness:

8.48

where denotes the identity operator on . This witness now detects :

8.49

8.5.1.2 Positive Maps

8 Introduction and Definitions

So far we have only considered states belonging to a Hilbert space and operators acting on the Hilbert space. However, the space of operators has also a Hilbert‐space structure. We now look at transformations of operators, the so‐called maps, which can be regarded as superoperators. As we will see, this will lead us to an important characterization of entangled and separable states. We start by defining linear maps.

For brevity, we only write “linear map,” instead of “linear self‐adjoint map.” The following definitions help to further characterize linear maps.

Positive maps have, therefore, the property of mapping positive operators onto positive operators. It turns out that by considering maps that are a tensor product of a positive operator acting on subsystem , and the identity acting on subsystem , one can learn about the properties of the composite system.

As an example for this map, consider the time‐evolution of a density matrix. It can be written as , that is, in the form given above. Clearly, this map is linear, self‐adjoint, positive and trace‐preserving. It is also completely positive, because for ,

8.58

where is unitary. But then , if and only if (since positivity is not changed by unitary evolution).

8 Positive Maps and Entangled States

Partial transposition can be regarded as a particular case of a map that is positive but not completely positive. We have already seen that this particular positive but not completely positive map gives us a way to discriminate entangled states from separable states. The theory of positive maps provides with stronger conditions for separability, as shown in Ref. (9).

Theorem 8.10 can also be recast into the following form.

Note that Eq. 8.66 can never hold for maps, , that are completely positive, and for nonpositive maps, it may hold even for separable states. Hence, any positive but not completely positive map can be used to detect entanglement.

8 Choi–Jamiołkowski Isomorphism

In order to complete the proof of Theorem 8.10, we introduce first the Choi–Jamiołkowski isomorphism (37) between operators and maps. Given an operator , and an orthonormal product basis , we define a map by

8.67

or in short form,

8.68

This shows how to construct the map from a given operator . To construct an operator from a given map, we use the state

8.69

(where ) to get

8.70

This isomorphism between maps and operators results in the following properties.

To indicate further how this equivalence between maps and operators works, we develop here a proof for the “only if” direction of the second statement. Let be an entanglement witness, then . By the Jamiołkowski isomorphism, the corresponding map is defined as where . We have to show that

8.71

Since acts on Bob's space, using the spectral decomposition of , , leads to

8.72

where all . Then

8.73

We can now prove the direction of Theorem 8.10 or, equivalently, the direction of Theorem 8.11. We thus have to show that if is entangled, there exists a positive map , such that is not positive definite. If is entangled, then there exists an entanglement witness such that

8.74

for all separable . is an entanglement witness (which detects ) if and only if (note the complete transposition!) is also an entanglement witness (which detects ). We define a map by

8.75

8.76

where . Then

8.77

where we have used Lemma 8.1, and that . To complete the proof, one has to show that , which can be done by showing that , where , with being an orthonormal basis.

8.5.1.3 Range Criterion

The range criterion (6) gives a nonoperational condition for separability, which is based on the range of the density matrix and is, in particular, independent of the partial transposition criterion. The range criterion may not detect inseparability in some states for which the partial transposition criterion succeeds but it works efficiently in many cases, especially for the bound entangled states, where the other one fails.

It can be easily shown that for a density matrix in a Hilbert space having spectral decomposition

8.78

where , the set of vectors spans the range of , . The range criteria of separability is given by the following theorem.

As an example, let us consider a state in , given by

8.81

with . The partial transposition of this density matrix, turns out to be positive. In Ref. (6), it was demonstrated that is entangled, which can be successfully detected by the range criterion. For , one can find all product vectors in the range of . It was shown that the partial complex conjugation with respect to , that is, cannot span the range of , thus violating Condition 2 of Theorem 8.13.

8.6 Bell Inequalities

The concept of locality with respect to shared quantum states was first brought into light by Einstein, Podolsky, and Rosen (EPR) in their seminal paper in 1935 (38). They argued that since nonclassical correlations of entangled states of the form cannot be explained by any physical theory satisfying the notions of “locality” and “realism,” quantum mechanics must be incomplete. In 1964, Bell (39) provided a formulation of the problem that made the assumptions of locality and realism more precise and, more importantly, showed that the assumptions are actually testable in experiments. He derived a mathematical inequality that must be satisfied by any physical theory of nature, which is local as well as realistic.

As we shall see, Bell inequalities are essentially a special type of entanglement witness. An additional property of Bell inequalities is that any entangled state detected by them is nonclassical in a particular way: It violates “local realism.” The inequality is actually a constraint on a linear function of results of certain experiments. Modulo some so‐called loopholes (see, e.g., (40)), these inequalities have been shown to be actually violated in experiments (see e.g. (41) and references therein). In this section, we first derive a Bell inequality⁰⁶ and then show how this inequality is violated by the singlet state.

Consider a two spin‐1/2 particle state where the two particles are far apart. Let the particles be called and . Let projection‐valued measurements in the directions and be done on and , respectively. The outcomes of the measurements performed on the particles and in the directions and , are, respectively, and . The measurement result ( ), whose values can be , may depend on the direction ( ) and some other uncontrolled parameter , which may depend on anything, that is, may depend upon system or measuring device or both. Therefore, we assume that ( ) has a definite premeasurement value ( ). Measurement merely uncovers this value. This is the assumption of reality. is usually called a hidden variable, and this assumption is also termed as the hidden variable assumption. Moreover, the measurement result at ( ) does not depend on what measurements are performed at ( ). That is, for example does not depend upon . This is the assumption of locality, also called the Einstein's locality assumption. The parameter is assumed to have a probability distribution, say . Therefore, satisfies the following:

8.82

The correlation function of the two spin‐1/2 particle state for a measurement in a fixed direction for particle and for particle is then given by (provided the hidden variables exist)

8.83

Here

8.84

because the measurement values were assumed to be .

Let us now suppose that the observers at the two particles and can choose their measurements from two observables , and , , respectively, and the corresponding outcomes are , and , , respectively. Then

8.85

Now and can only be and 0, or 0 and , respectively. Consequently,

8.86

This is the well‐known CHSH inequality. Note here that in obtaining the above inequality, we have never used quantum mechanics. We have only assumed Einstein's locality principle and an underlying hidden variable model. Consequently, a Bell inequality is a constraint that any physical theory that is both local and realistic has to satisfy. Below, we will show that this inequality can be violated by a quantum state. Hence, quantum mechanics is incompatible with an underlying local realistic model.

8.6.1 Detection of Entanglement by Bell Inequality

Let us now show how the singlet state can be detected by a Bell inequality. This additionally will indicate that quantum theory is incompatible with local realism. For the singlet state , the quantum mechanical prediction of the correlation function is given by

8.87

where and similarly for . , where , and are the Pauli spin matrices. And is the angle between the two measurement directions and .

So for the singlet state, one has

8.88

The maximum value of this function is attained for the directions , , , on a plane, as given in Figure 8.4, and in that case

8.89

This clearly violates the inequality in Eq. 8.86. But Eq. 8.86 was a constraint for any physical theory, which has an underlying local hidden variable model. As the singlet state, a state allowed by the quantum mechanical description of nature, violates the constraint 8.86, quantum mechanics cannot have an underlying local hidden variable model. In other words, quantum mechanics is not local realistic. This is the statement of the celebrated Bell theorem.

Moreover, it is easy to convince oneself that any separable state does have a local realistic description, so that such a state cannot violate a Bell inequality. Consequently, the violation of Bell inequality by the singlet state indicates that the singlet state is an entangled state. Further, the operator (cf. Eqs. 8.87 and 8.88)

8.90

can, by suitable scaling and change of origin, be considered as an entanglement witness for the singlet state, for , , , chosen as in Figure 8.4 (cf. (44)).

Note that violation of Bell inequalities is stronger than entanglement. For example, the Werner state is entangled for , but it violates CHSH inequality for ( 5,45).

8.7 Quantification of Entanglement

The entanglement content of a pure two‐party quantum state was initially quantified by the usefulness of the state in communication protocols, for example, quantum teleportation, quantum dense coding, and so on (46,47). Since entangled quantum states can be used to perform teleportation and dense coding with efficiencies exceeding those situations in which no entanglement is available, entanglement is considered to be a “resource.” Moreover, it was found that the singlet state can perform these tasks with the maximal possible efficiency, thus it was assumed that the singlet state or any other state connected to the singlet state by local unitary transformations is a maximally entangled state in . It was further assumed that maximally entangled states in has a unit amount of entanglement, or has 1 ebit (“entanglement bit”). What if one has a shared entangled state ? In that case, one can show that given many copies of , one can extract a fewer number of singlets using LOCC, which can thereafter be used in quantum communication schemes. Conversely, if one has a collection of singlets, then it can be converted into a collection of via LOCC. Bennett et al. showed that copies of an entangled state , shared between Alice and Bob can be reversibly converted, using only LOCC between Alice and Bob, into copies of singlets, where tends to and the fidelity of the conversion approaches unity for large (47). This led to the quantification of the entanglement content of a pure quantum state by the von Neumann entropy of its reduced density matrices (47):

8.91

This quantification also remains valid in higher dimensions. We refer to the quantity as the “entropy of entanglement” (or simply entanglement) of . Clearly, for a disentangled pure state , and are also pure states, for which the von Neumann entropies vanish, and . But if is an entangled state of the form with more than one nonzero , then we have and . In this case, we have , and is given by the Shannon entropy of the probability distribution . Entropy of entanglement ranges from zero for a product state to for a maximally entangled state in a Hilbert space of dimension . Clearly, for the singlet state , the entropy of entanglement .

Before extending the quantitative theory of entanglement to the more general situation in which Alice and Bob share a mixed state , we present essential conditions that any measure of entanglement has to satisfy (48–50).

1. Nonnegativity, that is, for any bipartite quantum state .
2. Entanglement vanishes for separable states, that is, if is separable.
3. Invariance under local unitary transformations, that is, .
4. Entanglement cannot increase under local operations and classical communication, that is, for a given LOCC , .

Condition 1 is there by convention. In any resource theory, the quantification of the resource must be done by a quantity that does not increase under the free operations. Moreover, the quantity must be zero for the states that can be created by these free operations. In the resource theory of entanglement, the free operations are the LOCC, and thus entanglement measures cannot increase under LOCC and separable states must not have any entanglement. This accounts for Conditions 2 and 4. Condition 3 arises as local unitary transformations represent only a local change of basis and do not change any correlation. A quantity that satisfies these conditions can be called an entanglement measure and is eligible for the quantification of the entanglement content of a quantum state. They are also often referred to as entanglement monotones. Some authors also impose convexity and additivity properties for entanglement measures:

• Entanglement is a convex function, that is, , for an ensemble .
• Entanglement is additive, that is, .

Below, we briefly discuss a few measures of entanglement.

8.7.1 Entanglement of Formation

One way to widen the theory of entanglement measures to the mixed state regime is by the convex roof extension of pure state entanglement measures (51). The first measure introduced by this technique was the entanglement of formation (26).

Clearly, the entanglement of formation for a pure state collapses to the corresponding entropy of entanglement. Using the singlet as the basic unit of entanglement, one can perceive the operational meaning of the entanglement of formation in the following manner:

1. Decompose into a pure state ensemble as .
2. Choose the state according to the corresponding probability .
3. Prepare the state from singlets via local operations and classical communication.
4. Finally, forget the identity of the chosen ensemble state.

In this way, one needs, on average, singlets, and then one can choose the pure state ensemble for which the average is minimum. This minimum number of singlets required to prepare in this procedure gives the entanglement of formation of .

The convex roof optimization given in Eq. 8.92 is formidable to compute for general mixed states. However, the exact closed form of entanglement of formation is known for two‐qubit mixed states in terms of the “concurrence.”

8.7.1.1 Concurrence

Concurrence for pure states was first introduced in Ref. (52). For a two‐qubit pure state, , the concurrence is defined as

8.93

where , with being the complex conjugate of in the standard computational basis . For two‐qubit mixed states, a closed form expression of the convex roof extension of concurrence can be obtained (53). For a two‐qubit density matrix , let us first define the spin‐flipped density matrix as , and the operator . The convex roof extended concurrence of is then given by

8.94

where the 's are the eigenvalues of in decreasing order. A computable formula for entanglement of formation of two‐qubit quantum states can be expressed in terms of the concurrence ( 52, 53).

Since is a monotonically increasing function of and goes from 0 to 1 as goes from 0 to 1, we can also consider the concurrence as a measure of entanglement in .

8.7.1.2 Entanglement Cost

As we have seen earlier, the entropy of entanglement of a pure state quantifies the average number of singlets needed to asymptotically construct via LOCC. We went over to the mixed‐state scenario by using the concept of entanglement of formation. However, the definition of entanglement of formation consists of a combination of asymptotic and nonasymptotic LOCC transformations. Let us now present a purely asymptotic entanglement measure, known as entanglement cost.

Hayden et al. have shown than entanglement cost is equal to the regularized entanglement of formation (54), given by

8.97

Clearly, if entanglement of formation is additive, entanglement cost will be equal to the entanglement of formation. For pure states, entanglement cost reduces to the entropy of entanglement.

8.7.3 Relative Entropy of Entanglement

A qualitatively different way to quantify entanglement is based on the geometry of quantum states. It is defined as the distance between an entangled state and its closest separable state (58). If is the set of all separable states, then a distance‐based measure of entanglement for a bipartite shared state is given by ( 48 58–60)

8.99

where is a suitably chosen distance measure. For to be a “good” measure of entanglement, the distance measure can be required to satisfy the following properties.

1. for any two states and ; equality holds iff .
2. Invariance under unitary operations. .
3. is nonincreasing under every completely positive and trace‐preserving map , that is, .

The reason for the distance measure to satisfy these properties is that they imply Conditions 1–4 for entanglement measures mentioned earlier in this section.

One of the most famous members of this family of distance‐based measures is the relative entropy of entanglement, where we take the von Neumann relative entropy, which is defined in analogy with the classical Kullback–Leibler distance, as the distance measure. For two density matrices, and , it is defined as (61)

8.100

It is to be noted that the relative entropy is not symmetric in its arguments, and .

We now state two important theorems on relative entropy of entanglement.

Although computation of the relative entropy of entanglement for arbitrary mixed states is quite hard, one can characterize the set of entangled states for all of whom a given separable state is the closest separable state, when relative entropy is considered as the distance measure (62).

Based on other distance measures, several “geometric” entanglement measures have also been introduced (see Section 8.9.1).

8.7.4 Negativity and Logarithmic Negativity

The partial transposition criterion for entanglement, mentioned in Section 8.4.1, provides another quantity to quantify the entanglement content of a given quantum state. This quantity is known as the negativity (63–66), given by the absolute sum of negative eigenvalues of the partial transposed density matrix. In other words, the negativity of a shared quantum state is defined as

8.102

where is the matrix trace norm. Although satisfies the convexity property, it is not additive. Based on negativity, one can define an additive entanglement measure, known as logarithmic negativity, and is given by

8.103

is a monotone under deterministic LOCC operations. However, it fails to be a convex function. It was also shown to be an upper bound of distillable entanglement (64).

A major advantage of negativity and logarithmic negativity is that they are easy to compute for general, possibly mixed, quantum states of arbitrary dimensions. Clearly, for PPT bound entangled states, and are zero and cannot be used to quantify entanglement. But in and , their nonzero values are necessary and sufficient for detecting entanglement (see Theorem 8.3).

8.8 Classification of Bipartite States with Respect to Quantum Dense Coding

Up to now, we have been interested in splitting the set of all bipartite quantum states into separable and entangled states. However, one of the main motivations behind the study of entangled states is that some of them can be used to perform certain tasks, which are not possible if one uses states without entanglement. It is, therefore, important to find out which entangled states are useful for a given task. We discuss here the particular example of quantum dense coding (2).

Suppose that Alice wants to send two bits of classical information to Bob. Then, a general result known as the Holevo bound (to be discussed below) shows that Alice must send two qubits (i.e., 2 two‐dimensional quantum systems) to Bob, if only a noiseless quantum channel is available. However, if additionally Alice and Bob have previously shared entanglement, then Alice may have to send less than two qubits to Bob. It was shown by Bennett and Wiesner (2) that by using a previously shared singlet (between Alice and Bob), Alice will be able to send two bits to Bob, by sending just a single qubit.

The protocol of dense coding (2) works as follows. Assume that Alice and Bob share a singlet state

8.104

The crucial observation is that this entangled two‐qubit state can be transformed into four orthogonal states of the two‐qubit Hilbert space by performing unitary operations on just a single qubit. For instance, Alice can apply a rotation (the Pauli operations) or do nothing to her part of the singlet, while Bob does nothing, to obtain the three triplets (or the singlet):

8.105

where

8.106

are the Bell states and is the qubit identity operator. Suppose that the classical information that Alice wants to send to Bob is , where . Alice and Bob previously agree on the following correspondence between the operations applied at Alice's end and the information that she wants to send:

8.107

Depending on the classical information she wishes to send, Alice applies the appropriate rotation on her part of the shared singlet, according to the above correspondence. Afterward, Alice sends her part of the shared state to Bob, via the noiseless quantum channel. Bob now has in his possession the entire two‐qubit state, which is in any of the four Bell states . Since these states are mutually orthogonal, he will be able to distinguish between them and hence find out the classical information sent by Alice.

To consider a more realistic scenario, usually two avenues are taken. One approach is to consider a noisy quantum channel, while the additional resource is an arbitrary amount of shared bipartite pure state entanglement (see e.g. (67,68); see also (69,70)). The other approach is to consider a noiseless quantum channel, while the assistance is by a given bipartite mixed entangled state (see e.g. ( 69–74)).

Here, we consider the second approach, and derive the capacity of dense coding in this scenario, for a given state, where the capacity is defined as the number of classical bits that can be accessed by the receiver, per usage of the noiseless channel. This will lead to a classification of bipartite states according to their ability to assist in dense coding. In the case where a noisy channel and an arbitrary amount of shared pure entanglement is considered, the capacity refers to the channel (see e.g. ( 67, 68)). However, in our case when a noiseless channel and a given shared (possibly mixed) state is considered, the capacity refers to the state. Note that the mixed shared state in our case can be thought of as an output of a noisy channel. A crucial element in finding the capacity of dense coding is the Holevo bound (75), which is a universal upper bound on classical information that can be decoded from a quantum ensemble. Below we discuss the bound, and subsequently derive the capacity of dense coding.

8.8.1 The Holevo Bound

The Holevo bound is an upper bound on the amount of classical information that can be accessed from a quantum ensemble in which the information is encoded. Suppose therefore that Alice ( ) obtains the classical message that occurs with probability , and she wants to send it to Bob ( ). Alice encodes this information in a quantum state , and sends it to Bob. Bob receives the ensemble , and wants to obtain as much information as possible about . To do so, he performs a measurement, which gives the result , with probability . Let the corresponding postmeasurement ensemble be . The information gathered can be quantified by the mutual information between the message index and the measurement outcome (76):

8.108

Note that the mutual information can be seen as the difference between the initial disorder and the (average) final disorder. Bob will be interested to obtain the maximal information, which is maximum of for all measurement strategies. This quantity is called the accessible information:

8.109

where the maximization is over all measurement strategies.

The maximization involved in the definition of accessible information is usually hard to compute, and hence the importance of bounds ( 75,77). In particular, in Ref. (75), a universal upper bound, the Holevo bound, on is given as follows:

8.110

See also (78–80). Here is the average ensemble state, and is the von Neumann entropy of .

The Holevo bound is asymptotically achievable in the sense that if the sender Alice is able to wait long enough and send long strings of the input quantum states , then there exists a particular encoding and a decoding scheme that asymptotically attains the bound. Moreover, the encoding consists in collecting certain long and “typical” strings of the input states, and sending them all at once (81,82).

8.8.2 Capacity of Quantum Dense Coding

Suppose that Alice and Bob share a quantum state . Alice performs the unitary operation with probability , on her part of the state . The classical information that she wants to send to Bob is . Subsequent to her unitary rotation, she sends her part of the state to Bob. Bob then has the ensemble , where

The information that Bob is able to gather is . This quantity is bounded above by , and is asymptotically achievable. The “one‐capacity” of dense coding for the state is the Holevo bound for the best encoding by Alice:

8.111

The superscript reflects the fact that Alice is using the shared state once at a time, during the asymptotic process. She is not using entangled unitaries on more than one copy of her parts of the shared states . As we will see below, encoding with entangled unitaries does not help her to send more information to Bob.

In performing the maximization in Eq. 8.111, first note that the second term in the right‐hand side (rhs) is , for all choices of the unitaries and probabilities. Second, we have

where is the dimension of Alice's part of the Hilbert space of , and , . Moreover, , as nothing was done at Bob's end during the encoding procedure. (In any case, unitary operations does not change the spectrum, and hence the entropy, of a state.) Therefore, we have

But the bound is reached by any complete set of orthogonal unitary operators , to be chosen with equal probabilities, which satisfy the trace rule , for any operator . Therefore, we have

The optimization procedure above sketched essentially follows that in Ref. (74). Several other lines of argument are possible for the maximization. One is given in Ref. (72) (see also (83)). Another way to proceed is to guess where the maximum is reached (maybe from examples or by taking the most symmetric option), and then perturb the guessed result. If the first‐order perturbations vanish, the guessed result is correct, as the von Neumann entropy is a concave function and the maximization is carried out over a continuous parameter space.

Without using the additional resource of entangled states, Alice will be able to reach a capacity of just bits. Therefore, entanglement in a state is useful for dense coding if . Such states will be called dense‐codable (DC) states. Such states exist, an example being the singlet state.

Note here that if Alice is able to use entangled unitaries on two copies of the shared state , the capacity is not enhanced (see Ref. (84)). Therefore, the one‐capacity is really the asymptotic capacity, in this case. Note however that this additivity is known only in the case of encoding by unitary operations. A more general encoding may still have additivity problems (see e.g. (70)). Here, we have considered unitary encoding only. This case is both mathematically more accessible and experimentally more viable.

A bipartite state is useful for dense coding if and only if . It can be shown that this relation cannot hold for PPT entangled states (69) (see also (83)). Therefore a DC state is always NPT. However, the converse is not true: There exist states that are NPT, but not useful for dense coding. Examples of such states can be obtained by the considering the Werner state (5).

The discussions above leads to the following classification of bipartite quantum states:

1. Separable states: These states are, of course, not useful for dense coding. They can be prepared by LOCC.
2. PPT entangled states: These states, despite being entangled, cannot be used for dense coding. Moreover, their entanglement cannot be detected by the partial transposition criterion.
3. NPT non‐DC states: These states are entangled, and their entanglement can be detected by the partial transposition criterion. However, they are not useful for dense coding.
4. DC states: These entangled states can be used for dense coding.

The above classification is illustrated in Figure 8.5. A generalization of this classification has been considered in Refs. ( 83, 84).

8.9 Multipartite States

The discussion about detection of bipartite entanglement presented earlier is of course quite far from complete. And yet, in this section, we present a few remarks on multipartite states and multipartite entanglement.

The case of detection of entanglement of pure states is again simple, although there are different types of entanglement present in a multipartite system. One quickly realizes that a multipartite pure state is entangled if and only if it is entangled in at least one bipartite splitting. So, for example, the Greenberger–Horne–Zeilinger (GHZ) state (85), shared between three parties , , and , is entangled, because it is entangled in the : bipartition (as also in all others), whereas the state is entangled in the : and : bipartite splits but not in the : one.

8.9.1 ‐Separable, Fully‐Separable, and Genuine Multipartite Entangled States

Among multipartite states, there exists a hierarchical structure of states with respect to their entanglement quality. An ‐party pure quantum state is called ‐separable ( ), if it is separable in at least bipartite splitting. Similarly, an ‐party pure quantum state is ‐separable or fully separable if it is separable in all bipartite splittings. A pure quantum state possesses genuine multipartite entanglement if and only if it is entangled in all possible bipartite cuts. For example, the state is biseparable or 2‐separable, whereas the GHZ state is genuinely multipartite entangled.

The case of mixed states is more involved. A possibly mixed quantum state of parties is ‐separable, if in every pure state decomposition of , there exists at least one ‐separable pure state and no other state with separability lesser than . Similarly, a possibly mixed quantum state is genuinely multipartite entangled, if it has at least one genuine multipartite entangled pure state in every pure state decomposition of it. For example, in the three‐qubit case, the equal mixture of the W state (86,87) and its “complement” is genuinely multipartite entangled (88), and the equal mixture of and is bi‐separable. Figure 8.6 depicts the schematic geometric picture of this hierarchical structure of multipartite entanglement.

One avenue to quantify the degree of such multipartite entanglement relies on the above geometric structure of multipartite entangled states ( 58,89,90). Given a distance functional , that satisfies Conditions (1)–(3) given in Section 8.7.3, the quantity

8.112

gives a measure of ‐inseparable multipartite entanglement in the state . As two special cases, for , Eq. 8.112 gives the minimum distance from fully separable states, and thus quantifies the “total” multipartite entanglement, while for , gives a measure of genuine multipartite entanglement. Optimization in Eq. 8.112 is a formidable problem for general multipartite states. But there exist forms of the geometric measure for various families of states (pure and mixed) corresponding to certain distance measures (91–93). For example, in the case of pure states, if we take the following distance measure

8.113

we get the “geometric measures” of multipartite entanglement for pure states ( 58 90–92). In case the minimum distance is from biseparable states, the corresponding measure has been termed as the generalized geometric measure ( 91, 92),

8.114

which measures the genuine multiparty entanglement in . In this case, we get a computable form of the measure for an arbitrary ‐party pure state, , shared between , in arbitrary dimensions, given by (92)

8.115

where is the maximum Schmidt coefficient in each possible bipartition split of the type of the given state .

Until now, in the case of multiparty mixed states, we have considered only the distance‐based measures. However, it is also possible to use the convex‐roof approach to define entanglement measures for multiparty mixed states, after choosing a certain measure for pure states (94,95).

8.9.2 Three‐Qubit Case: GHZ‐Class versus ‐Class

For three‐qubit pure states, the above classification of multipartite states boils down to three categories:

1. Fully separable states of the form
2. Biseparable states of three types: : , : , and : , where : type states are separable in the : splitting but not in others and so on
3. Finally, genuine tripartite entangled states.

Another classification is possible by considering interconversion of states through stochastic local operations and classical communication (SLOCC) (96), i.e, through LOCC but with a nonunit probability. In this scenario, we call two states and to be equivalent if there is a nonvanishing probability of success when trying to convert into as well as in the opposite direction through SLOCC. For example, in the two‐qubit case, every entangled state is equivalent to any other entangled states, and the entropies of entanglement quantify these conversion rates (see Section 8.7). This ideal situation is absent already in the case of pure three‐qubit states (86). It turns out that any genuine three‐qubit pure entangled state can be converted into either the GHZ state or the W state, but not both, using SLOCC. This divides the set of genuine three‐qubit pure entangled states into two sets that are incompatible under SLOCC. In other words, if a state is convertible into and another state is convertible into via SLOCC, then one cannot transform to or vice versa, with any nonzero probability. These two sets of genuine three‐qubit pure entangled states are termed as the GHZ‐class and the ‐class, respectively. Figure 8.7 shows these different classes of three‐qubit pure states and possible SLOCC transformations between the different classes.

Dür et al. (86) presented general forms of the members of each class. A member of the GHZ‐class can be expressed as

8.116

where

8.117

is a normalization factor, and , and . Here, , , and so on. Similarly, a member state of the ‐class, up to a local unitary transformation, can be written as

8.118

where , and .

For multipartite states with , there exist infinitely many inequivalent kinds of such entanglement classes under SLOCC (86). See Refs. (97,98) for further results.

8.9.3 Monogamy of Quantum Entanglement

The concept of monogamy ( 47 99–101) is an inherent feature of multipartite quantum correlations, and, in particular, of sharing of two‐party entanglements in multiparty quantum states. Unlike classical correlations, quantum entanglement cannot be freely shared among many parties. For example, given three parties , , and , if party is maximally entangled with party , for example, if they share a singlet state , then cannot be simultaneously entangled with party . In other words, there exist a trade‐off between 's entanglement with and its entanglement with . In principle, and in its simplest form, for a two‐party entanglement measure and a three‐party system shared between , , and , any relation providing an upper bound to the sum that is stronger than the sum of individual maxima of and , can be termed as a monogamy relation for . However, in Ref. (100), an intuitive reasoning for the validity of the relation

8.119

is given. As in Ref. (100), we will call an entanglement measure monogamous in a certain three‐party system, if the relation 8.119 is valid for all quantum states in that system.

It was shown later that holds also for arbitrary mixed three‐qubit states (102), where is defined via convex‐roof extension. Using Theorem 8.17, one can define a positive quantity in terms of squared concurrence, named tangle, or three‐tangle, for three‐qubit pure states, as (100)

8.120

The tangle , also called “residual entanglement,” is independent of the choice of the “node” or “focus,” which is the party here. It has been argued that the tangle gives a quantification of three‐qubit entanglement. The generalization of the tangle to mixed states can, for example, be obtained by the convex roof extension, which is difficult to compute. The tangle is a proper entanglement monotone, as it does not increase, on average, under LOCC (86). It also successfully distinguishes the two inequivalent SLOCC classes in three‐qubit pure state scenario, namely the GHZ‐class and the ‐class. It has been shown that tangle vanishes for states in the ‐class, whereas it is always nonzero for states in the GHZ‐class (86). Therefore, to quantify entanglement content of states from the ‐class, one has to look for other multipartite entanglement measures, different from tangle.

In the ‐party scenario, generalization of inequality 8.119 can be written as

8.121

In the same spirit as for the definition of the tangle in Eq. 8.120, we can define the “monogamy score” (103) corresponding to a bipartite entanglement measure as

8.122

with party as “nodal.” Like for the tangle, it has been argued that the monogamy score, , can act as a measure of multiparty entanglement ( 100, 103), obtained by subtracting the bipartite contributions in the “total” entanglement in the partition. Unlike the tangle, monogamy scores for certain entanglement measures can possess negative values for some ‐party quantum states. For further information on recent works about the monogamy of quantum entanglement and correlations, see Ref. (101) and references therein.

For further results about entanglement criteria, detection, and classification of multipartite states, see e.g. ( 32 104–114), and references therein.

Exercises

1. 14.1 Show that the singlet state has nonpositive partial transposition.
2. 14.2 Consider the Werner state in , where (5). Find the values of the mixing parameter , for which entanglement in the Werner state can be detected by the partial transposition criterion.
3. 14.3 Show that in , the partial transposition of a density matrix can have at most one negative eigenvalue.
4. 14.4 Given two random variables and , show that the Shannon entropy of the joint distribution cannot be smaller than that of either.
5. 14.5 Prove Theorem 8.5.
6. 14.6 Consider the following state in :
   8.123

   where , , , and (115). Find the ranges of the parameter , for which entanglement in the state can be detected by the majorization and the cross‐norm criteria.

7. 14.7 Prove Corollaries 8.1 and 8.2.
8. 14.8 Prove Lemma 8.1.
9. 14.9 Prove Theorem 8.10.
10. 14.10 Consider the following set of orthogonal product states in (116):
    8.124

    Using the range criterion, show that the quantum state

    8.125

    is entangled, where denotes the identity operator on .

11. 14.11 Prove Theorem 8.15.
12. 14.12 Find the relative entropy of entanglement for Werner state in , where (5).
13. 14.13 Show that each of the shells depicted in Figure 8.5 is nonempty and of nonzero measure. Show also that all the boundaries are convex.
14. 14.14 Show that the entanglement of formation is nonmonogamous for three‐qubit states.
15. 14.15 Consider the geometric measure of genuine multipartite entanglement given in Eq. 8.114, and then show that it assumes the computable closed form given in Eq. 8.115.

Acknowledgments

ML acknowledges financial support from the John Templeton Foundation, the EU grants OSYRIS (ERC‐2013‐AdG Grant No. 339106), QUIC (H2020‐FETPROACT‐2014 No. 641122), and SIQS (FP7‐ICT‐2011‐9 No. 600645), the Spanish MINECO grants FOQUS (FIS2013‐46768‐P), FÏSICATEAMO (FIS2016‐79508‐P), and “Severo Ochoa” Programme (SEV‐2015‐0522), the Generalitat de Catalunya support (2014 SGR 874) and CERCA/Programme, and Fundació Privada Cellex. AS acknowledges financial support from the Spanish MINECO projects FIS2013‐40627‐P,FIS2016‐80681‐P the Generalitat de Catalunya CIRIT (2014‐SGR‐966).

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