Many-body physics from a quantum information perspective

Many-body physics from a quantum information perspective

R. Augusiak Remigiusz Augusiak Fernando M. Cucchietti ICFO–Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain, Maciej Lewenstein ICFO–Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain, ICREA–Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain
66email: maciej.lewenstein@icfo.es
   F. M. Cucchietti Remigiusz Augusiak Fernando M. Cucchietti ICFO–Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain, Maciej Lewenstein ICFO–Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain, ICREA–Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain
66email: maciej.lewenstein@icfo.es
   and M. Lewenstein Remigiusz Augusiak Fernando M. Cucchietti ICFO–Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain, Maciej Lewenstein ICFO–Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain, ICREA–Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain
66email: maciej.lewenstein@icfo.es
Abstract

The quantum information approach to many-body physics has been very successful in giving new insights and novel numerical methods. In these lecture notes we take a vertical view of the subject, starting from general concepts and at each step delving into applications or consequences of a particular topic. We first review some general quantum information concepts like entanglement and entanglement measures, which leads us to entanglement area laws. We then continue with one of the most famous examples of area-law abiding states: matrix product states, and tensor product states in general. Of these, we choose one example (classical superposition states) to introduce recent developments on a novel quantum many-body approach: quantum kinetic Ising models. We conclude with a brief outlook of the field.

1 Introduction

There has been an explosion of interest in the interface between quantum information (QI) and many-body systems, in particular in the fields of condensed matter and ultracold atomic gases. Remarkable examples are Ref. Jaksch99 (), which proposed using ultracold atomic gases in optical lattices for QI (and stimulated interest in distributed quantum information processing), and Refs. Osborne02a (); Osborne02b (); Osterloh02 (), who discussed the first connections between entanglement and quantum phase transitions (QPT). Overall, the confluence of ideas has opened fundamentally deep questions about QPT’s, as well as practical questions about how to use QI ideas in numerical simulations of many-body quantum systems. Here, we will (partially) review these two major themes. We will first introduce some basic notions and tools of quantum information theory, focusing on entanglement and entanglement measures. We shall then discuss area laws, i.e. laws that characterize correlations and entanglement in physically relevant many-body states, and allow to make general statements about computational complexity of the corresponding Hamiltonians. Afterwards, we will explore the concept of matrix product states (MPS) and their generalizations (projected entangled pairs states, PEPS, and tensor networks states). These states provide not only a very useful ansatz for numerical applications, but also a powerful tool to understand the role of entanglement in the quantum many-body theory. We will review one particular example of a state with a straightforward MPS representation: the classical superposition state. The introduction of its parent Hamiltonian will lead us to the final subject of these lectures: quantum kinetic Ising models — an analytically solvable generalization of the popular classical many-body model described by a master equation.

2 Aspects of Quantum Information

Quantum theory contains elements that are radically different from our everyday (“classical”) description of Nature: a most important example are the quantum correlations present in quantum formalism. Classically, complete knowledge of a system implies that the sum of the information of its subsystems makes up the total information for the whole system. In the quantum world, this is no longer true: there exist states of composite systems about which we have complete information, but we know nothing about its subsystems. We may even reach paradoxical conclusions if we apply a classical description to such “entangled” states—whose concept can be traced back to 1932 in manuscripts of E. Schrödinger.

What we have just realized during the last two decades is that these fundamentally nonclassical states (from hereon “entangled states”) can provide us with more than just paradoxes: They may be used to perform tasks that cannot be achieved with classical states. As landmarks of this transformation in our view of such nonclassical states, we mention the spectacular discoveries of (entanglement-based) quantum cryptography Ekert91 (), quantum dense coding Bennett92 (), and quantum teleportation Bennett93 (). Even though our knowledge of entanglement is still far from complete, significant progress has been made in the recent years and very active research is currently underway (for a recent and very complete review see Horodecki09 ()).

In the next section, we will focus on bipartite composite systems. We will define formally what entangled states are, present some important criteria to discriminate entangled states from separable ones, and show how they can be classified according to their capability to perform some precisely defined tasks. However, before going into details, let us introduce the notation. In what follows we will be mostly concerned with bipartite scenarios, in which traditionally the main roles are played by two parties called Alice and Bob. Let denote the Hilbert space of Alice’s physical system, and that of Bob’s. Our considerations will be restricted to finite-dimensional Hilbert spaces, so we can set and . Thus, the joint physical system of Alice and Bob is described by the tensor product Hilbert space . Finally, will denote the set of bounded linear operators from the Hilbert space to .

2.1 Bipartite pure states: Schmidt decomposition

We start our study with pure states, for which the concepts are simpler. Pure states are either separable or entangled states according to the following definition:

Definition 1

Consider a pure state from . It is called separable if there exist pure states and such that . Otherwise we say that is entangled.

The most famous examples of entangled states in are the maximally entangled states, given by

(1)

where the vectors and form bases (in particular they can be the standard ones) in and , respectively. In what follows, we also use to denote the projector onto . The reason why this state is called maximally entangled will become clear when we introduce entanglement measures.

In pure states, the separability problem — the task of judging if a given quantum state is separable — is easy to handle using the concept of Schmidt decomposition:

Theorem 2.1

Let with . Then can be written as a Schmidt decomposition

(2)

where and form a part of an orthonormal basis in and , respectively, , , and .

Proof

A generic pure bipartite state can be written in the standard basis of as , where, in general, the coefficients form a matrix obeying . Using singular-value decomposition, we can write , where and are unitary () and is diagonal matrix consisting of the eigenvalues of . Using this we rewrite as

(3)

where denotes the rank of . By reshuffling terms, and defining and we get the desired form [Eq. (2)]. To complete the proof, we notice that due to the unitarity of and , vectors and satisfy , and constitute bases of and , respectively. In fact, and are eigensystems of the first and second subsystem of . Moreover, since it holds that .

The numbers are called the Schmidt coefficients, and the Schmidt rank of . One can also notice that and are eigensystems of the first and second subsystem of , and that the Schmidt rank denotes the rank of both subsystems. Then, comparison with definition 2.1 shows that bipartite separable states are those with Schmidt rank one. Thus, to check if a given pure state is separable, it suffices to check the rank of one of its subsystems. If (the corresponding subsystem is in a pure state) then is separable; otherwise it is entangled. Notice that the maximally entangled state (1) is already written in the form (2), with and all the Schmidt coefficients equal to .

2.2 Bipartite mixed states: Separable and entangled states

The easy-to-handle separability problem in pure states complicates considerably in the case of mixed states. In order to understand the distinction between separable and entangled mixed states — first formalized by Werner in 1989 Werner89 () — let us consider the following state preparation procedure. Suppose that Alice and Bob are in distant locations and can produce and manipulate any physical system in their laboratories. Moreover, they can communicate using a classical channel (for instance a phone line). However, they do not have access to quantum communication channels, i.e. they are not allowed to exchange quantum states. These two capabilities, i.e. local operations (LO) and classical communication (CC), are frequently referred to as LOCC.

Suppose now that in each round of the preparation scheme, Alice generates with probability a random integer , which she sends to Bob. Depending on this number, in each round Alice prepares a pure state , and Bob a state . After many rounds, the result of this preparation scheme is of the form

(4)

which is the most general one that can be prepared by Alice and Bob by means of LOCC. In this way we arrive at the formal definition of separability in the general case of mixed states.

Definition 2

We say that a mixed state acting on is separable if and only if it can be represented as a convex combination of the product of projectors on local states as in Eq. (4). Otherwise, the mixed state is said to be entangled.

The number of pure separable states necessary to decompose any separable state according to Eq. (4) is limited by the Caratheodory theorem as (see Refs. Horodecki97 (); Horodecki09 ()). No better bound is known in general, however, for two-qubit and qubit-qutrit systems it was shown that a () and b (), respectively.

By definition, entangled states cannot be prepared locally by two parties even after communicating over a classical channel. To prepare entangled states, the physical systems must be brought together to interact111Due to entanglement swapping Zukowski93 (), one must suitably enlarge the notion of preparation of entangled states. So, an entangled state between two particles can be prepared if and only if either the two particles (call them A and B) themselves come together to interact at a time in the past, or two other particles (call them C and D) do the same, with C having interacted beforehand with A and D with B.. Mathematically, a nonlocal unitary operator222A unitary operator on is said to be “nonlocal” if it is not of the form , where is a unitary operator acting on and acts on . must necessarily act on the physical system described by to produce an entangled state from an initial separable state.

The question whether a given bipartite state is separable or not turns out to be quite complicated. 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 paragraphs.

2.3 Entanglement criteria

An operational necessary and sufficient criterion for detecting entanglement still does not exist. However, over the years the whole variety of criteria allowing for detection of entanglement has been worked out. Below we review some of the most important ones, while for others the reader is referred to Ref. GuhneTothReview (). Note that, even if we do not have necessary and sufficient separability criteria, there are numerical checks of separability: semidefinite programming was used to show that separability can be tested in a finite number of steps, although this number can become too large for big systems Doherty02 (); Hulpke05 (). In general —without a restriction on dimensions— the separability problem belongs to the NP-hard class of computational complexity Gurvits03 ().

2.4 Partial Transposition

Let us start with an easy–to–apply necessary criterion based on the transposition map recognized by Choi Choi82 () and then independently formulated directly in the separability context by Peres Peres96 ().

Let be a state on the product Hilbert space , and a transposition map with respect to some real basis in , defined through for any from . Let us now consider an extended map called hereafter partial transposition, where is the identity map acting on the second subsystem. When applied to , the map transposes the first subsystem leaving the second one untouched. More formally, writing as

(5)

where and are real bases in Alice and Bob Hilbert spaces, respectively, we have

(6)

Similarly, one may define partial transposition with respect to the Bob’s subsystem (denoted by ). Although the partial transposition of depends upon the choice of the basis in which is written, its eigenvalues are basis independent. The applicability of the transposition map in the separability problem can be formalized by the following statement Peres96 ().

Theorem 2.2

If a state is separable, then and .

Proof

Since is separable, according to definition 2 it has the form (4). Then, performing the partial transposition with respect to the first subsystem, we have

(7)

In the second step we used that for all . The above shows that is a proper (and also separable) density matrix implying that . The same reasoning leads to the conclusion that , finishing the proof.

Due to the identity , and the fact that global transposition does not change eigenvalues, partial transpositions with respect to the and subsystems are equivalent from the point of view of the separability problem.

In conclusion, we have a simple criterion (partial transposition criterion) for detecting entanglement. More precisely, if the spectrum of one of the partial transpositions of contains at least one negative eigenvalue then is entangled. As an example, let us apply the criterion to pure entangled states. If is entangled, it can be written as (2) with . Then, the eigenvalues of will be and . So, an entangled of Schmidt rank has partial transposition with negative eigenvalues violating the criterion stated in theorem 2.2.

The partial transposition criterion allows to detect in a straightforward manner all entangled states that have non–positive partial transposition (hereafter called NPT states). However, even if this is a large class of states, it turns out that —as pointed out in Refs. Horodecki97 (); Horodecki98 ()— there exist entangled states with positive partial transposition (called PPT states) (cf. Fig. 2). Moreover, the set of PPT entangled states does not have measure zero Zyczkowski98 (). It is, therefore, important to have further independent criteria that identify entangled PPT states. Remarkably, PPT entangled states are the only known examples of bound entangled states, i.e., states from which one cannot distill entanglement by means of LOCC, even if the parties have an access to an unlimited number of copies of the state Horodecki98 (); Horodecki09 (). The conjecture that there exist NPT “bound entangled” states is one of the most challenging open problems in quantum information theory DiVincenzo00 (); Duer00 (). Note also that both separable as well as PPT states form convex sets.

Theorem 2.2 is a necessary condition of separability in any arbitrary dimension. However, for some special cases, the partial transposition criterion is both a necessary and sufficient condition for separability Horodecki96 ():

Theorem 2.3

A state acting on or is separable if and only if .

We will prove this theorem later. Also, we will see that Theorem 2.2 is true for a whole class of maps (of which the transposition map is only a particular example), which also provide a sufficient criterion for separability. Before this, let us discuss the dual characterization of separability via entanglement witnesses.

2.5 Entanglement Witnesses from the Hahn-Banach theorem

Central to the concept of entanglement witnesses is the corollary from the Hahn–Banach theorem (or Hahn–Banach separation theorem), which we will present here limited to our needs and without proof (which the reader can find e.g. in Ref. alt:1985 ()).

Theorem 2.4

Let be a convex compact set in a finite–dimensional Banach space. Let be a point in this space, however, outside of the set (). Then there exists a hyperplane333A hyperplane is a linear subspace with dimension smaller by one than the dimension of the space itself. that separates from .

Figure 1: Schematic picture of the Hahn-Banach theorem. The (unique) unit vector orthonormal to the hyperplane can be used to define right and left with respect to the hyperplane by using the sign of the scalar product.

The statement of the theorem is illustrated in figure 1. In order to apply it to our problem let denote now the set of all separable states acting on . This is a convex compact subset of the Banach space of all the linear operators . The theorem implies that for any entangled state there exists a hyperplane separating it from .

Let us introduce a coordinate system located within the hyperplane (along with an orthogonal vector chosen so that it points towards ). Then, every state can be characterized by its “distance” from the plane, here represented by the Hilbert–Schmidt scalar product444Let be some Hilbert space. Then the set of linear bounded operators acting on is also a Hilbert space with the Hilbert-Schmidt scalar product (). . According to our choice of the coordinate system (see Fig. 1), for any such hyperplane every separable state has a positive “distance”, while there are some entangled states with a negative “distance”. More formally, theorem (2.4) implies the following seminal result Horodecki96 ().

Theorem 2.5

Let be some entangled state acting on . Then there exists a Hermitian operator such that and for all separable .

It is then clear that all the operators representing such separating hyperplanes deserve special attention as they are natural candidates for entanglement detectors. That is, given some Hermitian , if and simultaneously for all separable , we know that is entangled. One is then tempted to introduce the following definition TerhalPLA00 ().

Definition 3

We call the Hermitian operator an entanglement witness if for all separable and there exists an entangled state such that .

Example 1

Let us discuss how to construct entanglement witnesses for all NPT states. If is NPT then its partial transposition has at least one negative eigenvalue. Let denote the eigenstates of corresponding to its negative eigenvalues . Then the Hermitian operator has negative mean value on , i.e., . Simultaneously, using the identity obeyed by any pair of matrices and , it is straightforward to verify that for all and separable . One notices also that any convex combination of and in particular itself are also entanglement witnesses.

Let us comment shortly on the properties of entanglement witnesses. First, it is clear that they have negative eigenvalues, as otherwise their mean value on all entangled states would be positive. Second, since entanglement witnesses are Hermitian, they can be treated as physical observables — which means that separability criteria based on entanglement witnesses are interesting from the experimental point of view. Third, even if conceptually easy, entanglement witnesses depend on states in the sense that there exist entangled states that are only detected by different witnesses. Thus, in principle, the knowledge of all entanglement witnesses is necessary to detect all entangled states.

2.6 Positive maps and the entanglement problem

Transposition is not the only map that can be used to deal with the separability problem. It is rather clear that the statement of theorem 2.2 remains true if, instead of the transposition map, one uses any map that when applied to a positive operator gives again a positive operator (a positive map). Remarkably, as shown in Ref. Horodecki96 (), positive maps give not only necessary but also sufficient conditions for separability and entanglement detection. Moreover, via the Jamiołkowski-Choi isomorphism, theorem 2.5 can be restated in terms of positive maps. To see this in more detail we need to review a bit of terminology.

We say that a map is linear if for any pair of operators acting on and complex numbers . We also say that is Hermiticity–preserving (trace–preserving) if () for any Hermitian .

Definition 4

A linear map is called positive if for all positive the operator is positive.

As every Hermitian operator can be written as a difference of two positive operators, any positive map is also Hermiticity–preserving. On the other hand, a positive map does not have to be necessarily trace–preserving.

It follows immediately from the above definition that positive maps applied to density matrices give (usually unnormalized) density matrices. One could then expect that positive maps are sufficient to describe all quantum operations (as for instance measurements). This, however, is not enough, as it may happen that the considered system is only part of a larger one and we must require that any quantum operation on our system leaves the global system in a valid physical state. This requirement leads us to the notion of completely positive maps:

Definition 5

Let be a positive map and let denote an identity map. Then, we say that is completely positive if for all the extended map is positive.

Let us illustrate the above definitions with some examples.

Example 2

(Hamiltonian evolution of a quantum state) Let and let be defined as for any , with being some unitary operation acting on . Since unitary operations do not change eigenvalues when applied to , it is clear that is positive for any such . Furthermore, is completely positive: an application of the extended map to gives , where denotes identity acting on . Therefore, the extended unitary is also unitary. Thus, if , then . The commonly known example of is the unitary evolution of a quantum state .

Example 3

(Transposition map) The second example of a linear map is the already considered transposition map . It is easy to check that is Hermiticity and trace–preserving. However, the previously discussed example of partially transposed pure entangled states shows that it cannot be completely positive.

To complete the characterization of positive and completely positive maps let us just mention the Choi–Kraus–Stinespring representation. Recall first that any linear Hermiticity–preserving (and so positive) map can be represented as GoriniKossakowskiSudarshan76 ():

(8)

where , , and are orthogonal in the Hilbert–Schmidt scalar product . In this representation, completely positive maps are those (and only those) that have for all . As a result, by replacing (which preserves the orthogonality of ), we arrive at the aforementioned form for completely positive maps Choi75 (); Kraus83 (); Stinespring55 ().

Theorem 2.6

A linear map is completely positive iff admits the Choi–Kraus–Stinespring form

(9)

where and , called usually Kraus operators, are orthogonal in the Hilbert–Schmidt scalar product.

Finally, let us recall the so–called Choi–Jamiołkowski isomorphism Jamiolkowski72 (); Choi75 (): every linear operator acting on can be represented as with some linear map . With this isomorphism, entanglement witnesses correspond to positive maps. Notice also that the dual form of this isomorphism reads ).

Equipped with new definitions and theorems, we can now continue with the relationship between positive maps and the separability problem. It should be clear by now that theorem 2.2 is just a special case of a more general necessary condition for separability: if acting on is separable, then is positive for any positive map . In a seminal paper in 1996 Horodecki96 (), the Horodeckis showed that positive maps also give a sufficient condition for separability. More precisely, they proved the following Horodecki96 ():

Theorem 2.7

A state is separable if and only if the condition

(10)

holds for all positive maps .

Proof

The “only if” part goes along exactly the same lines as proof of theorem 2.2, where instead of the transposition map we put . On the other hand, the “if” part is much more involved. Assuming that is entangled, we show that there exists a positive map such that . For this we can use theorem 2.5, which says that for any entangled there always exists entanglement witness detecting it, i.e., . Denoting by a positive map corresponding to the witness via the the Choi–Jamiołkowski isomorphism, i.e., , we can rewrite this condition as

(11)

As is positive it can be represented as in Eq. (8), and hence the above may be rewritten as with called the dual map of . One immediately checks that dual maps of positive maps are positive. This actually finishes the proof since we showed that there exists a positive map such that .

In conclusion, we have two equivalent characterizations of separability in bipartite systems, in terms of either entanglement witnesses or positive maps. However, on the level of a particular entanglement witness and the corresponding map, both characterizations are no longer equivalent. This is because usually maps are stronger in detection than entanglement witnesses (see Ref. Horodecki01c ()). A good example comes from the two qubit case. On one hand, theorem 2.3 tell us that the transposition map detects all the two–qubit entangled states. On the other hand, it is clear that the corresponding witness, the so–called swap operator (see Ref. Werner89 ()) does not detect all entangled states — as for instance .

Let us also notice that an analogous theorem was proven in Ref. Horodecki01c (), which gave a characterization of the set of the fully separable multipartite states

(12)

in terms of multipartite entanglement witnesses. Here, however, instead of positive maps one deals with maps which are positive on products of positive operators.

2.7 Positive maps and entanglement witnesses: further characterization and examples

We discuss here the relationship between positive maps (or the equivalent entanglement witnesses) and the separability problem.

Definition 6

Let be a positive map. We call it decomposable if it admits the form555By we denote the composition of two maps , i.e., a map that acts on a given operator as . , where are some completely positive maps. Otherwise is called indecomposable.

It follows from this definition that decomposable maps are useless for detection of PPT entangled states. To see this explicitly, assume that is PPT entangled. Then it holds that , where is some quantum state. Since are completely positive, both terms are positive and thus for any decomposable and PPT entangled .

The simplest example of a decomposable map is the transposition map, with both being just the identity map. It is then clear that, from the point of view of entanglement detection, the transposition map is also the most powerful example of a decomposable map. Furthermore, as shown by Woronowicz Woronowicz (), all positive maps from and to are decomposable. Therefore, the partial transposition criterion is necessary and sufficient in two-qubit and qubit-qutrit systems as stated in theorem 2.3.

Using the Jamiołkowski-Choi isomorphism we can check the form of entanglement witnesses corresponding to the decomposable positive maps. One immediately sees that they can be written as , with and being some positive operators. Following the nomenclature of positive maps, such witnesses are called decomposable.

It is then clear that PPT entangled states can only be detected by indecomposable maps, or, equivalently indecomposable entanglement witnesses (cf. Fig. 2). Still, however, there is no criterion that allows to judge unambiguously if a given PPT state is entangled.

Figure 2: Schematic view of the Hilbert-space with two states and and two witnesses and . is a decomposable EW, and it detects only NPT states like . is an indecomposable EW, and it detects also some PPT states like . Note that none of the witnesses detect all entangled states.

To support the above discussion, we give particular examples of positive maps and corresponding entanglement witnesses.

Example 4

Let be the so-called reduction map map defined through for any . It was introduced in Ref. TanahashiTomiyama88 () and considered first in the entanglement context in Refs. Horodecki00 (); Cerf99 (). One immediately finds that is positive, but not completely positive, as it detects entanglement of . Moreover, , where is a completely positive map with Kraus operators (cf. theorem 2.6) given by , meaning that the reduction map is decomposable.

Example 5

Let be the so-called extended reduction map BreuerPRL06 (); HallJPA06 () defined by , where obeys and . It is obviously positive but not completely positive. However, unlike the reduction map, this one is indecomposable as examples of PPT entangled states detected by can be found BreuerPRL06 (); HallJPA06 ().

Let us summarize our considerations with the following two theorems. First, using the definitions of decomposable and indecomposable entanglement witnesses, we can restate the consequences of the Hahn-Banach theorem in several ways Woronowicz76 (); Choi82 (); Horodecki96 (); Lewenstein00 (); Terhal01 ():

Theorem 2.8

The following statements hold.

  1. A state is entangled iff there exists an entanglement witness such that .

  2. A state is PPT entangled iff there exists an indecomposable entanglement witness such that .

  3. A state is separable iff for all entanglement witnesses.

Notice that the Jamiołkowski-Choi isomorphism between positive maps and entanglement witnesses allows to rewrite immediately the above theorem in terms of positive maps. From a theoretical point of view, the theorem is quite powerful. However, it does not give any insight on how to construct for a given state , the appropriate witness operator.

Second, the relations between maps and witnesses can be collected as follows Jamiolkowski72 (); Woronowicz76 (); Horodecki96 (); Lewenstein00 (); Terhal01 ().

Theorem 2.9

Let be a Hermitian operator and map defined as . Then the following statements hold.

  1. iff is a completely positive map.

  2. is an entanglement witness iff is a positive map.

  3. is a decomposable entanglement witness iff is decomposable map.

2.8 Entanglement measures

The criteria discussed above allow to check if a given state is entangled. However, in general they do not tell us directly how much is entangled. In what follows we discuss several methods to quantify entanglement of bipartite states. This quantification is necessary, at least partly because entanglement is viewed as a resource in quantum information theory. There are several complementary ways to quantify entanglement (see Refs. Bennett96b (); Vedral97 (); DiVincenzo98 (); Laustsen03 (); Nielsen99 (); Vidal00 (); Jonathan99 (); Horodecki04 (); Horodecki01b (); Plenio07 (); Horodecki09 () and references therein). We will present here three possible ways to do so.

Let us just say few words about the definition of entanglement measures666For a more detailed axiomatic description, and other properties of entanglement measures, the reader is encouraged to consult, e.g., Refs. Horodecki01b (); Plenio07 (); Horodecki09 ().. The main ingredient in this definition is the monotonicity under LOCC operations. More precisely, if denotes some LOCC operation, and our candidate for the entanglement measure, has to satisfy

(13)

or

(14)

where are states resulting from the LOCC operation appearing with probabilities (as in the case of e.g. projective measurements). Both requirements follow from the very intuitive condition saying that entanglement should not increase under local operations and classical communication. It follows also that if is convex, then the condition (14) implies (13), but not vice versa — therefore (14) gives a stronger condition for the monotonicity. For instance, the three examples of measures presented below satisfy this condition. Finally, notice that from the monotonicity under LOCC operations one also concludes that is invariant under unitary operations, and gives a constant value on separable states (see e.g. Ref. Horodecki09 ()).

Entanglement of formation

Consider a bipartite pure state shared between Alice and Bob. As shown by Bennett et al. Bennett96a (), given copies of the maximally entangled state, Alice and Bob can by LOCC transform them into copies of , if is large. Here

(15)

with and being the local density matrices of and stands for the von Neumann entropy of given by . It clearly follows from theorem 2.1 that is zero iff is separable, while its maximal value is attained for the maximally entangled states (1).

For the two-qubit maximally entangled state , the function gives one: an amount of entanglement also called ebit. With this terminology, one can say that has ebits. Since is the number of singlets required to prepare a copy of the state , it is called entanglement of formation of . We are therefore using the amount of entanglement of the singlet state as our unit of entanglement.

Following Ref. Bennett96b (), let us now extend the definition of entanglement of formation to all bipartite states. By definition, any mixed state is a convex combination of pure states, i.e., , where probabilities and pure states (not necessarily orthogonal) constitute what is called an ensemble. A particular example of such an ensemble is the eigendecomposition of . Thus, it could be tempting to define the entanglement of formation of as an averaged cost of producing pure states from the ensemble, i.e., . One knows, however, that there exist an infinite number of ensembles realizing any given . A natural solution is then to minimize the above function over all such ensembles — with which we arrive at the definition of entanglement of formation for mixed states Bennett96b ():

(16)

with the minimum taken over all ensembles such that .

In general, the above minimization makes the calculation of entanglement of formation extremely difficult. Nevertheless, it was determined for two-qubits Hill97 (); Wootters98 (), or states having some symmetries, as the so–called isotropic Terhal00 () and Werner VollbrechtWerner () states. In the first case it amounts to

(17)

where is the binary entropy function. The function is given by

(18)

with the eigenvalues of the Hermitian matrix in decreasing order, and . Note that the complex conjugation over is taken in the eigenbasis, and denotes the well-known Pauli matrix777In the standard basis is given by .. The function , called concurrence, can also be used to quantify entanglement of more general quantum states. Although Eq. (18) gives the explicit form of concurrence only for two-qubit states, it can also be defined for arbitrary bipartite states — as we shall discuss in the following section.

Concurrence

For any we define concurrence as where is one of the subsystems of (note that the value of does not depend on the choice of subsystems) Rungta01 (). In the case , one sees that its value for pure states ranges from for separable states to for the maximally entangled state.

The extension to mixed states goes in exactly the same way as in the case of entanglement of formation,

(19)

where again the minimization is taken over all the ensembles that realize . For the same reason, as in the case of EOF, concurrence is calculated analytically only in few instances like two-qubit states Hill97 (); Wootters98 () and isotropic states RungtaCaves03 ().

Seemingly, the only difference between and lies in the function taken to define both measures for pure states. However, the way concurrence is defined enables one to determine it experimentally for pure states AolitaMintertPRL06 (); WalbornNat06 (), provided that two copies of the state are available simultaneously.

Negativity and logarithmic negativity

Based on the previous examples of entanglement measures, one may get the impression that all of them are difficult to determine. Even if this is true in general, there are entanglement measures that can be calculated for arbitrary states. The examples we present here are negativity and logarithmic negativity. The first one is defined as Zyczkowski98 (); Vidal02b ():

(20)

The calculation of even for mixed states reduces to determination of eigenvalues of , and amounts to the sum of the absolute values of negative eigenvalues of . This measure has a disadvantage: partial transposition does not detect PPT entangled states; therefore is zero not only for separable states but also for all PPT states.

The logarithmic negativity is defined as Vidal02b ():

(21)

It was shown in Ref. Plenio05a () that it satisfies condition (14). Moreover, logarithmic negativity is additive, i.e., for any pair of density matrices and , which is a desirable feature. However, this comes at a cost: is not convex Plenio05a (). Furthermore, for the same reason as negativity it cannot be used to quantity entanglement of PPT entangled states. Finally, let us notice that these measures range from zero for separable states, to for negativity and for logarithmic negativity.

3 Area laws

Area laws play a very important role in many areas of physics, since generically relevant states of physical systems described by local Hamiltonians (both quantum and classical) fulfill them. This goes back to the seminal work on the free Klein–Gordon field Bombelli86 (); Srednicki93 (), where it was suggested that the area law of geometric entropy might be related to the physics of black holes, and in particular the Bekenstein-Hawking entropy that is proportional to the area of the black hole surface Bekenstein73 (); Bekenstein04 (); Hawking74 (). The related holographic principle Bousso02 () says that information about a region of space can be represented by a theory which lives on a boundary of that region. In recent years there has been a wealth of studies of area laws, and there are excellent reviews EisertCramerPlenioReview () and special issues Calabrese10 () about the subject. As pointed out by the authors of Ref. EisertCramerPlenioReview (), the interest in area laws is particularly motivated by the four following issues:

  • The holographic principle and the entropy of black holes,

  • Quantum correlations in many-body systems,

  • Computational complexity of quantum many-body systems,

  • Topological entanglement entropy as an indicator of topological order in certain many-body systems

3.1 Mean entanglement of bipartite states

Before we turn to the area laws for physically relevant states let us first consider a generic pure state in the Hilbert space in (). Such a generic state (normalized, i.e. unit vector) has the form

(22)

where the complex numbers may be regarded as random variables distributed uniformly on a hypersphere, i.e. distributed according to the probability density

(23)

with the only constraint being the normalization. As we shall see, such a generic state fulfills on average a “volume” rather than an area law. To this aim we introduce a somewhat more rigorous description, and we prove that on average, the entropy of one of subsystems of bipartite pure states in () is almost maximal for sufficiently large . In other words, typical pure states in are almost maximally entangled. This “typical behavior” of pure states happens to be completely atypical for ground states of local Hamiltonians with an energy gap between ground and first excited eigenstates.

Rigorously speaking, the average with respect to the distribution (23) should be taken with respect to the unitarily invariant measure on the projective space . It is a unique measure generated by the Haar measure on the unitary group by applying the unitary group on an arbitrarily chosen pure state. One can show then that the eigenvalues of the first subsystem of a randomly generated pure state are distributed according to the following probability distribution Lubkin78 (); LloydPagels88 (); Page93 () (see also Ref. BengtssonZyczkowski ()):

(24)

where the delta function is responsible for the normalization, and the normalization constant reads (see e.g. Ref. BengtssonZyczkowski ())

(25)

with being the Euler gamma function888In general the gamma function is defined through (26) For being positive integers the gamma function is related to the factorial function via .

Theorem 3.1

Let be a bipartite pure state from drawn at random according to the Haar measure on the unitary group and be its subsystem acting on . Then,

(27)
Proof

Let us give here just an intuitive proof without detailed mathematical discussion (which can be found e.g. in Refs. Lubkin78 (); LloydPagels88 (); Page93 (); Foong94 (); Sen96 (); Sanchez–Ruiz95 (); BengtssonZyczkowski ()).

Our aim is to estimate the following quantity

(28)

where the probability distribution is given by Eq. (24). We can always write the eigenvalues as , where and . This allows us to expand the logarithm into the Taylor series in the neighborhood of as

(29)

which after application to Eq. (28) gives the following expression for the mean entropy

(30)

Let us now notice that , and therefore . This, after substitution in the above expression, together with the fact that for sufficiently large we can omit terms with higher powers of (cf. Lubkin78 ()), leads us to

(31)

One knows that denotes the purity of . Its average was calculated by Lubkin Lubkin78 () and reads

(32)

Substitution in Eq. (31) leads to the desired results, completing the proof.

Two remarks should be made before discussing the area laws. First, it should be pointed out that it is possible to get analytically the exact value of . There is a series of papers Foong94 (); Sen96 (); Sanchez–Ruiz95 () presenting different approaches leading to

(33)

with being the bigamma function999The bigamma function is defined as and for natural it takes the form (34) with being the Euler constant, of which exact value is not necessary for our consideration as it vanishes in Eq. (33).. Using now the fact that , and the asymptotic properties of bigamma function, , we get (27).

Second, notice that the exact result of Lubkin (32) can be estimated by relaxing the normalization constraint in the distribution (23), and replacing it by a product of independent Gaussian distributions, , with , and . The latter distribution, according to the central limit theorem, tends for to a Gaussian distribution for centered at 1, with width . One obtains then straightforwardly , and after a little more tedious calculation , which agrees asymptotically with the Lubkin result for .

3.2 Area laws in a nutshell

In what follows we shall be concerned with lattices in spatial dimensions, . At each site we have a -dimensional physical quantum system (one can, however, consider also classical lattices, with a -dimensional classical spin at each site with the configuration space ) at each site101010For results concerning other kind of systems one can consult Ref. EisertCramerPlenioReview ().. The distance between two sites and of the lattice is defined as

(35)

Accordingly, we define the distance between two disjoint regions and of as the minimal distance between all pairs of sites , where and ; i.e., . If is some region of , we define its boundary as the set of sites belonging to whose distance to (the complement of ) is one. Formally, . Finally, by we denote number of sites (or volume) in the region (see Figure 3).

Figure 3: Schematic representation of a lattice system , an arbitrary region (denoted in light grey background), and its boundary (denoted in dark grey background).

Now, we can add some physics to our lattice by assuming that interactions between the sites of are governed by some hamiltonian . We can divide the lattice into two parts, the region and its complement