Many body physics from a quantum information perspective
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.
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. (1), which proposed using ultracold atomic gases in optical lattices for QI (and stimulated interest in distributed quantum information processing), and Refs. (2); (3); (4), 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.
0.2 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 (5), quantum dense coding (6), and quantum teleportation (7). 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 (8)).
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 .
0.2.1 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:
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
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:
Let with . Then can be written as a Schmidt decomposition
where and form a part of an orthonormal basis in and , respectively, , , and .
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
where denotes the rank of . By reshuffling terms, and defining and we get the desired form [Eq. (0.2)]. 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 0.2.1 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 (0.1) is already written in the form (0.2), with and all the Schmidt coefficients equal to .
0.2.2 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 (9) — 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
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.
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. (0.4). Otherwise, the mixed state is said to be entangled.
The number of pure separable states necessary to decompose any separable state according to Eq. (0.4) is limited by the Caratheodory theorem as (see Refs. (10); (8)). No better bound is known in general, however, for two-qubit and qubit-qutrit systems it was shown that (11) and (12), 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 interact
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.
0.2.3 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. (14). 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 (15); (16). In general —without a restriction on dimensions— the separability problem belongs to the NP-hard class of computational complexity (17).
0.2.4 Partial Transposition
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
where and are real bases in Alice and Bob Hilbert spaces, respectively, we have
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 (19).
If a state is separable, then and .
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 (0.2) 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 0.2.2.
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. (10); (20)— there exist entangled states with positive partial transposition (called PPT states) (cf. Fig. 0.2). Moreover, the set of PPT entangled states does not have measure zero (21). 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 (20); (8). The conjecture that there exist NPT “bound entangled” states is one of the most challenging open problems in quantum information theory (22); (23). Note also that both separable as well as PPT states form convex sets.
Theorem 0.2.2 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 (24):
A state acting on or is separable if and only if .
We will prove this theorem later. Also, we will see that Theorem 0.2.2 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.
0.2.5 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. (25)).
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
The statement of the theorem is illustrated in figure 0.1. 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 product
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 (26).
We call the Hermitian operator an entanglement witness if for all separable and there exists an entangled state such that .
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.
0.2.6 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 0.2.2 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. (24), positive maps give not only necessary but also sufficient conditions for separability and entanglement detection. Moreover, via the Jamiołkowski-Choi isomorphism, theorem 0.2.5 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 .
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:
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.
(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 .
(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 (27):
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 (28); (29); (30).
A linear map is completely positive iff admits the Choi–Kraus–Stinespring form
where and , called usually Kraus operators, are orthogonal in the Hilbert–Schmidt scalar product.
Finally, let us recall the so–called Choi–Jamiołkowski isomorphism (31); (28): 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 0.2.2 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 (24), the Horodeckis showed that positive maps also give a sufficient condition for separability. More precisely, they proved the following (24):
A state is separable if and only if the condition
holds for all positive maps .
The “only if” part goes along exactly the same lines as proof of theorem 0.2.2, 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 0.2.5, 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
As is positive it can be represented as in Eq. (0.8), 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. (32)). A good example comes from the two qubit case. On one hand, theorem 0.2.3 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. (9)) does not detect all entangled states — as for instance .
Let us also notice that an analogous theorem was proven in Ref. (32), which gave a characterization of the set of the fully separable multipartite states
in terms of multipartite entanglement witnesses. Here, however, instead of positive maps one deals with maps which are positive on products of positive operators.
0.2.7 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.
be a positive map. We call it decomposable if it admits the
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 (33), 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 0.2.3.
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. 0.2). Still, however, there is no criterion that allows to judge unambiguously if a given PPT state is entangled.
To support the above discussion, we give particular examples of positive maps and corresponding entanglement witnesses.
Let be the so-called reduction map map defined through for any . It was introduced in Ref. (34) and considered first in the entanglement context in Refs. (35); (36). 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 0.2.6) given by , meaning that the reduction map is decomposable.
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 (39); (18); (24); (40); (41):
The following statements hold.
A state is entangled iff there exists an entanglement witness such that .
A state is PPT entangled iff there exists an indecomposable entanglement witness such that .
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.
Let be a Hermitian operator and map defined as . Then the following statements hold.
iff is a completely positive map.
is an entanglement witness iff is a positive map.
is a decomposable entanglement witness iff is decomposable map.
0.2.8 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. (42); (43); (44); (45); (46); (47); (48); (49); (50); (51); (8) and references therein). We will present here three possible ways to do so.
Let us just say few words about the definition of entanglement
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 (0.14) implies (0.13), but not vice versa — therefore (0.14) 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. (8)).
Entanglement of formation
Consider a bipartite pure state shared between Alice and Bob. As shown by Bennett et al. (52), given copies of the maximally entangled state, Alice and Bob can by LOCC transform them into copies of , if is large. Here
with and being the local density matrices of and stands for the von Neumann entropy of given by . It clearly follows from theorem 0.2.1 that is zero iff is separable, while its maximal value is attained for the maximally entangled states (0.1).
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. (42), 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 (42):
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 (53); (54), or states having some symmetries, as the so–called isotropic (55) and Werner (56) states. In the first case it amounts to
where is the binary entropy function. The function is given by
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 matrix
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) (57). 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,
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 (53); (54) and isotropic states (58).
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 (21); (61):
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 (61):
It was shown in Ref. (62) that it satisfies condition (0.14). 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 (62). 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.
0.3 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 (63); (64), 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 (65); (66); (67). The related holographic principle (68) 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 (69) and special issues (70) about the subject. As pointed out by the authors of Ref. (69), 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
0.3.1 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
where the complex numbers may be regarded as random variables distributed uniformly on a hypersphere, i.e. distributed according to the probability density
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 (0.23) 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 (71); (72); (73) (see also Ref. (74)):
where the delta function is responsible for the normalization, and the normalization constant reads (see e.g. Ref. (74))
with being the Euler gamma function
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,
Our aim is to estimate the following quantity
where the probability distribution is given by Eq. (0.24). 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
which after application to Eq. (0.28) gives the following expression for the mean entropy
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. (71)), leads us to
One knows that denotes the purity of . Its average was calculated by Lubkin (71) and reads
Substitution in Eq. (0.31) 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 (75); (76); (77) presenting different approaches leading to
with being the bigamma function
Second, notice that the exact result of Lubkin (0.32) can be estimated by relaxing the normalization constraint in the distribution (0.23), 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 .
0.3.2 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 site
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 0.3).
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 . Roughly speaking, we
aim to understand how the entropy of the subsystem scales with
its size. In particular, we are interested in the entropy of the
state reduced from a ground state or a thermal state
of the Hamiltonian . We say that the entropy satisfies an area
law if it scales at most as the boundary area
Let us start with the simplest case of one-dimensional lattices, . Let be a subset of consisting of contiguous spins starting from the first site, i.e., with . In this case the boundary of the region contains one spin for open boundary conditions, and two for periodic ones. Therefore, in this case the area law is extremely simple: