Towards an Entanglement Measure for Mixed States in CFTs
Based on Relative Entropy
Tadashi Takayanagi111takayana@yukawa.kyoto-u.ac.jp, Tomonori Ugajin222tomonori.firstname.lastname@example.org, and Koji Umemoto333koji.email@example.com
Center for Gravitational Physics,
Yukawa Institute for Theoretical Physics (YITP), Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwano-ha, Kashiwa, Chiba 277-8582, Japan
Okinawa Institute of Science and Technology,
Tancha, Kunigami gun, Onna son, Okinawa 1919-1
Relative entropy of entanglement (REE) is an entanglement measure of bipartite mixed states, defined by the minimum of the relative entropy between a given mixed state and an arbitrary separable state . The REE is always bounded by the mutual information because the latter measures not only quantum entanglement but also classical correlations. In this paper we address the question of to what extent REE can be small compared to the mutual information in conformal field theories (CFTs). For this purpose, we perturbatively compute the relative entropy between the vacuum reduced density matrix on disjoint subsystems and arbitrarily separable state in the limit where two subsystems A and B are well separated, then minimize the relative entropy with respect to the separable states. We argue that the result highly depends on the spectrum of CFT on the subsystems. When we have a few low energy spectrum of operators as in the case where the subsystems consist of finite number of spins in spin chain models, the REE is considerably smaller than the mutual information. However in general our perturbative scheme breaks down, and the REE can be as large as the mutual information.
- 1 Introduction and Summary
- 2 Properties of Relative Entropy of Entanglement
- 3 The Calculation of the Relative Entropy
- 4 Minimization
- 5 Next Leading Order
- 6 Contribution from the Next Lightest Primary
- 7 Conclusions and Discussions
- A Calculation of
- B On a replacement rule
1 Introduction and Summary
Quantum entanglement is one of the central ideas in modern theoretical physics. It does not only play crucial roles in quantum information theory but also has a broader range of applications, from condensed matter physics to string theory.
When we consider a bipartite pure state , we call the state does not have any quantum entanglement when it is represented by a direct product state . For pure states, the amount of quantum entanglement can correctly be measured by the entanglement entropy (or von Neumann entropy): , where is the reduced density matrix. This is because the entanglement entropy essentially counts the number of Bell pairs which can be distilled from a given pure state by local operations and classical communication (LOCC). In LOCC, we can act quantum operations on and separately and allow classical communications between and at the same time. It is important that the LOCC procedures, which convert a given state into Bell pairs, are reversible for pure states in an asymptotic sense444Instead of considering a given state itself, one sometimes discusses the procedures on copies of the original state followed by the asymptotic () limit. The argument about LOCC reversibility should be correctly taken into account in this regime.. Namely, after distilling the Bell pairs, one can reproduce the original pure state by performing LOCC on the given Bell pairs. In general, an amount of entanglement quantified by an appropriate entanglement measure has to be always less than the number of Bell pairs necessary to produce a given state by LOCC, and also to be greater than that of Bell pairs distillable from a given state by LOCC. Thus the reversibility guarantees that there is only one measure of quantum entanglement, namely the entanglement entropy . Refer to the reviews [2, 3, 4, 5, 6] for studies of entanglement entropy in quantum field theories and holography.
Next let us turn to a bipartite mixed state, which is described by a density matrix . A mixed state has no entanglement if is separable i.e.
where are positive coefficients such that and each of is a density matrix, which is hermitian and non-negative operator with the unit trace. However, the beautiful story which we find for pure states is missing for mixed states because the LOCC procedures of the conversion between a mixed state and Bell pairs is irreversible in general. Nevertheless, we can define an entanglement measure by a quantity which is monotonically decreasing under LOCC with a few more optional properties such as the asymptotic continuity. We write an entanglement measure for a given bipartite state as . Such an entanglement measure is far from unique as is clear from the irreversibility (for entanglement measures of mixed states refer to e.g. [7, 8] for excellent reviews).
So far, few calculations of genuine entanglement measures for mixed states have been performed for quantum field theories. The main reasons for this is that the known entanglement measures, such as the entanglement of formation , the relative entropy of entanglement and the squashed entanglement , all involve very complicated minimization procedures. A correlation measure for mixed state, called entanglement of purification , involves a slightly simpler minimization procedure, though it is not an entanglement measure. Recently a holographic dual of this quantity has been proposed in [10, 11] and computations of this quantity in field theories and spin chains have been performed in [11, 12] (for more progresses refer to [13, 14, 15, 16, 17, 18, 19]). There is another interesting quantity called the logarithmic negativity , which does not need any minimizations and thus has been successfully computed in two dimensional CFTs [21, 22, 23]. Though this quantity is monotone under LOCC, the asymptotic continuity condition and convexity are not satisfied. Thus it does not coincide with the entanglement entropy when the system is pure.
The main purpose of this paper is to initiate calculations of a true entanglement measure for mixed state in conformal field theories (CFTs). In particular, we focus on the relative entropy of entanglement [24, 25] among entanglement measures, motivated by recent progresses of computational techniques in CFTs of relative entropies [26, 27, 28, 29, 30, 31]. Several bounds for REE in quantum field theories have been obtained in [32, 33] via an algebraic quantum field theory approach555 In , an upper bound of in CFT is given: , where is the conformal dimension of lightest primary operator (except the identity) and is its degeneracy. This follows from Thm 14, Remark 5 of . Note that when , we can approximate in (235) in  by our via a conformal transformation . Our result in this paper is consistent with this bound and is actually stronger because the REE is at least bounded by the mutual information as in (5). (refer to  for an excellent review).
The relative entropy of entanglement (REE) is defined as follows. We can measure a distance between two density matrices and by the relative entropy:
A basic property of the relative entropy is , where the equality holds iff .
The REE is defined as the shortest distance in the sense of the relative entropy between a given bipartite state and an arbitrary separable state as follows:
where Sep denotes all separable states. It is obvious that iff is separable. Moreover, when is pure, coincides with the entanglement entropy .
In this paper we will study the REE for the vacuum reduced density matrix of CFTs on two disjoint subsystems in any dimensions. This REE quantifies how much two subsystems and are quantum mechanically entangled in a CFT vacuum. We will analyse the REE assuming the subsystems and are far apart in terms of power series of , where is the size of and , while is the geometrical distance between and .
Another useful measure of correlations between and is the mutual information:
Obviously from the definition of REE, we have the inequality
This upper bound can also be intuitively understood because the REE measures the amount of quantum entanglement, while the mutual information measures not only quantum entanglement but also classical correlations. When and are far apart, the mutual information for a CFT vacuum (its reduced density matrix is written as ) is approximated by the square of vacuum two point function of the (non-trivial) primary operator with the lowest conformal dimension (regardless to the positions of operators or the shapes of subsystems):
For example, the free massless Dirac fermion CFT in two dimensions corresponds to . Thus in our limit , the REE is at least as small as , as can be seen from its upper bound (5). Below we are interested in whether the REE can be much smaller than .
For general mixed states and , if is very small, the relative entropy becomes symmetric . Therefore, we will first calculate the relative entropy for arbitrary separable density matrices , and then take the infinitum with respect to the ensemble . The necessary ingredients for the calculation have been obtained in the previous paper  written by the one of the authors, including the vacuum modular Hamiltonian as well as the von Neumann entropy for any separable density matrices, assuming .
In this paper we first compute the contribution of the lightest primary operator to the relative entropy, then minimizing it by assuming it gives the dominant contribution in the large separation limit, as in case of the mutual information. We are able to show that we can make this contribution always vanish by appropriately choosing the separable state at any order of the perturbation. We also give an explanation why the separable state is indistinguishable from from the viewpoint of local observables.
However, the minimization becomes much more complicated when we include the effects of other operators with higher conformal dimensions. In this case, we find that our perturbative calculation is not enough, since we cannot suppress the expectation value of higher dimensional operators in general.
From these observations we argue that the behavior of REE is highly dependent on the operator spectrum of CFT in the subsystems. For a CFT with few low energy states such as the case where the subsystems consist of finite number of spins in spin chain models, the perturbative analysis is enough and we find that there is tiny quantum entanglement as . We can check this statement by having an independent argument in spin chain models.
However, in generic setups our perturbative expansion gets uncontrollable and this implies that the REE can be as large as the mutual information . Especially we expect for holographic CFTs, as the operator spectrum does not seem to allow us to optimize the minimizations in the definition of REE. On the other hand, since integrable CFTs such as the rational CFTs in two dimensions, have simple operator spectrum and algebra, there might be a chance that the REE can be smaller than the mutual information even when the subsystems are much larger than the lattice spacing. For further investigations, we probably need to develop methods which does not rely on perturbations.
The organization of this paper is as follows: In section 2, we review basic properties of the relative entropy of entanglement. In section 3, after explaining the basic set up, we compute the relative entropy between the vacuum reduced density matrix and an arbitrary separable state in the leading order of the large distance limit , based on results of . In section 4, we minimize the relative entropy with respect to the separable states. We find there alway be a separable state whose relative entropy is vanishing therefore at the quadratic order of perturbative expansions. In section 5 we take into account of higher order perturbative corrections, and argue they do not change our result under certain conditions. In section 6, we discuss the contribution from the next lightest primary, which shows the result of REE is very sensitive to the operator spectrum. In section 7, we will compare our results with other known results and discuss future problems. In the appendix we explain the details of our calculations.
2 Properties of Relative Entropy of Entanglement
The relative entropy of entanglement is defined by (3) for a bipartite quantum state , i.e. the shortest distance between and the set of separable states measured by the relative entropy.
2.1 Properties of REE
The properties of REE is summarized as follows (for more details, refer to [7, 8])666In this section we deal with the finite dimensional Hilbert space for simplicity. Most of the properties and the inequalities are also proven in the infinite dimensional setup, refer to [32, 36] for recent discussion.:
(i) Faithfulness: and if and only
if is separable.
(ii) Monotonicity: is monotonically decreasing under (stochastic)
(iii) Convexity: is convex i.e.
for any .
(iv) Continuity: is continuous respect to i.e. if and are close in trace distance, then the value of approaches that of 777There are many variations of the continuity of entanglement measures. In particular, REE is also asymptotic continuous, which is described by the limit of many copies and an important property in the axiomatic approach of entanglement measures. :
where is the Hilbert space and act on .
(v) Subadditivity: always satisfies the subadditivity . Note that it does not satisfy the additivity in general.
(vi) When is pure, reduces to the entanglement entropy . To see this, consider a pure state with the Schmidt decomposition
Indeed, one can easily check that of these states reduces to the entanglement entropy:
Above properties indicate that REE is a good generalization of entanglement entropy to a genuine entanglement measure for mixed states.
There are several upper/lower bounds for REE: As we have already mentioned, is bounded from above by the mutual information as , which follows directly from the definition of REE. Another upper bound is given the entanglement of formation , which is also a good measure of entanglement for mixed states. On the other hand, a lower bound is given by the distillable entanglement , which counts the number of EPR pairs extractable from a given state by LOCC. This bound also leads to an entropic inequality 888This inequality can be rewritten in terms of conditional entropy as , which was firstly derived in . by virtue of the hashing inequality . It may also be worth noting that there is no generic inequality relationship between REE and the negativity .
2.2 Quadratic Approximations
In the present paper we will deal with rather than for technical simplicity, where represents a separable state. This does not change the main results at the quadratic order of small perturbation of quantum state. Consider the case where and are very closed to each other
If we expand up to the quadratic order of , we find (see e.g.)
From this expression, it is clear that coincides with the reversed one up to the quadratic order
One can also understand this symmetry as a consequence from positivity and non-degeneracy of the relative entropy.
As an illustration, consider the case where and are density matrices, expressed as:
and treat and as infinitesimally small real parameters. We require for positivity of density matrix. If we only keep up to quadratic terms of them, we can confirm the equivalence (13) explicitly as follows:
In , an entanglement measure so-called the reversed REE was introduced in the same spirit of REE with reversed components:
where the minimization is restricted to a class of separable states locally identical to i.e. . This quantity also satisfies many properties of a good entanglement measure, especially the additivity. However, when is pure, generically diverges (or trivially vanishes) and thus it can not be regarded as an appropriate generalization of entanglement entropy for mixed states.
3 The Calculation of the Relative Entropy
3.1 Set up
We begin with a CFT on a dimensional flat space , and two ball shaped regions and , with the radius and the distance . In this section we estimate the relative entropy between the vacuum reduced density matrix on defined by,
and an arbitrary separable density matrix , in the large distance limit . 999Precisely speaking, in the actual computation we regard this set up as a particular limit of the system on a cylinder . Let be the radius of the spacial sphere , then the large distance limit in is equivalent to the double scaling limit on the cylinder, (18)
It is convenient to split the relative entropy into two parts:
where is the von Neumann entropy of the separable density matrix and is the modular Hamiltonian of ,
3.2 The calculation of
In this subsection we explain how to compute the von Neumann entropy, for a separable state . This is a slight generalization of the previous calculation done in [28, 29]. Here we only outline the calculation, and leave details in appendix A.
For this purpose, we employ the usual replica trick,
This Rényi entropy can be expanded as
We first compute the right hand side of (22) for reduced density matrices of global excitations, ( corresponds to the vacuum: )
on cylinder with the metric,
We then read off the result for arbitrary from it. We take both subsystems to be isomorphic to the ball shaped region on the spatial sphere ,
Also it is important to notice that in this calculation we do not need to specify the distance between two regions.
State operator correspondence allows us to write the quantities in the right hand side in terms of the 2n point correlation functions on the covering space ,
where is the local operator corresponding to the global state and there is a similar relation for the subsystem and the global state ; also denotes the vacuum partition function on . The correlation functions are normalized such that .
The covering space is equipped with the metric,
and the locations of the local operators are given by
The small subsystem size limit corresponds to choose the particular channel of these correlation functions. There one can expand them by OPE. By picking up the contribution of the lightest primary operator with the conformal dimension . By taking the analytic continuation of the Rényi entropy, we finally obtain101010 We choose the component to be reduced density matrices of the vacuum, ie (29)
where is the vacuum modular Hamiltonian on the region . In a CFT vacuum on a ball shaped region, is given by a simple integral of stress tensor. We do not need its precise form, as it is always canceled with other contributions in the relative entropies.
Meanwhile, the von Neumann entropy of a reduced density matrix on the single subsystem is given by (see for example  )
and is the OPE coefficient of the primary .
Our result indicates the von Neumann entropy of gets factorized
up to order, and the effect of the classical correlation first enters at order. If we write the correlation part in terms of original separable density matrix
therefore this part is basically the square of the connected part of the two point function evaluated on .
This can be compared with the mutual information of a reduced density matrix at this order ,
and the two results are related by the exchange . Indeed, as is clear from the discussion in the appendix B, the derivations of the two results are identical to each other, once we identify the two correlation functions .
3.3 Modular Hamiltonian and Calculation of
Having calculated the von Neumann entropy part, let us move on to the modular Hamiltonian part,
It was shown in , takes following form,
and in the large distance limit , we have
This was obtained by starting from the expression of von Neumann entropy for a generic state which is related to the mutual information (35), and applying the “first law trick”, which will be reviewed in section 5. More details of the discussion can be again found in . in (38) denotes the constant part of the modular Hamiltonian. We need this part in order to make sure the relation
and coincides with the value of the vacuum mutual information (6). Then,
3.4 Net result
Notice that there are higher order corrections. We will discuss on this in section 5.
In the previous section we computed the relative entropy between the vacuum reduced density matrix and an arbitrary separable density matrix in the large distance limit keeping only the contributions from the lightest primary operator. In this section, we would like to find the separable density matrix that minimizes the relative entropy and compute the relative entropy of entanglement . We choose the separable state to be in the form:
where is a small parameter and . In addition, are arbitrary density matrices with non-vanishing one-point function of the primary , which is defined to be
We would like to keep only quadratic perturbations to so that we have as in (13). To implement this, we define the small perturbations and by
Our perturbations are parameterized by the following two small parameters:
It will be useful to note that the mutual information (35) when and are far apart is at the quadratic order. Indeed, we have
In this parametrization, our result in the small interval expansion (41) is expresses as follows up to the quadratic order of and :
By varying (or equally ) to minimize the relative entropy, we obtain
Next we vary the choice of the state so that the one-point function (43) gets larger such that is still very small. It is obvious that we can define such a state with an arbitrary large in the continuous limit of field theories. In the limit,
we find that the infimum of the relative entropy is vanishing
up to the quadratic order. Note that at this infimum, the separable state is locally vacuum on the region and , i.e. .
This manifestly shows that the REE is much smaller than the mutual information
in the limit where and are far apart. However, notice again that in this calculation we only keep contributions from the lightest primary operator.
4.1 An Interpretation
There is an intuitive way to understand why the separable density matrix is indistinguishable from the vacuum reduced density matrix .
It is useful to write the separable density matrix,
Notice that this separable density matrix reproduces all correlation functions of on the disjoint region , as it should be. In our small subsystem limit, if we truncate the spectrum to the lightest primary operator, we only need to reproduce one and two point functions of :
We can easily see that this is indeed the case,
As we will see in the final section, this result corresponds to a critical spin chain example where the subsystem and consist of finite number of spins.
Furthermore, this observation makes it clear that for disjoint subsystems the separable density matrix which minimize the analogous relative entropy is given by
One can easily see that the density matrix reproduce all k point functions of
4.2 An example of the separable density matrix in 2d CFT
One can indeed construct a one parameter family of density matrices , of which realizes the infimum in a class of two dimensional conformal field theory. Suppose that the lightest primary operator of the 2d CFT in question is the stress tensor . The we can take defined by
where is a boundary state of the CFT, and is the normalization factor. Then its stress tensor expectation value is