Towards an entanglement measure for mixed states in CFTs based on relative entropy

Relative entropy of entanglement (REE) is an entanglement measure of bipartite mixed states, defined by the minimum of the relative entropy S(ρAB ||σAB ) between a given mixed state ρAB and an arbitrary separable state σAB . The REE is always bounded by the mutual information IAB = S(ρAB ||ρA ⊗ ρB) 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 ρAB0 on disjoint subsystems A ∪ B and arbitrarily separable state σAB 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.


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 |Ψ AB , we call the state does not have any quantum entanglement when it is represented by a direct product state |Ψ 1 A ⊗ |Ψ 2 B .For pure states, the the amount of quantum entanglement can correctly be measured by the entanglement entropy (or von Neumann entropy): S(ρ A ) = S(ρ B ) ≡ −tr[ρ A log ρ A ], where ρ A ≡ tr B |Ψ Ψ| 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 |Ψ AB by local operations and classical communication (LOCC).In LOCC, we can act quantum operations on A and B separately and allow classical communications between A and B at the same time.It is important that the LOCC procedures, which convert a given pure state into Bell pairs, are reversible in an asymptotic sense.Thus there is only one measure of quantum entanglement, namely the entanglement entropy.Refer to the reviews [1,2,3,4,5] 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 ρ AB .A mixed state σ AB has no entanglement if σ AB is separable i.e.
where p a are positive coefficients such that a p a = 1 and each of ρ a A,B 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 ρ AB as E # (ρ AB ).Such an entanglement measure is far from unique as is clear from the irreversibility (for entanglement measures of mixed states refer to e.g.[6,7] 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 E F , the relative entropy of entanglement E R and the squashed entanglement E Sq , all involve very complicated minimization procedures.A correlation measure for mixed state, called entanglement of purification [8], involves a slightly simpler minimization procedure, though it is not an entanglement measure.Recently a holographic dual of this quantity has been proposed in [9,10] and computations of this quantity in field theories and spin chains have been performed in [10,11] (for more progresses refer to [12,13,14,15,16,17,18]).There is another interesting quantity called the logarithmic negativity [19], which does not need any minimizations and thus has been successfully computed in two dimensional CFTs [20,21,22].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 S(ρ A ) when the system AB 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 E R (ρ AB ) [23,24] among entanglement measures, motivated by recent progresses of computational techniques in CFTs of relative entropies [25,26,27,28,29,30].Several bounds for REE in quantum field theories have been obtained in [31,32] via an algebraic quantum field theory approach4 (refer to [34] for an excellent review).
The relative entropy of entanglement (REE) is defined as follows.We can measure a distance between two density matrices ρ and σ by the relative entropy: A basic property of the relative entropy is S(ρ||σ) ≥ 0, where the equality holds iff ρ = σ.
The REE is defined as the shortest distance in the sense of the relative entropy between a given bipartite state ρ AB and an arbitrary separable state σ AB as follows: where Sep denotes all separable states.It is obvious that E R (ρ AB ) = 0 iff ρ AB is separable.Moreover, when ρ AB is pure, E R (ρ AB ) coincides with the entanglement entropy S(ρ A ).
In this paper we will study the REE E R for the vacuum reduced density matrix ρ 0 AB of CFTs on two disjoint subsystems A ∪ B(≡ AB) in any dimensions.This REE quantifies how much two subsystems A and B are quantum mechanically entangled in a CFT vacuum.We will analyse the REE assuming the subsystems A and B are far apart in terms of power series of l/R ≪ 1, where l is the size of A and B, while R is the geometrical distance between A and B.
Another useful measure of correlations between A and B 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 A and B are far apart, the mutual information for a CFT vacuum (its reduced density matrix is written as ρ 0 AB ) is approximated by the square of vacuum two point function O A O B of the (non-trivial) primary operator O with the lowest conformal dimension ∆: For example, the free massless Dirac fermion CFT in two dimensions corresponds to ∆ = 1/2.Thus in our limit l/R ≪ 1, the REE is at least as small as (l/R) 4∆ , as can be seen from its upper bound (5).Below we are interested in whether the REE can be much smaller than (l/R) 4∆ .For general mixed states ρ and σ, if ρ−σ is very small, the relative entropy becomes symmetric S(σ||ρ) ≃ S(ρ||σ).Therefore, we will first calculate the relative entropy S(σ AB ||ρ AB ) for arbitrary separable density matrices σ AB , and then take the infinitum with respect to the ensemble {p a , ρ a A , ρ a B }.The necessary ingredients for the calculation have been obtained in the previous paper [28] written by the one of the authors, including the vacuum modular Hamiltonian K AB = − log ρ AB as well as the von Neumann entropy S(σ AB ) for any separable density matrices, assuming l/R ≪ 1.
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 ρ 0 AB 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 depending 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 E R (ρ 0 AB ) ≪ I(ρ 0 AB ).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 I AB .Especially we expect E R (ρ 0 AB ) ≃ I(ρ 0 AB ) 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 S(σ AB ||ρ 0 AB ) between the vacuum reduced density matrix ρ 0 AB and an arbitrary separable state σ AB in the leading order of the large distance limit l/R → 0, based on results of [28].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 E R (ρ 0 AB ) = 0 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.

Properties of Relative Entropy of Entanglement
The relative entropy of entanglement E R (ρ AB ) is defined by (3) for a bipartite quantum state ρ AB , i.e. the shortest distance between ρ AB and the set of separable states measured by the relative entropy.

Properties of REE
The properties of REE is summarized as follows (for more details, refer to [6,7]): (i) Faithfulness: E R (ρ AB ) ≥ 0 and E R (ρ AB ) = 0 if and only if ρ AB is separable.
(iv) Continuity: E R (ρ AB ) is continuous respect to ρ AB i.e. if ρ AB and σ AB are close in trace distance, then the value of E R (ρ AB ) approaches that of E R (σ AB )5 : (vi) When ρ AB is pure, E R (ρ AB ) reduces to the entanglement entropy S(ρ A )(= S(ρ B )).To see this, consider a pure state ρ AB = |ψ ψ| AB with the Schmidt decomposition where λ i ≥ 0, i λ i = 1.Then it is shown that the closest separable state of ρ AB which reaches the minimization in (3) is given by a simple form [24,35] Indeed, one can easily check that S(ρ AB ||σ AB ) 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, E R (ρ AB ) is bounded from above by the mutual information I(ρ AB ) = S(ρ AB ||ρ A ⊗ ρ B ) as E R (ρ AB ) ≤ I(ρ AB ), which follows directly from the definition of REE.Another upper bound is given the entanglement of formation E R (ρ AB ) ≤ E F (ρ AB ), which is also a good measure of entanglement for mixed states.On the other hand, a lower bound is given by the distillable entanglement E D (ρ AB ) ≤ E R (ρ AB ), which counts the number of EPR pairs extractable from a given state ρ AB by LOCC.This bound also leads to an entropic inequality E R (ρ AB ) ≥ max[S(ρ A ), S(ρ B )] − S(ρ AB )6 by virtue of the hashing inequality [36].It may also be worth noting that there is no generic inequality relationship between REE and the negativity [37].

Quadratic Approximations
In the present paper we will deal with S(σ AB ||ρ AB ) rather than S(ρ AB ||σ AB ) for technical simplicity, where σ AB 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 ρ and σ are very closed to each other If we expand S(ρ||σ) up to the quadratic order of δρ, we find (see e.g.[29]) From this expression, it is clear that S(ρ||σ) coincides with the reversed one S(σ||ρ) 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 σ and ρ are 2 × 2 density matrices, expressed as: and treat δ 1 and δ 2 as infinitesimally small real parameters.We require 0 < α < 1 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 [38], 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 ρ AB i.e. tr B (σ AB ) = ρ A , tr A (σ AB ) = ρ B .This quantity also satisfies many properties of a good entanglement measure, especially the additivity.However, when ρ AB is pure, E RR (ρ AB ) 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

Set up
We begin with a CFT on a d dimensional flat space R d , and two ball shaped regions A and B, with the radius l and the distance R. In this section we estimate the relative entropy S(σ AB ||ρ 0 AB ) between the vacuum reduced density matrix on A ∪ B defined by, and an arbitrary separable density matrix σ AB , in the large distance limit l/R → 0. 7 7 Precisely speaking, in the actual computation we regard this set up as a particular limit of the system on a cylinder R × S d−1 .Let L be the radius of the spacial sphere S d−1 , then the large distance limit in R is equivalent to the double scaling limit on the cylinder, It is convenient to split the relative entropy into two parts: where S(σ AB ) is the von Neumann entropy of the separable density matrix and K AB is the modular Hamiltonian of ρ 0 AB ,

The calculation of S(σ AB )
In this subsection we explain how to compute the von Neumann entropy, S(σ AB ) for a separable state σ AB .This is a slight generalization of the previous calculation done in [27,28].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, |X a , |Y a (a = 0 corresponds to the vacuum: on cylinder R × S d−1 with the metric, We then read off the result for arbitrary ρ a A , ρ a B from it.We take both subsystems A, B 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 X a k (w k ) is the local operator corresponding to the global state |X a k and there is a similar relation for the subsystem B and the global state |Y a k ; also Z (n) A denotes the vacuum partition function on Σ n .The correlation functions are normalized such that 1 Σn = 1.
The covering space Σ n is equipped with the metric, and the locations of the local operators are given by The small subsystem size limit l → 0 corresponds to choose the particular channel w k → ŵk of these correlation functions.There one can expand them by OPE.By picking up the contribution of the lightest primary operator O with the conformal dimension ∆.By taking the analytic continuation n → 1 of the Rényi entropy, we finally obtain where K 0 A is the vacuum modular Hamiltonian on the region A. In a CFT vacuum on a ball shaped region, K 0 A 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 ρ A on the single subsystem A is given by (see for example [27] ) with and C OOO is the OPE coefficient of the primary O.
Our result indicates the von Neumann entropy of σ AB gets factorized up to l 3∆ order, and the effect of the classical correlation first enters at l 4∆ order.If we write the correlation part in terms of original separable density matrix σ AB therefore this part is basically the square of the connected part of the two point function This can be compared with the mutual information I AB (ρ 0 AB ) of a reduced density matrix ρ 0 AB at this l 4∆ order [28], and the two results are related by the exchange σ AB ↔ ρ 0 AB .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 It was shown in [28], K AB takes following form, and in the large distance limit l R → 0, we have This was obtained by starting from the expression of von Neumann entropy S(ρ AB ) for a generic state ρ AB which is related to the mutual information (34), and applying the "first law trick", which will be reviewed in section 5.More details of the discussion can be again found in [28].I AB in (37) denotes the constant part of the modular Hamiltonian.We need this part in order to make sure the relation and I AB coincides with the value of the vacuum mutual information (6).Then,

Net result
Combining (29) (39), the relative entropy we would like to minimize is 4 Minimization In the previous section we computed the relative entropy S(σ AB ||ρ 0 AB ) between the vacuum reduced density matrix and an arbitrary separable density matrix σ AB in the large distance limit l R → 0 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 E R (ρ 0 AB ).We choose the separable state σ AB to be in the form: where ǫ is a small parameter and ρ 0 A = tr B ρ 0 AB .In addition, ρ 1 A,B are arbitrary density matrices with non-vanishing one-point function of the primary O, which is defined to be We would like to keep only quadratic perturbations to S(σ AB ||ρ 0 AB ) so that we have S(σ AB ||ρ 0 AB ) ≃ S(ρ 0 AB ||σ AB ) as in (13).To implement this, we define the small perturbations δρ 0 and δρ 1 by such that Our perturbations are parameterized by the following two small parameters: It will be useful to note that the mutual information (34) when A and B are far apart is at the quadratic order.Indeed, we have In this parametrization, our result in the small interval expansion ( 40) is expresses as follows up to the quadratic order of Z and W : By varying Z (or equally ǫ) to minimize the relative entropy, we obtain at Z = a 2∆ x 2 2a ∆ +a 2∆ x 2 W . Next we vary the choice of the state ρ 1 A,B so that the one-point function (42) gets larger such that Z = ǫx 2 is still very small.It is obvious that we can define such a state with an arbitrary large x 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 A and B, i.e. tr A,B σ AB = tr A,B ρ AB .Finally, by employing the relation ( 13) up to the quadratic order of our perturbation (44), we obtain the estimation of REE: This manifestly shows that the REE is much smaller than the mutual information in the limit (l/R) → 0 where A and B are far apart.However, notice again that in this calculation we only keep contributions from the lightest primary operator.

An Interpretation
There is an intuitive way to understand why the separable density matrix σ AB is indistinguishable from the vacuum reduced density matrix ρ 0 AB .It is useful to write the separable density matrix, Notice that this separable density matrix σ AB reproduces all correlation functions of ρ AB on the disjoint region A ∪ B, 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 {1, O}: 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 A and B consist of finite number of spins.Furthermore, this observation makes it clear that for m disjoint subsystems A 1 ∪ • • • A m the separable density matrix which minimize the analogous relative entropy is given by with One can easily see that the density matrix reproduce all k (≤ m)point functions of O

An example of the separable density matrix σ AB in 2d CFT
One can indeed construct a one parameter family of density matrices {ρ β }, β → 0 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 O = T zz .The we can take ρ 1 A defined by where |B is a boundary state of the CFT, and N is the normalization factor.Then its stress tensor expectation value is and x β → ∞ when β → 0. This implies that if we define ρ β by then the density matrix, is indistinguishable from the vacuum reduced density matrix ρ 0 AB , at least in the l R 8 order.
If we consider a discretized lattice model such as spin chains and introduce the lattice spacing a, then the minimum possible value of the parameter β is O(a).In more general, we expect that for a generic operator with the dimension ∆, the maximal value of x will behave like 5 Next Leading Order In the previous section we found the relative entropy of entanglement E R (ρ 0 AB ) is vanishing up to l R 4∆ order.It is natural to ask whether higher order corrections can modify this result or not.Motivated by this question, in this section we compute S(σ AB ||ρ 0 AB ) up to l R 6∆ by again assuming the lightest primary plays still a dominant role at this order.We also use the fact that the one point functions of the separable state σ AB must be vanishing, in order to reproduce the vacuum one point functions.Restricting σ AB to be in this class of states drastically simplifies the computation below.Notice that from (30) this in particular implies that S(

S(σ AB )
The von Neumann entropy S(σ AB ) can be computed along the line of section 3.2 by further expanding the correlator (22), in particular allowing 3 O s to propagate in the internal lines of it.The final result of the cubic order is given by (see appendix A for more details): and we can write S(σ AB ) where

AB
Next let us compute the expectation value of the modular Hamiltonian at this order.First of all, the von Neumann entropy of a reduced density matrix ρ AB satisfying the locally vacuum condition (63), (64) (but not necessary a separable state) is directly related to its mutual information, where ρ 0 A,B is the vacuum reduced density matrix on the region A, B respectively.This mutual information can be computed either directly by a correlator with twist operators in the replica trick or indirectly from S(σ AB ) by the replacement in (66) 8I AB (ρ AB ) where We can use this expression of mutual information for ρ AB satisfying the locally vacuum condition to read off the form of vacuum modular Hamiltonian K 0 AB at l 6∆ order, by using the first law trick.Imagine starting from the vacuum reduced density matrix ρ 0 AB , and slightly deform it ρ 0 AB → ρ AB = ρ 0 AB + δρ 0 AB , then the value of mutual information I(ρ AB ) as well as entanglement entropy I(ρ AB ) are changed by the deformation.In particular the first order change satisfies the first law.If we know the form of S(ρ AB ) for any ρ AB , we can read off the form of modular Hamiltonian from the above equation.In our current case it goes like, with W = W (ρ 0 AB ).Since this is true for any δρ AB satisfying the locally vacuum condition, we derive the form of modular Hamiltonian at this order where a AB is the constant part of the modular Hamiltonian, fixed by the relation By plugging these expressions, we get Again notice that the form of S(ρ AB ) is not generic, and valid only when ρ AB satisfies the locally vacuum condition.Therefore the form of modular Hamiltonian we derive from the expression is only true when it is acted on the space of reduced density matrix satisfying the condition.However it is sufficient for our purpose of computing the expectation value of vacuum modular Hamiltonian with respect to a separable σ AB which satisfies the condition.
A more rigorous argument is as follows.Again consider the change of the density matrix ρ 0 AB → σ AB = ρ 0 AB + δρ 0 AB , then From this we can read off the value which we want as follows in the derivation we do not need to use the precise form of the modular Hamiltonian.

Minimization
Combining these results, (66) and ( 74), we obtain the expression of relative entropy up to this order l 6∆

S(σ
This function again has a minima at Z = W , where S(σ AB ||ρ 0 AB ) is vanishing.One may worry that this relative entropy negatively diverges in Z → ∞ limit.Of course this is just an artifact of our truncation the perturbative expansion, and the local minima Z = W should be the global minima, as is clear from the argument found in section 4.1.
As long as we assume that only the primary operator O is relevant, the above argument of vanishing S(σ AB ||ρ 0 AB ) at Z = W continues to be true in all orders in the perturbative expansion with respect to Z and W . First, in this expansion the von Neumann entropy S(σ AB ) is expressed as where b n s are unknown coefficients depending on ∆ and C OOO , though we do not need their precise values in the argument below.The modular Hamiltonian expectation value tr σ AB K 0 AB can again be read off from the mutual information of locally vacuum state, which is related to (76) by replacing Z to the corresponding two point function, Finally the relative entropy is given by By taking derivative with respect to Z, we see that each term in the expansion has the minimum at W = Z where the relative entropy vanishes.
In this section we have shown that under the assumption that the primary O, which as the lowest conformal dimension, gives dominant contributions in each order of l R expansions, the minimum of relative entropy S(σ AB ||ρ 0 AB ) vanishes.Even though we cannot use the relation (13) for perturbations higher than quadratic order, the vanishing relative entropy shows that the vacuum reduced density matrix ρ 0 AB is very closed to the separable states at each order of perturbation.Therefore our result here suggests that the reversed one S(ρ 0 AB ||σ AB ) and the REE E R (ρ 0 AB ) vanishes in each perturbative order.

Contribution from the Next Lightest Primary
So far, we have been discussing possible higher order corrections due to the exchanges of the lightest primary operator.There is another type of corrections to the relative entropy, which is coming from exchanges of heavier operators.To get some intuitions for this, here we study the effect of the next lightest primary O N L with the conformal dimension ∆ N L .
If we assume the locally vacuum condition, the contribution of O N L to the relative entropy first enters at l 2∆+2∆ NL order.From the replica calculation we find the expression of S(σ AB ), up to this order, Similarly the mutual information of generic ρ AB up to this order is Notice however the second term vanishes once we set ρ AB = ρ 0 AB thus the modular Hamiltonian part does not receive correction at this order.
The net result of the relative entropy up to this order is therefore We then minimize this relative entropy.If we can regard second term of (81) as a perturbative correction to the first term of order l 2∆ NL , then the first order correction to the minimum value of the relative entropy is evaluated just by substituting the separable density matrix (53) that minimizes the relative entropy at the leading order.The value of Z 1 for this separable state is given by In order for this to work, we need to require x ≫ x N L .However it seems difficult to find such ρ 1 in general especially when we need to take x to be large.If we naively construct such ρ 1 with large x, we fail.This is because the maximal value of x and x N L scales as in (62) in terms of the lattice spacing a: x ∼ (l/a) ∆ and x N L ∼ (l/a) ∆ NL .Thus we generically expect x N L ≫ x, assuming l ≫ a.
From the above analysis of the contribution from the next lightest operator, it does not seem to be possible to reduce the relative entropy S(σ AB ||ρ 0 AB ) in generic CFTs, by fine-tuning the separable state σ AB as far as we assume our perturbative analysis.

Conclusions and Discussions
In this paper, we considered the relative entropy of entanglement (REE) E R (ρ 0 AB ) for CFT vacua.We focus on the case where the subsystem A and B are largely separated compared with their sizes.In this limit we can employ the OPE expansions in terms of operators localized in A and B.

Lightest Operator Dominant Case and Spin Chain Example
In the first part of this paper, we assumed that the lightest primary operator gives the dominant contribution.Under this assumption we were able to show that E R (ρ 0 AB ) gets much smaller than the mutual information I(ρ AB ) as in (51) and (52).This means that the vacuum reduced density matrix ρ AB is an almost separable state.Moreover, under the assumption that the lightest primary is always dominant, we showed that S(σ AB ||ρ 0 AB ) for a certain separable state σ AB , is vanishing at each order of power expansions of l R and this strongly suggests that the REE E R (ρ 0 AB ) also vanishes in the same way.Thus we find that the correlations between A and B are classical in this case.
We expect that the assumption of taking into account only the lightest primary can be justified when we consider a critical spin chain model and the subsystems consist of finite numbers of spins.For this, let us consider a S = 1/2 spin chain at a quantum critical point and choose the subsystem A and B to be the p-th and (p + R)-th spin, denoted by σ A i and σ B i , where i = 1, 2, 3 i.e. the Pauli matrices, which satisfy the relation Tr[σ i σ j ] = 2δ ij .The correlation function looks like where ∆ is the dimension of the spin operator.Note that when the distance R between two spins are large the magnitude γ gets very small.In this setup, the reduced density matrix for AB is given by In the 4 × 4 matrix form this reads The requirement of positivity of density matrix is expresses as −1 < γ < 1/3.If γ is small as we consider, this condition is clearly satisfied.Since the dimension H A ⊗H B is less than six, we know that the condition of separability is equivalent to the PPT criterion (positivity under partial transposition) [39].The density matrix under the partial transposition (transposition w.r.t B) reads In this case the PPT criterion says that ρ AB is separable if and only if − 1 3 < γ < 1.In summary ρ AB is separable when −1/3 < γ < 1/3 and is not separable (i.e. is entangled) when −1 < γ < −1/3.Thus, in our spin chain example, when the distance R between A and B are large (i.e.γ is very small), we can conclude that ρ AB is separable and the logarithmic negativity defined by E = log |(ρ AB ) T B | is vanishing, where T B is transposition only for B (called partial transposition).
For a larger spin S ≥ 1, or for larger subsystems A and B, the PPT criterion and separability are not equivalent.However, still it is known that the state (in a finite dimensional Hilbert space) which is very closed to the maximally mixed state ρ = I N N is separable [40].Therefore if two spins are far apart and their correlation functions are small, we can apply this theorem to find that ρ AB is separable.Indeed, the above results for spin chains are consistent with our field theoretic result that the REE is vanishing in our perturbation theory.

Generic Cases and Holographic CFTs
In the later part of this paper, we estimated the contribution from the next lightest primary.This analysis tells us that the higher dimensional operators can give substantial contributions to the relative entropy in general, which violates our perturbation theory.The main reason for this is that if we want to choose a state ρ 1 with a very large expectation value of the lightest primary, then the expectation value of a heavier operator for the same state also inevitably gets larger.
For example, if we consider holographic CFTs, the lightest primary is typically a single trace operator.The double trace operator has the contribution x double = x 2 single and thus cannot be negligible.This suggests that in holographic CFT, we have E R (ρ 0 AB ) ≃ I AB (ρ 0 AB ), i.e. the correlations between A and B origin from quantum entanglement. 9omputations of the REEs for integrable CFTs, such as rational CFTs in two dimensions, will need careful treatments.Interestingly, in [20,21,22], the logarithmic negativity in the same setup as ours was computed in two dimensional CFTs and spin chains and was shown to be much smaller than any powers of l/R for rational CFTs.The logarithmic negativity is known to be monotone under LOCC and is vanishing for all separable states, though can be zero even for non-separable states.In this sense, the relation between the REE and logarithmic negativity is not straightforward.However, this result strongly implies that the quantum entanglement is highly reduced.In our analysis of REE, since the primary operator spectrum and its OPE algebra are simple, it might be possible that the argument for generic CFTs in the above cannot be applied.If so, the REE can be smaller.To completely answer this question, we need to develop calculations of relative entropy beyond our perturbation theory, which is an interesting future problem.For a moment we consider the density matrices coming from tracing out global excited states |X a , |Y a on cylinder (23), so that their Rényi entropies are computed by (after applying several conformal mappings) the corresponding correlation function on n sheet covering space where the locations of these operators w k , ŵk are defined in (28).Note also that the correlation functions are normalized such that 1 Σn = 1.In the small subsystem size limit 2l → 0, w k → ŵk .Also we have From this we have an expression of the Rényi entropy in terms of correlation functions, A Z (1) (w ′ k , ŵ′ k ) are again the locations of the local operators for the subsystem B. The strategy to calculate the right hand side of (89) is as usual, expanding the correlation functions by using OPEs where α k s are the operators propagating the internal line, and by ∆ α k we denote the scaling dimension of α k .We also have similar expansion of Y 's Using these formulae A Z (1) where (93) When the subsystem size l is small, α k can only be either identity 1 or the first non trivial primary O with the scaling dimension ∆, α k ∈ {1, O}, and similarly, β k ∈ {1, O}.This implies that we have the following expansion of tr σ n AB (92) in terms of l ∆ , tr σ n AB • (Z A Z (1) In the next few subsections we calculate these coefficients.
A.1 L (n) 0 : the first law part Only the trivial operator configuration can contribute to the coefficient where K 0 A , K 0 B is vacuum modular Hamiltonian of region A and B respectively.This part is just an analog of the first law part of excited state entanglement entropy.

A.2 L (n) 1
Configurations in which only one non trivial operator is present are not allowed because every vacuum one point function vanishes.Therefore L In this case two types of operator configuration can contribute to the coefficient.One is and In both cases there are two non trivial operators.
(99) We are only interested in n → 1 limit.In this case we can set n = 1 in J O1 as the sum of two point function j O(w q 1 )O(w j ) is already proportional to n − 1 [43], Combining them, we conclude, In this term again we have two types of contributions (104) As in the case of L (n) 2 , the first contribution generates the cubic order of the von Neuman entrpy on region A, S(σ A ) which was explained in (30) , and similarly the second contribution generates the cubic order of S(σ B ). Therefore we conclude, In this case we have The precise form of I j,k q 1 ,q 2 highly depends on the value of the indices.For example, when (j = q 2 , k = q 1 ), We can compare this expression to (51) of [28] .They can be identified by the replace- Again this can be compare to (59) of [28], and they are identified by O α O β → J O1 J 1O .
The strategy to calculate the sum (106) is almost same as the calculation of Appendix A of [28] ie, first computing the sum with respect to j, k with fixed q 1 , q 2 , then performing the sum with respect to q 1 , q 2 .Indeed, we can easily convince ourself that we can derive the result of the sum (106) from (69) of Appendix A of [28] , just by the replacing O α O β in [28] to J OO and O α O β to J O1 J 1O .Therefore the final result is We similarly have L (n) 5 term.This term can be relevant in section 5 in which we compute the relative entropy up to l 6∆ term by assuming the locally vacuum condition.However if we assume this condition, L (n) 5 term is vanishing, therefore we can ignore this term.

A.7 L (n) 6
We can also compute the one more higher term L n 6 once we assume the locally vacuum condition 63.
From the OPE expansion (92) and the condition 63, the result is, It is hard to directly perform the sum in right hand side and analytically continue the result in n.However we can read off the outcome from (5.15) of [30] where they computed the entangle entropy of an excited state at cubic order, lim In our case (112) we have lim ( We can see that up to the order of l 2∆ the entropy splits, S = S( p a ρ a A ) + S( p a ρ a B ). However this no longer holds at the l 4∆ order.

B On a replacement rule
In the body of the paper, we used the fact that S(ρ AB ) is related to S(σ AB ) by the replacement, In this appendix we prove this prescription.For simplicity we consider the case where ρ AB is the reduced density matrix of a pure state, and for σ AB , (23).
The Rényi entropy trρ n AB has an expression in terms of a correlation function of the twist defect D n [44], trρ the correlation function is evaluated on the cyclic orbifold (CF T ) ⊗n /Z n of the original CFT.Here we take V (∞)V (0) = 1.In the small subsystem size limit |A|, |B| → 0 one can expand the twist defect in terms of local operators, with (93) (129) In the n → 1 limit, these two expressions (127),(128) are related by the identification,

Figure 1 :
Figure 1: The choice of subsystem A and B to define the REE E R (ρ AB ).

3. 3
Modular Hamiltonian and Calculation of tr σ AB K 0 AB Having calculated the von Neumann entropy part, let us move on to the modular Hamiltonian part, tr

Figure 2 :
Figure 2: A graphical representation of the expansion the Renyi entropy