Capacity of entanglement in random pure state

We compute the capacity of entanglement in the bipartite random pure state model using the replica method. We find the exact expression of the capacity of entanglement which is valid for a finite dimension of the Hilbert space. We argue that in the gravitational path integral, the capacity of entanglement receives contributions only from the sub-leading saddle points corresponding to the partially connected geometries.


Introduction
In recent papers [1,2], the Page curve of the Hawking radiation [3] is reproduced from the replica computation of the entanglement entropy (see also [4] for a review).As argued by Page in [5], the entropy computation of the Hawking radiation is nicely modeled by the random pure state |Ψ in a bipartite Hilbert space Here the subsystems A and B correspond to the Hawing radiation and the black hole, respectively.From the reduced density matrix ρ A of the subsystem A we can compute the entanglement entropy S A where the bracket • • • denotes the ensemble average over the random pure state |Ψ .The exact form of S A (4.5) is obtained in [5] as a function of the dimensions d A , d B of the Hilbert spaces (1.1) As discussed in [3], S A in the random pure state model exhibits a similar behavior as the Page curve for the Hawking radiation from an evaporating black hole.In a recent paper [6], it is argued that the capacity of entanglement C A introduced in [7] is a useful quantity to diagnose the phase transition around the Page time.C A is defined by where K = − log ρ A is the modular Hamiltonian.In other words, C A measures the fluctuation of the modular Hamiltonian.See e.g.[8][9][10] for the study of C A in various models.
In this paper, we compute the capacity of entanglement in the random pure state model using the replica method.From the ensemble average of Tr ρ n A over the random pure state, we can compute the entanglement entropy S A and the capacity of entanglement C A as the derivative with respect to the replica number n at n = 1 (1.6) We find the exact form of C A as a function of the dimensions d A , d B of the Hilbert spaces H A , H B .The exact expression of C A in (4.9) is the main result of this paper.
As discussed in [1,2], fully disconnected or fully connected geometries dominate in the replica computation of the entropy S A , and their contributions exchange dominance around the Page time.In the case of the capacity C A , it turns out that the leading saddle point from the fully disconnected or fully connected geometries does not contribute to C A , and it receives non-zero contributions only from the sub-leading saddle points corresponding to the partially connected geometries.This is consistent with the result of [10] that C A is a measure of the partial entanglement.
This paper is organized as follows.In section 2, we review the random pure state model and the known exact result of Tr ρ n A .We find a new formula of Tr ρ n A in terms of the Narayana number (2.13), which is useful for the replica computation of the entanglement entropy S A and the capacity of entanglement C A .In section 3, we compute S A and C A in the planar limit using the replica method.Our computation shows that the leading saddle point does not contributes to C A and it receives contributions only from sub-leading saddle points.In section 4, we compute the exact S A and C A using the replica method.Finally we conclude in section 5.

Random pure state model
In this section, let us briefly review the random pure state model.We consider a pure state |Ψ in the bipartite Hilbert space H = H A ⊗ H B .This models the black hole evaporation where A corresponds to the Hawking radiation while B corresponds to the black hole.In the model discussed in [1], A and B correspond to the end of the world brane and the bulk JT gravity, respectively.We can expand the state |Ψ in terms of the orthonormal basis of where N is the normalization factor to ensure the unit norm of |Ψ It is useful to regard the coefficient X iα in (2.1) as a component of the Then the normalization factor N is written as Tr(XX † ) . (2.4) We are interested in the reduced density matrix ρ A defined in (1.2) obtained by tracing out B. In terms of the matrix X in (2.3), ρ A is written as a d A × d A matrix where W = XX † .The ensemble average over the random pure state |Ψ can be defined by the Gaussian integral over the matrix X As a distribution of the matrix W = XX † , this is known as the Wishart-Laguerre ensemble.See [11] for a nice review on this subject.The matrix integral (2.6) can be written as the eigenvalue integral by diagonalizing the matrix W .In the original paper by Page [5], the entropy S A was computed by evaluating the eigenvalue integral of log ρ A directly.In this paper, we will compute the entropy S A and the capacity C A using the replica method (1.6).To do this, we need the expectation value of the moment Tr ρ n A .Fortunately, the exact result of Tr where [a] n = a(a − 1) • • • (a − n + 1) denotes the falling factorial.For instance, the first few terms of Tr which agree with the known exact results of Tr ρ n A [15,16].Note that Tr ρ n A in (2.8) is symmetric under the exchange of d A and d B , which implies that S A and C A are also symmetric functions of (2.10) The exact result of Tr W n in the Wishart-Laguerre ensemble was obtained in [13,14], which is equivalent to (2.8) via the relation (2.7).
In what follows, we will assume d A ≤ d B without loss of generality.S A and C A in the opposite regime d A > d B can be obtained by exchanging d A and d B using the symmetry (2.10).
When n is a positive integer, the summation of j in (2.8) can be extended to j = ∞ since the summand vanishes for j > n.Then (2.8) becomes The last expression makes sense for non-integer n and it defines an analytic continuation of Tr ρ n A away from the integer n.One can in principle compute the derivative of the last expression in (2.11) with respect to n to find S A and C A .However, it is not straightforward to simplify the derivative of the hypergeometric function 3 F 2 in (2.11).
It turns out that it is useful to rewrite (2.11) as the following form using the identity (2.12) We find that this is expanded as where N n,k is the Narayana number Indeed, one can show that the summation in (2.13) reproduces the hypergeometric function in (2.12).This expression (2.13) makes contact with the planar limit of Tr ρ n A where the Narayana number naturally appears from the number of non-crossing permutations [17].When n is a positive integer, the summation of k in (2.13) is truncated at k = n and one can easily check that (2.13) reproduces the result (2.9) for small n.
Using the analytic continuation of the Narayana number by the last expression in (2.14), we can define a natural analytic continuation of Tr ρ n A in (2.13) for non-integer n.When d A ≤ d B and d A is a positive integer, the summand in (2.13) vanishes for k > d A and hence (2.13) becomes In section 4, we will use this expression of Tr ρ n A for the replica computation of the exact S A and C A .

Planar limit
Before discussing the exact of C A , in this section we will compute C A in the planar limit We will assume α ≤ 1 without loss of generality.C A in the opposite regime α > 1 can be obtained by sending α → α −1 using the symmetry (2.10).The computation of C A in this limit (3.1) has been already done in [6] using the planar eigenvalue density of the Wishart-Laguerre ensemble, known as the Marchenko-Pastur distribution.Here we will use the replica method to compute C A , which clarifies the role of replica wormholes in C A .
In the planar limit (3.1), the exact result of Tr ρ n A in (2.15) reduces to [17] Tr When n is a positive integer, the sum over k is truncated to One can see that (3.3) is obtained from the planar limit of the exact result (2.9), as expected.
When n is a positive integer, (3.2) is expanded as In the gravitational path integral, the first term d A planar is written as where f (n, α) is given by In other words, in the gravitational picture f (n, α) summarizes all contributions from the sub-dominant, partially connected geometries.Note that f (n, α) vanishes at n = 1 by our definition in (3.6) Plugging (3.5) into (1.6)we find where the prime in f and f denotes the derivative with respect to n.Note that the capacity is completely determined by the sub-leading contributions f (n, α) in (3.5).In fact, if we use the leading approximation of the trace the capacity vanishes (3.10) The same conclusion holds in the opposite regime d A > d B as well if we use the leading approximation Tr ρ n A planar ≈ d 1−n B .This implies that the dominant saddle point of gravitational path integral does not contribute to the capacity.In other words, the capacity of entanglement is sensitive to the sub-dominant saddle points corresponding to the partially connected geometries.Thus, C A is a useful probe of the contributions of replica wormholes which are not fully connected nor fully disconnected geometries, but some "intermediate" geometries.This is consistent with the result in [10] that C A takes a non-zero value for partially entangled states and C A vanishes for the pure state or a maximally entangled state.Namely, C A is a measure of partial entanglement [10].
Let us evaluate f (1, α) and f (1, α).From (3.6) they are written as (3.11) Thus, we need to compute the derivative of Narayana number N n,k at n = 1.From (2.14), one can easily show that N n,k is expanded around n = 1 as This implies that the first derivative ∂ n N n,k at n = 1 vanishes unless k = 2, and In a similar manner, from (3.12) we that f (1, α) in (3.11) becomes This agrees with the result in [6] obtained from the Marchenko-Pastur distribution.Our replica computation reveals the importance of the sub-leading contribution f (n, α) to the capacity of entanglement.We note in passing that C A,planar in (3.15) takes the maximal value at α = 1, or 4 Exact capacity of entanglement at finite d A , d B In this section we will compute the exact S A and C A using the exact result of Tr ρ n A in (2.15). 3Here we assume that d A and d B are both integers and Let us first compute the entanglement entropy S A .Plugging (2.15) into the definition of S A in (1.6) we find where ψ(z) denotes the digamma function From the behavior (3.12) of the Narayana number N n,k near n = 1, (4.1) becomes Using the property of the digamma function with γ being the Euler's constant, we arrive at the exact entanglement entropy S A for This agrees with the famous Page's result [5]. 4 S A in the opposite regime d A > d B is obtained from (4.5) by exchanging the role of d A and d B .Note that the first term of (4.5) is written as where H m = m k=1 1/k denotes the harmonic number.Next, let us compute the capacity of entanglement C A the replica method (1.6).To do this, we need to compute the second derivative of Tr ρ n A at n = 1 where ψ 1 (z) = d dz ψ(z) denotes the trigamma function.Using the relation and the behavior (3.12) of N n,k near n = 1, after some algebra we find the exact result of capacity (4.9)This is our main result.C A in the opposite regime d A > d B is obtained from (4.9) by exchanging d A and d B using the symmetry (2.10).Note that the first term of (4.9) is written as where H (2) m = m k=1 1/k 2 denotes the generalized harmonic number of the 2nd order.This is similar to the first term of S A in (4.5), but the other terms in C A are more complicated than S A .
One can easily check that (4.9) reduces to C A,planar in (3.15) in the planar limit (3.1).Also, one can check that C A (d A , d B ) in (4.9) for d A , d B = 2, 3 agree with the result in [10]  From the exact result (4.9), we find the small d A and the large (4.12) Our exact C A in (4.9) takes the maximal value at In the large d B limit this is expanded as where the first two terms agree with the maximal value of capacity in the planar limit (3.16).For finite d A , d B , one can show that the exact C A is bounded from above In Fig. 2, we show the plot of the exact capacity C A in (4.9).We can see that C A vanishes for d A = 1 and approaches zero at large d A d B .This is qualitatively similar to the result of the planar limit found in [6].We emphasize that our result (4.9) is exact at finite d A , d B and (4.9) includes all the non-planar corrections.As we argued in the previous section, C A is sensitive to the sub-leading terms in Tr ρ n A corresponding to the partially connected geometries.Indeed, C A vanishes at the early and late "time" t = log d A where the fully connected or fully disconnected geometry is dominant.C A takes a non-zero value near the Page time t = log d B (or d A = d B ) which is interpreted that the partially connected geometries give substantial contributions to C A near the Page time.

Conclusions and outlook
In this paper, we have computed the exact capacity of entanglement C A (4.9) at finite d A , d B using the replica method (1.6).At the technical level, the important ingredient in our computation is the new exact formula (2.15) of Tr ρ n A written in terms of the Narayana number N n,k .This (2.15) makes the relation to the planar limit manifest.We argued that C A vanishes for the fully connected or fully disconnected geometries, and C A is sensitive to the sub-leading contributions to Tr ρ n A coming from the partial connected geometries in the gravitational path integral.This suggests that C A is a good probe of the partial entanglement, as discussed in [10].
There are several open questions.The capacity of entanglement is introduced in [7] as an analogue of the heat capacity.Indeed, if we introduce the modular Hamiltonian K = − log ρ A , the moment Tr ρ n A looks like the partition function and n plays the role of the inverse temperature β.In this picture, our definition (1.6) of S A and C A is based on the "annealed" free energy log Z n One could consider the quenched version of S A and C A as well We leave the computation of the quenched version of S A and C A as an interesting future problem.
It would be interesting to study the gravitational picture of the capacity of entanglement.C A is related to the quantum fluctuation of the modular Hamiltonian and the prescription of the gravitational computation of C A is proposed in [8].We have argued that C A receives contributions only from the sub-leading partially connected geometries in the replica computation.It would be interesting to related this picture to the prescription in [8].
1−n A and the last term d 1−n B come from fully disconnected and fully connected geometries, respectively.If we assume d A < d B , the dominant contribution is the first term d 1−n A and Tr ρ n

Figure 1 :
Figure 1: Plot of the entanglement entropy S A as a function of log d A .We have set d B = 20 in this figure.The blue solid curve represents the exact result of S A in (4.5).The orange dashed curve is the leading approximation S A = log d A (d A ≤ d B ) and S A = log d B (d A > d B ).

Figure 2 :
Figure 2: Plot of the capacity of entanglement C A as a function of log d A .We have set d B = 20 in this figure.The blue solid curve is the exact result of C A in (4.9).The orange dashed curves represent the asymptotic behavior of C A in (4.12).The dashed vertical line is at d A = d B where C A becomes maximal.