The Page Curve for Reflected Entropy

We study the reflected entropy $S_R$ in the West Coast Model, a toy model of black hole evaporation consisting of JT gravity coupled to end-of-the-world branes. We demonstrate the validity of the holographic duality relating it to the entanglement wedge cross section away from phase transitions. Further, we analyze the important non-perturbative effects that smooth out the discontinuity in the $S_R$ phase transition. By performing the gravitational path integral, we obtain the reflected entanglement spectrum analytically. The spectrum takes a simple form consisting of superselection sectors, which we interpret as a direct sum of geometries, a disconnected one and a connected one involving a closed universe. We find that area fluctuations of $O(\sqrt{G_N})$ spread out the $S_R$ phase transition in the canonical ensemble, analogous to the entanglement entropy phase transition. We also consider a Renyi generalization of the reflected entropy and show that the location of the phase transition varies as a function of the Renyi parameter.


Introduction
The black hole information problem has served as a beacon guiding us in the quest to understand quantum gravity [1][2][3][4]. Although a complete resolution still eludes us and might require a better understanding of the UV-complete theory of quantum gravity, significant progress has been made in recent years simply by taking the gravitational path integral seriously [5,6].
A commendable milestone in this endeavour is the calculation of the "Page Curve" using the semiclassical theory [7,8]. In fact, in the so-called West Coast Model [7], a toy model of black hole evaporation consisting of Jackiw-Teitelboim (JT) gravity coupled to end-of-the-world (ETW) branes, the detailed curve including effects near the phase transition were computed. Figure 1: The Lorentzian description of the state we consider in the West Coast Model, a JT gravity black hole with an ETW brane. The ETW brane carries two sub-flavours, denoted black and green, that are entangled (dashed, coloured lines) with radiation systems R 1 and R 2 respectively. The extremal surface is denoted in purple and the island that dominates after the Page time is coloured gray.
Away from phase transitions, the Page curve can be computed using the quantum extremal surface (QES) formula [9][10][11] which reads where S(A) is the entanglement entropy of subsystem A. Σ A is a partial Cauchy slice, whose domain of dependence is the entanglement wedge of A, such that ∂Σ A = A ∪ γ A . In the West Coast Model, one considers a state with ETW branes carrying flavour indices that are entangled with an auxiliary radiation system R = R 1 ∪ R 2 . 1 The model consists of two parameters: the horizon area S BH and the number of ETW brane flavour indices k = k 1 k 2 . Tuning these parameters simulates black hole evaporation, and by applying Eq. (1.1) to R, one finds the Page curve to be S(R) = min(S BH , log k), (1.2) where the candidate quantum extremal surfaces are the bifurcation surface and the trivial surface respectively, as shown in Fig. (1). While the entanglement entropy is essentially the unique measure for bipartite entanglement in pure states, one would also like to understand the multipartite entanglement structure of the radiation. Various distinct measures of entanglement have been proposed for this purpose. Among these we shall focus on the reflected entropy S R (A : B), which seems to be particularly interesting for holography [12,13].
Given a density matrix ρ AB , the reflected entropy S R (A : B) is defined by considering the canonically purified state | √ ρ AA * BB * in a doubled Hilbert space where A * and B * are the mirror copies of A and B. The reflected entropy is then defined as S R (A : B) = S(AA * ) | √ ρ . (1. 3) The reflected entropy is proposed to be holographically computed by the entanglement wedge cross section [12] and corresponding quantum corrections [14,15], i.e., where Γ A:B is a surface dividing the entanglement wedge of AB into portions a and b, which are homologous to A and B respectively (see Fig. (2)). The reflected entropy has already proven useful in understanding the requirement of tripartite entanglement in holographic states [13,15]. We analyze the reflected entropy between the radiation subsystems R 1 and R 2 . Simply applying Eq. (1.4) leads to 2 min(log k 1 , log k 2 ) k > exp(S BH ) (1.5) Figure 3: The main result: (a) The reflected entanglement spectrum of ρ AA * in the canonically purified state | √ ρ AB is a mixture of two superselection sectors: a single pole of weight p d corresponding to the disconnected purification, and a mound of approximately k 2 1 eigenvalues (assuming k 1 < k 2 ) with weight p c corresponding to a connected purification involving a closed universe with the entanglement wedge cross section denoted in orange. (b) The probability of the connected purification p c as we vary k across the Page transition. We show analytic plots for the microcanonical and canonical ensemble. The latter shows a spread in the phase transition of O( √ G N ). We also show plots for p c when considering the (m, 1)-Renyi reflected entropy; these undergo sharp transitions at m-dependent locations in the canonical ensemble. This follows from the two possible saddles obtained upon canonical purification: the disconnected geometry and the connected geometry depicted in Fig. (3).
In this paper we compute the reflected entropy precisely, demonstrating the validity of the holographic formula away from phase transitions. The fact that the gravitational path integral can be performed exactly allows us to compute the detailed behaviour of S R (R 1 : R 2 ) including corrections to Eq. (1.5) near the phase transition. We term this behavior the 'Page curve for reflected entropy.' This is the main result of this paper, summarized by Fig. (3).
In fact, we compute the entire reflected entanglement spectrum, which takes a very simple form: it consists of a mixture of two superselection sectors corresponding to the disconnected and connected geometries shown in Fig. (3). The probabilities p d and p c of the two sectors are computable functions, that vary as we change k, leading to the phase transition in Eq. (1.5). Moreover, we find corrections to Eq. (1.5) in a window of ∆ log k = O( 1 √ G N ) near the phase transition that arise from fluctuations in the horizon area 2 (equivalently thermal fluctuations, in this model). This is analogous to a similar effect found in the entanglement entropy Page curve [5,[16][17][18][19], and we expect it to be a universal feature of holographic reflected entropy.

Overview
In Sec. (2), we set up the stage for our analysis. In Sec. (2.1), we review the West Coast Model. In Sec. (2.2), we discuss basic aspects of the reflected entropy in this model.
The proposed holographic answer arises from a leading saddle computation assuming replica symmetry. This calculation suffers from various issues, which motivates us to analyze the problem in more detail.
In Sec.
(3), we review the computation of the entanglement spectrum in the West Coast Model. This serves as a warm-up for the resolvent trick which is used to obtain the spectrum in Sec. (3.1). We then use the spectrum to compute the Renyi entropies in this model in Sec. (3.2). Apart from being of interest on their own, the results for the Renyi entropies serve as an input for the reflected entropy calculation.
In Sec. (4), we analyze the reflected entanglement spectrum in the West Coast Model. We describe the resolvent trick in Sec. (4.1) to obtain a Schwinger-Dyson equation for the reflected entanglement spectrum. In Sec. (4.2), we solve this equation to obtain the spectrum and use it to analyze the reflected entropy. In Sec. (4.3), we go on to analyze a two-parameter Renyi generalization of the reflected entropy, called the (m, n)-Renyi reflected entropies.
In Sec. (5), we discuss these results and future directions. Note: This paper provides a complementary analysis to Ref. [20] where we analyze the reflected entropy phase transition in random tensor networks.

Preliminaries
In this section, we set up the background for our analysis. We will first review the West Coast Model in Sec. (2.1). In Sec. (2.2), we review aspects of the reflected entropy and discuss the proposed holographic dual.

West Coast Model
The West Coast Model is a toy model that was used to derive the Page curve of entanglement entropy. We briefly describe the model here, for more details refer to Ref. [5]. The model consists of JT gravity coupled to ETW branes. The action is given by (2.1) where S 0 is the extremal entropy and µ is the mass of the brane. We will take both the above parameters to be large in our analysis. The ETW branes also possess a large number of flavours k, which we artificially divide into two sub-flavour indices that number k 1 and k 2 respectively. We consider a state where the ETW brane is entangled with two radiation bath systems R 1 and R 2 as where |ψ ij B is the state of the black hole system B with the ETW brane chosen to be of sub-flavours i and j respectively. The state can be prepared using a Euclidean path integral, and the Lorentzian description is obtained by analytic continuation as shown in Fig. (1). A diagrammatic description of the boundary conditions that compute the overlap between two such states is where the red line represents an asymptotic AdS boundary, whereas the black and green lines correspond to sub-flavour indices of the ETW brane. 3 In this paper, we will consider two choices of boundary conditions imposed at the asymptotic boundary: a) microcanonical, where we impose a fixed energy E, and b) canonical, where we fix the renormalized length β corresponding to the inverse temperature. The computation in Eq. (2.4) is done by performing the gravitational path integral over geometries consistent with these boundary conditions, e.g., where the ETW brane has a definite flavour and thus, makes the diagram proportional to δ i 1 ,i 2 δ j 1 ,j 2 .

Reflected Entropy
The reflected entropy S R (A : B) is a function defined for a density matrix ρ AB on a bipartite quantum system AB. One first considers the canonical purification of ρ AB on a doubled Hilbert space where the space of linear maps End(H A ) acting on H A itself forms a Hilbert space with inner product X|Y = tr A (X † Y ). In general End(H) is isomorphic to the doubled copy H ⊗ H * . In other words, define | √ ρ AB by finding the unique positive matrix square root of ρ AB and interpret the result as a state in End(H A ⊗ H B ). This procedure is a generalization of the much more familiar procedure of purifying the thermal density matrix by considering the thermofield double state, discussed in the context of holography by Ref. [21]. The reflected entropy is then defined as [12]: In fact, it has a two parameter Renyi generalization based on the following state: 4 The Renyi generalization, which we call the (m, n)-Renyi reflected entropy, is then given by In the West Coast Model, we are interested in calculating the reflected entropy S R (R 1 : R 2 ). To motivate the proposed holographic answer, we can first consider preparing the Renyi generalization of the canonically purified state given in Eq. (2.8). For even integer m, this can be analyzed using the replica trick. Consider the Euclidean path integral that computes the norm of the state ψ (m) |ψ (m) = tr(ρ m R 1 R 2 ), it is diagramatically represented by the boundary conditions: (e.g. m = 4) (2.10) The boundary conditions have a dihedral symmetry composed of Z m rotations and a Z reflection. We now consider two natural saddle points that respect this replica symmetry [12], and applying the saddle point approximation gives us where Z p is the p-boundary partition function and the purple dot indicates the location of the extremal surface. The disconnected saddle dominates before the Page transition, i.e. k ≪ exp(S 0 ), while the connected saddle dominates after the Page transition. Each of these saddles has a Z symmetric slice which can be used to analytically continue the Euclidean saddle to a Lorentzian solution, giving us two candidate geometries corresponding to |ψ (m) : We refer to these geometries as the disconnected and connected purifications respectively. Here, m is simply a parameter that fixes the area of the extremal surface and thus, the analytic continuation in m is straightforward. As m → 1, these geometries can be interpreted as arising from gluing together two copies of the entanglement wedge of R 1 ∪ R 2 [22,23]. 5 Within the respective phases, we can apply the QES formula once we know the geometry corresponding to |ψ (m) . For the disconnected saddle, the QES is trivial and thus, the reflected entropy vanishes. On the other hand, for the connected saddle we have two non-trivial candidate QESs represented by the pink and orange cuts in Eq. (2.13). This leads to the proposed formula While this saddle point approximation is expected to work far away from the Page transition, there are various issues with this proposal which will require a more detailed analysis that we carry out in Sec. (4). In particular, the saddle point calculation suffers from an order of limits issue when analytically continuing m, n → 1. Further, Eq. (2.14) has a discontinuous jump at the phase transition which is resolved by non-perturbative effects. Once these effects are included, we will find significant corrections to this formula near the phase transition.

Entanglement Spectrum
Before we go to a more careful analysis of the reflected entropy, we first discuss the simpler case of entanglement entropy. In Sec. (3.1), we first review the resolvent trick that was used to compute the entanglement spectrum in Ref. [5]. This serves as a warm-up for the analogous calculation for the reflected spectrum. Further, we analyze the Renyi entropies in the West Coast Model in Sec. (3.2) which will be useful for our later calculations.

Resolvent Trick
The technique we use here was presented in Ref. [5] (see also [25][26][27]). This powerful approach enables us to write down a Schwinger-Dyson equation for the resolvent of ρ R 1 R 2 , which then gives full information about the entanglement spectrum. Consider the resolvent matrix R ij (λ) for the density matrix ρ R 1 R 2 defined formally as where the i, j indices run over both R 1 and R 2 labels and take values from 1 to k = k 1 k 2 .
We then define the resolvent as the trace R(λ) = tr R ij (λ). From the resolvent, we can obtain the density of eigenvalues D(λ) using To evaluate R(λ), we expand the matrix inverse around λ = ∞ as sum the series and then analytically continue to λ ∈ [0, 1] on the real axis where D(λ) is non-zero. In terms of diagrams, this leads to the boundary conditions: , where uncontracted indices have been used to represent a matrix equation for R ij and each red line carries a power of 1 kZ 1 for normalization. These boundary conditions are then filled in with planar disk geometries since crossing (higher genus) geometries are suppressed by powers of 1 k (e −S 0 ). Now denote F ij (λ) to be the connected part of the resolvent, defined by , (3.5) where the last term denotes the matrix (λ−F ) −1 . We can now write down a Schwinger-Dyson equation for R ij (λ) and F ij (λ): , where we note that a self consistent solution is to consider every term to be proportional to δ ij , and thus F ij (λ) = F (λ) k δ ij , where we have defined F (λ) = tr F ij (λ). After taking the trace of Eq. (3.6), we obtain the following algebraic equation We could now consider boundary conditions that correspond to either the microcanonical or canonical ensemble. The microcanonical case is identical to a random state and is discussed in detail in Ref. [20], 6 so our primary focus here will be on the canonical ensemble. For this case, we can use an integral representation for the n-boundary partition function Z n given by where ρ(s) is the density of states, and y(s) is the Boltzmann weight for the thermal spectrum given by It is convenient to define the normalized Boltzmann weight w(s) = y(s) Z 1 . Using this representation in Eq. (3.7), we obtain Finally, combining this with Eq. (3.5), we obtain an integral equation for R(λ): Solving this equation will give us the entanglement spectrum and corresponding entropies.
Approximate Entanglement Spectrum Figure 4: (a) The thermal spectrum of the JT black hole. (b) The approximate spectrum of ρ R 1 R 2 is a cutoff thermal spectrum which is obtained by truncating the spectrum to the k largest eigenvalues and shifting it by λ 0 to make it normalized.
The resolvent equation was solved approximately in Ref. [5] to obtain an approximate entanglement spectrum. It was shown that the spectrum of ρ R 1 R 2 can be well approximated by a cutoff thermal spectrum (see Fig. (4)), i.e., where the parameters s k and λ 0 can be computed by k = s k 0 ds ρ(s) (3.14) For more details about the calculation, refer to Appendix F of Ref. [5]. Here, we will simply use the result and discuss the interpretation of this spectrum. Intuitively, the thermal spectrum of the black hole is divided into two portions -a part that has already undergone the Page transition, and a part that hasn't. For states corresponding to sufficiently high energy in the thermal spectrum, the Page transition has still not occurred since they correspond to larger black holes. On the other hand, low energy states are past the Page transition and in this part of the wavefunction, the radiation system contains an island in the black hole interior. Thus, it is natural to expect that details of the spectrum such as the probability distribution of the horizon area should be observable. This natural ansatz is in fact a good approximation and leads to a cutoff thermal spectrum where s k roughly corresponds to the threshold dividing the pre-Page and post-Page parts of the wavefunction. The post-Page portion of the spectrum is not normalized on its own and thus, λ 0 is the additive constant contributed by the pre-Page spectrum that shifts the spectrum of ρ R 1 R 2 to make it appropriately normalized. Further, this ansatz captures the feature that the rank of ρ R 1 R 2 cannot exceed k even far past the Page transition, since the dimension of the radiation Hilbert space is k.

Renyi Entropy
Given this approximate description of the spectrum, we will now analyze the Renyi entropies in this model, for different values of m. To simplify the analysis, we choose µ ≫ 1 β . In this limit, the µ-dependent terms drop out from w(s). Further, we also work in the semiclassical limit β → 0. 7 Using Eq. (3.13), we obtain Note that β should be replaced with βG N to restore explicit dependence on the Newton constant. With these ingredients in hand, we can now analyze Eq. (3.17). An important feature that appears in this model is that different Renyi entropies have transitions at different values of k. We will analyze them separately for m > 1, m = 1 and m < 1 since they have qualitatively different behaviour.
Since at the level of the saddle point approximation, this comparison is identical to Eq. (3.18), we see that tr(ρ m R 1 R 2 ) has a phase transition at s k = s (m) k . More so, the second term in Eq. (3.19) is the integral of a Gaussian with a width σ m = 1 √ mβ . Since s (m) k − s m ≫ σ m , the second term is almost unchanging in this region. Thus, we see that for m > 1, tr(ρ m R 1 R 2 ) has a rather sharp phase transition. The Renyi entropy takes the form as shown in Fig. (5).
The case m = 1 corresponds to the entanglement entropy which was analyzed in detail in Ref. [5]. In this case, s m = s For m < 1, we instead have s m > s (m) k > s 1 . Thus, we see that the phase transition in tr(ρ m R 1 R 2 ) starts happening at s k = s 1 when λ 0 starts decaying. This is qualitatively different from the m > 1 case and will require a more careful analysis. 8 In order to compute Eq. (3.17), we can use the following inequality on the integrand: where the lower bound follows from positivity, while the upper bound follows from concavity. Upon performing the integral for where there is a potential O(1) multiplicative uncertainty. Similarly in the regime s k = s m + O( 1 β ), the w(s) term dominates over the λ 0 term giving us where we have used the saddle point approximation. Combining these results, we obtain where we have ignored subleading corrections at s k = s 1 and s k = s m . The result is sketched in Fig. (5), along with the naive expectation from analytically continuing the results of the Z m symmetric saddles a la Lewkowycz-Maldacena [31,32]. We see that there are in fact large O( 1 β ) corrections to the naive holographic answer for Renyi index m < 1. This is consistent with expectations, hinted at in Ref. [17], based on chaotic behaviour. As shown, the transition happens over a parametrically large window of ∆ log k ≈ O( 1 β ). Before moving on, we analyze the problem a little more carefully in a window s k = s 1 ± O( 1 √ β ) since this will be relevant for the reflected entropy transition. We can use the inequality in Eq. (3.22) and perform the integrals, which are simple Gaussian integrals. In fact, we will include one-loop effects to find the answer to greater accuracy. Computing the relevant quantities, we obtain x e −x 2 dx is the error function. Thus, integrating Eq. (3.22) with the measure ρ(s) and using Eq. (3.27), we obtain where we have used the fact that the integral of ρ(s)w(s) m is suppressed by powers of β in the regime of interest.

Reflected Entanglement Spectrum
We now use a generalization of the resolvent trick in Sec.

Resolvent Trick
Our goal here is to find the resolvent for the reduced density matrix ρ (m) R 1 R * 1 obtained from the Renyi generalization of the canonically purified state |ψ (m) defined in Eq. (2.8), i.e., where the integer moments of the (normalized) density matrix can be computed using the replica trick on mn copies of the system [12,20]. In order to do so, we first set up a slightly more general problem. 9 Consider the following 2 × 2 "matrix" of resolvents: where ρ → ρ Γ is an involution defined to take a linear operator ρ on H R 1 R 2 to a linear operator from It is defined by re-arranging the incoming/outgoing legs in the obvious (and canonical/basis independent) way. Each insertion of (ρ m/2 has m/2 replicas of each of the bra and ket of the original state |ψ . These involuted density operators are related to the density matrices in the canonically purified state by Diagramatically this is given by the following boundary conditions (ρ m/2 where we have chosen m = 6 for example. As before, any calculation using the gravitational path integral in this model will amount to a sum over geometries of different topology connecting the various asymptotic boundaries, e.g, in computing Eq. (4.4) we get (ρ m/2 For convenience, we will use a shorthand representation (4.6) From these results we can infer an alternative expression for matrix R(λ): The resolvent for system R 1 ∪ R * 1 in the state |ρ m/2 R 1 R 2 is given by A single copy of (ρ (m/2) R 1 R 2 ) Γ is referred to as a puddle and given two contiguous sets of puddles, there are two classes of diagrams, either disconnected or connected. As before, consider the connected part of R that we call F. This corresponds to a sum over the sub-diagrams over all connected contractions . We have (4.10) Figure 6: Example geometries that give rise to the projector or the identity. All possible contributions are either proportional to the identity operator 1 R 1 R * 1 , 1 R 2 R * 2 , or one of the four projectors |ǫ R 1 ,R 2 ǫ R 1 ,R 2 |.
A useful fact is that all diagrams in F and R take the following form: (see Fig. (6)) where |ǫ R 1 = 1 √ k 1 |1 R 1 is the maximally mixed state on R 1 R * 1 (same for |ǫ R 2 ), e R 1 = |ǫ R 1 ǫ R 1 | and e R 2 = |ǫ R 2 ǫ R 2 | are normalized minimal projectors and F is a 2 × 2 matrix of scalars (not to be confused with F).
In order to write down a Schwinger-Dyson equation for F(λ), we can organize the different diagrammatic contributions to F by the number of R insertions, labelled k − 1, where k is the number of puddles in each diagram: (4.13) where the shorthand conn. indicates connected contractions. In the above figure, the bubbles of λR indicate insertions of the resolvent matrix given in Eq. (4.11) and the various components of the matrix are represented by coloured index lines. Thus, combining Eq. (4.13) and Eq. (4.10), we now have self-consistency equations that we can solve to obtain R and F, order by order in k.

Spectrum and Reflected Entropy
The solution to the Schwinger-Dyson equation is identical to that obtained in Ref. [20] for a random tripartite state. Thus, we only discuss the results here. The results follow from a conjecture that the important contributions to the matrix F come from the lowest two orders in k, i.e. F ≃ F (1) + F (2) , in the limit we are considering. Evidence for this conjecture in the case of the microcanonical ensemble is provided in Ref. [20], while here we simply apply the same conjecture, now to the case of the canonical ensemble, and obtain physically sensible results.
Since the solution to the Schwinger-Dyson equation is obtained by considering k = 1 and k = 2, the two parameters that determine the spectrum turn out to be given by (e.g. for m = 6) 14) where connected topologies are the ones which have geometries joining at least two asymptotic boundaries on either side of the vertical dashed line. We define the normalized quantitiesB m = Bm Z 1 andD m = Dm √ Z 1 . Further, these quantities are related by a sum rule: The solution to the resolvent is then given by [20] The spectrum obtained from Eq. (4.17) takes a simple form (see Fig. (7)), with two superselection sectors: a single pole of weight p d =D 2 m and a mound of eigenvalues with weight p c =B m . The mound is a Marchenko-Pastur (MP) distribution with support between λ ± ≃ B m (k −1 1 ± k −1 2 ) 2 and min(k 2 1 − 1, k 2 2 − 1) eigenvalues. I.e., the spectrum is given by Figure 7: The spectrum obtained from the resolvent Eq. (4.17). We take k 1 < k 2 here so the MP distribution has no zero eigenvalues.
where the sum rule then takes the form: and thus we see that these quantities act like two "classical probabilities".   Figure 9: (a) The canonical purification in the disconnected phase has radiation systems purifying themselves. (b) The canonical purification in the connected phase has a state of radiation systems entangled via a closed universe. The two candidate RT surfaces are denoted orange and pink.
Using Eq. (4.14), we obtain we interpret these superselection sectors as corresponding to the two different geometries shown in Fig. (9), a disconnected and connected purification respectively. Thus, the reflected spectrum depends on the Renyi entropies of ρ R 1 R 2 , which we computed in Sec. (3.2). For m = 1, the physically most interesting case, we have from Eq. (4.21) and Eq. (3.28), which can be interpreted as the Pre-Page probability as discussed earlier. Consequently, the connected probability p c is interpreted as the Post-Page probability. We plot this result and compare the phase transitions in the canonical and microcanonical ensemble in Fig. (8). Due to thermal fluctuations, the phase transition is spread out over a range of parameter space of size ∆s = O( 1 √ β ).
The reflected entropy is given by (assuming k 1 < k 2 ) This is the main result of the paper. The reflected entropy is given by: where we generally have the upper bound S R (R 1 : R 2 ) ≤ 2 min(S(R 1 ), S(R 2 )). Thus, Fig. (8) serves as an approximate plot of S R (R 1 : R 2 ) normalized by the upper bound.

Renyi Reflected Entropy
Having discussed the reflected entropy, we now analyze the (m, n)-Renyi reflected entropies. Since the spectrum is an analytic function of m, the Renyi entropies can be computed by using the sum of the moments of the two sectors in the spectrum [20] S n (R 1 R * 1 ) = where r = k 2 1 /k 2 2 and C n (q) is the q-Catalan number defined by C n (q) = n k=0 q k N(n, k), (4.27) where N(n, k) is the Narayana number. Notably, the probabilities are the only source of m-dependence. Thus, we need to analyze the behaviour of p d (m) as we cross the Page transition using the analysis in Sec. (3.2). For m ≥ 2, we obtain (4.28) There are two continuous transitions in p d and hence, also in the (m, n)-Renyi reflected entropy. However, the second transition is almost invisible since p d is already nonperturbatively small at s k = s (m/2) k and the reflected entropy is dominated by the connected entanglement wedge. For m ∈ (1, 2], the phase transition in p d is largely determined by the phase transition in tr(ρ m R 1 R 2 ) which occurs at s k = s (m) k < s 1 . Thus, we obtain where the value in the last line is difficult to compute analytically, although it is nonperturbatively suppressed and has a negligible effect on the reflected entropy. We plot the behavior for p d (m) for different values of m > 1 in Fig. (10). Finally, for the case m < 1, there is a multiplicative uncertainty in the calculation of tr(ρ m R 1 R 2 ) which makes it difficult to compute p d (q) analytically. We provide a numerical plot instead in Fig. (10). Qualitatively, the transition happens over a large region in parameter space corresponding to s k ∈ [s 1 , s m ].
To summarize, the phase transition in the (m, n)-Renyi reflected entropy comes from a shift in dominance between the two terms in Eq. (4.26). As n → 1, the transition is dictated by the phase transition in p d . This transition is qualitatively different for m > 1 and m < 1. For m > 1, it is a sharp transition at s k = s (m) k and for m < 1, it is spread out over s k ∈ [s 1 , s m ].

Summary and Discussion
In this paper we have analyzed the canonical purification and reflected entropy in the West Coast Model. We found that the holographic proposal is satisfied in the expected regimes: the canonical purification is related to a doubled spacetime, and the reflected entropy is related to the entanglement wedge cross section. Furthermore, by summing over the contributions from all relevant saddles, we have understood how the geometric picture evolves as we cross an entanglement phase transition. We now discuss certain interesting aspects of our calculation and potential directions for future research.

Relation to Petz Map
Consider a code subspace of bulk states encoded in the boundary Hilbert space. The Petz map provides an explicit realization in this setting of a reconstruction that maps a given bulk operator to a boundary operator [5,33,34]. In the West Coast Model, one can consider bulk operators that live on the ETW brane. Before the Page transition, they are reconstructable by the black hole B, whereas after the Page transition they are reconstructable by the radiation system R. More so, in Ref. [5], they computed the probability of the Petz map acting on the radiation to succeed. It turns out to be given by p c , precisely the probability that showed up in our calculation of the reflected spectrum (regardless of the choice of how to split R into R 1 R 2 ). This is true in both the microcanonical ensemble and the canonical ensemble (see Fig. (8) for our p c , which matches the Petz recovery probability computed in Ref. [5]).
This hints at a connection between the reflected entropy and the Petz map success probability. From one point of view, this seems completely sensible within the paradigm of entanglement wedge reconstruction. The reflected entropy S R (R 1 : R 2 ) is a sharp diagnostic of whether the entanglement wedge is connected, and only when it's connected does the entanglement wedge of R 1 R 2 include the black hole interior. Indeed, the probability p c is simply the probability of measuring the canonical purification of R 1 R 2 in the connected phase. It will be interesting to investigate a possible deeper connection between the reflected entropy and reconstruction maps.

Area Fluctuations
Here, we considered a holographic state in which the horizon area for B had area fluctuations. However, in order to do the calculation, we needed to consider a maximally entangled state between R and the ETW brane flavours. In the ER=EPR story, this can be interpreted as considering a three boundary wormhole with area fluctuations for the horizon of B but fixed area boundary conditions for the horizons of R 1 and R 2 respectively, see Figure 11. It would be interesting in the future to understand how the area fluctuations of R 1 and R 2 affect the reflected entropy. Presumably, it will affect the spectrum on R 1 R * 1 by including the effects of thermal fluctuations into the corresponding superselection sectors. One may attempt to analyze this using the techniques of the equilibrium approximation [35,36]. Figure 11: The three-boundary wormhole corresponding to our current setup.