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arXiv:2501.10944 · Journal of High Energy Physics

Rényi entanglement of purification and half Rényi reflected entropy in free scalar theory

Liangyu Chen

Published Jun 5, 2025• 0 citations
arXivHTMLPDF

Abstract

A bstract In the AdS/CFT context, the entanglement of purification (EoP, denoted as E P ) of CFT is conjectured to be dual to the entanglement wedge cross section (EWCS) in bulk. However, another quantity called reflected entropy S R is also supposed to be dual to two times the EWCS. A natural question is whether they are the same in holographic CFTs even though they are different in general. Previous studies have shown E P ≥ $$ \frac{1}{2}{S}_R^{(n)} $$ 1 2 S R n , n ≥ 2 for random tensor networks. In this paper, we study this inequality beyond n ≥ 2, and we focus on the range 0 < n < 2. However, the calculations of EoP are notoriously difficult in general. Thus, our calculations mainly focus on the free scalar theory which is close to the holographic CFTs. We generalized the previous strategy for EoP in [1] to the Rényi case. And we have also presented two methods for Rényi reflected entropy, one is using correlators, the other one is Gaussian wavefunction ansatz. Our calculations show that the inequality still holds for 0 < n < 2, and it may give us some insights into the equivalence of EoP and half reflected entropy in holographic CFTs. As byproducts of our research, we have also demonstrated the positivity of the Rényi Markov gap and the monotonicity of the Rényi reflected entropy in the free scalar theory.

DOI
10.1007/jhep06(2025)045

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A bstract In the AdS/CFT context, the entanglement of purification (EoP, denoted as E P ) of CFT is conjectured to be dual to the entanglement wedge cross section (EWCS) in bulk. However, another quantity called reflected entropy S R is also supposed to be dual to two times the EWCS. A natural question is whether they are the same in holographic CFTs even though they are different in general. Previous studies have shown E P ≥ 12SR(n)\frac{1}{2}{S}_R^{(n)}21​SR(n)​ 1 2 S R n , n ≥ 2 for random tensor networks. In this paper, we study this inequality beyond n ≥ 2, and we focus on the range 0 < n < 2. However, the calculations of EoP are notoriously difficult in general. Thus, our calculations mainly focus on the free scalar theory which is close to the holographic CFTs. We generalized the previous strategy for EoP in [1] to the Rényi case. And we have also presented two methods for Rényi reflected entropy, one is using correlators, the other one is Gaussian wavefunction ansatz. Our calculations show that the inequality still holds for 0 < n < 2, and it may give us some insights into the equivalence of EoP and half reflected entropy in holographic CFTs. As byproducts of our research, we have also demonstrated the positivity of the Rényi Markov gap and the monotonicity of the Rényi reflected entropy in the free scalar theory.
The entanglement of purification E P ( A ∶ B ) is a powerful correlation measure, but it is notoriously difficult to compute because it involves an optimization over all possible purifications. In this paper, we prove a new inequality: E P ( A ∶ B ) ≥ 1 2 S R ( 2 ) ( A ∶ B ) , where S R ( n ) ( A ∶ B ) is the Renyi reflected entropy. Using this, we compute E P ( A ∶ B ) for a large class of random tensor networks at large bond dimension and show that it is equal to the entanglement wedge cross section E W ( A ∶ B ) , proving a previous conjecture motivated from AdS/CFT. Published by the American Physical Society 2024
In this study, we calculate the m − 1 correction to the reflected entropy for two adjacent intervals on a half-infinite line within the AdS 3 / BCFT 2 framework, where m is a Renyi index for a canonical purification. We utilize the doubling trick and compute the leading terms in the large central charge expansion of correlation functions in the holographic boundary conformal field theory (BCFT). In the corresponding anti–de Sitter (AdS) space with an end of the world brane, we analyze the entanglement wedge cross section, the dual counterpart of reflected entropy. This AdS/BCFT setup allows us to explore a richer set of phases in the entanglement wedge cross section. The m − 1 correction in the holographic BCFT manifests as modifications in the entanglement wedge cross section induced by cosmic branes. For the adjacent intervals anchored to the boundary of BCFT, we show the duality between the entanglement wedge cross section with the backreaction from a cosmic brane and Renyi reflected entropy at all orders in m − 1 . Furthermore, by analyzing the entanglement wedge cross section for general adjacent intervals, we provide guidance for an ε expansion of five-point functions in the holographic conformal field theory (CFT), where ε is the rescaled conformal dimension by the central charge.
A bstract In AdS/CFT we consider a class of bulk geometric quantities inside the entanglement wedge called reflected minimal surfaces. The areas of these surfaces are dual to the entanglement entropy associated to a canonical purification (the GNS state) that we dub the reflected entropy . From the bulk point of view, we show that half the area of the reflected minimal surface gives a reinterpretation of the notion of the entanglement wedge cross-section. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using a recently introduced approach to holographic modular flow. We also consider an explicit holographic construction of the canonical purification, introduced by Engelhardt and Wall; the reflected minimal surfaces are simply RT surfaces in this new spacetime. We contrast our results with the entanglement of purification conjecture, and finally comment on the continuum limit where we find a relation to the split property: the reflected entropy computes the von Neumann entropy of a canonical splitting type-I factor introduced by Doplicher and Longo.
The aim of this thesis is to compile our study of a quantum information quantity, called the reflected entropy, which is a measure of quantum correlations in mixed states. Our work is motivated by, and generalizes, an earlier work on the thermo-field double purification of thermal states, in the purview of the AdS/CFT correspondence. Our main result is that in so-called holographic CFT states, the reflected entropy is given by the area of a class of bulk geometric quantities inside the entanglement wedge, called reflected minimal surfaces. From the bulk point of view, we show that half the area of the reflected minimal surface, in Planck units, gives a reinterpretation of the notion of the entanglement wedge cross-section. In doing so, we address the connection with a recent conjecture on the entanglement of purification. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using a recently introduced approach to holographic modular flow. We also consider an Engelhardt-Wall holographic construction of the canonical purification − the reflected minimal surfaces are simply Ryu-Takayanagi surfaces in this new spacetime. When we move to the continuum limit, we find a relation to the split property − the reflected entropy computes the von Neumann entropy of a canonical splitting type-I factor. In this limit, we show that the reflected entropy serves as a regulator for the UV divergent entanglement entropy. Finally, we consider the reflected entropy for free fermions in two dimensions. Working directly in the continuum theory, we extract the reflected entropy from the spectrum of a singular integral equation whose kernel is determined by the modular flowed free fermion correlation function. We find the spectrum numerically and analytically in certain limits, in one of which, the reflected entanglement spectrum rapidly approaches the spectrum of the thermal density matrix. This suggests that the reflected entanglement spectrum is well suited to the task of extracting physical data of the theory directly from the vacuum/ground state wave function.
A bstract In holographic theories, the reflected entropy has been shown to be dual to the area of the entanglement wedge cross section. We study the same problem in random tensor networks demonstrating an equivalent duality. For a single random tensor we analyze the important non-perturbative effects that smooth out the discontinuity in the reflected entropy across the Page phase transition. By summing over all such effects, we obtain the reflected entanglement spectrum analytically, which agrees well with numerical studies. This motivates a prescription for the analytic continuation required in computing the reflected entropy and its Rényi generalization which resolves an order of limits issue previously identified in the literature. We apply this prescription to hyperbolic tensor networks and find answers consistent with holographic expectations. In particular, the random tensor network has the same non-trivial tripartite entanglement structure expected from holographic states. We furthermore show that the reflected Rényi spectrum is not flat, in sharp contrast to the usual Rényi spectrum of these networks. We argue that the various distinct contributions to the reflected entanglement spectrum can be organized into approximate superselection sectors. We interpret this as resulting from an effective description of the canonically purified state as a superposition of distinct tensor network states. Each network is constructed by doubling and gluing various candidate entanglement wedges of the original network. The superselection sectors are labelled by the different cross-sectional areas of these candidate entanglement wedges.
A bstract We compute the entanglement of purification (EoP) in a 2d free scalar field theory with various masses. This quantity measures correlations between two subsystems and is reduced to the entanglement entropy when the total system is pure. We obtain explicit numerical values by assuming minimal gaussian wave functionals for the purified states. We find that when the distance between the subsystems is large, the EoP behaves like the mutual information. However, when the distance is small, the EoP shows a characteristic behavior which qualitatively agrees with the conjectured holographic computation and which is different from that of the mutual information. We also study behaviors of mutual information in purified spaces and violations of monogamy/strong superadditivity.