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Journal of High Energy Physics

Mixed state entanglement entropy in CFT

Xin Jiang, Peng Wei Wang, Houwen Wu, Haitang Yang

Published Sep 17, 2025• 3 citations
Publisher

Abstract

A bstract How to calculate the entanglement entropy between two subsystems for a mixed state has remained an important problem. In this paper, we provide a straightforward method, namely the subtraction approach, to solve this problem for generic covariant bipartite mixed states, where time dependence is explicitly included. We further demonstrate that the mixed state entanglement entropy S vN can be calculated in a more abstract yet powerful way. Within the context of the AdS 3 /CFT 2 and a configuration of AdS 5 /CFT 4 correspondences, we show that S vN exactly matches the corresponding entanglement wedge cross section in the AdS bulk, respectively.

DOI
10.1007/jhep09(2025)133

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A bstract How to calculate the entanglement entropy between two subsystems for a mixed state has remained an important problem. In this paper, we provide a straightforward method, namely the subtraction approach, to solve this problem for generic covariant bipartite mixed states, where time dependence is explicitly included. We further demonstrate that the mixed state entanglement entropy S vN can be calculated in a more abstract yet powerful way. Within the context of the AdS 3 /CFT 2 and a configuration of AdS 5 /CFT 4 correspondences, we show that S vN exactly matches the corresponding entanglement wedge cross section in the AdS bulk, respectively.
How to calculate the entanglement entropy between two subsystems for a mixed state has remained an important problem. In this paper, we provide a straightforward method, namely the subtraction approach, to solve this problem for generic covariant bipartite mixed states, where time dependence is explicitly included. We further demonstrate that the mixed state entanglement entropy SvNS_\text{vN}SvN​ can be calculated in a more abstract yet powerful way. Within the context of the AdS3_33​/CFT2_22​ and a configuration of AdS5_55​/CFT4_44​ correspondences, we show that SvNS_\text{vN}SvN​ exactly matches the corresponding entanglement wedge cross section in the AdS bulk, respectively.
We substantiate a covariant proposal for the holographic reflected entropy in CFTs dual to non-static AdS geometries from the bulk extremal entanglement wedge cross section in the literature with explicit computations in the AdS3/CFT2 scenario. In this context we obtain the reflected entropy for zero and finite temperature time dependent bipartite mixed states in CFT1+1s with a conserved charge dual to bulk rotating extremal and non-extremal BTZ black holes through a replica technique. Our results match exactly with the corresponding extremal entanglement wedge cross section for these bulk geometries in the literature. This constitutes a significant consistency check for the proposal and its possible extension to the corresponding higher dimensional AdS/CFT scenario.
Abstract We obtain the odd entanglement entropy (OEE) for bipartite mixed states in ( 1 + 1 ) -dimensional boundary conformal field theories (BCFT 2 s) utilizing an appropriate replica technique. For the generic configuration of two disjoint intervals, the large- c factorization of the twist field correlator leads to several distinct phases depending on the relative sizes and separations of the intervals. The resulting expressions match consistently with the entanglement wedge cross section (EWCS) for the bulk AdS 3 braneworld geometries, thereby extending the proposed holographic duality between the difference of the OEE and the entanglement entropy with the EWCS to BCFT setups. Furthermore, we obtain the OEE for these mixed states in related holographic moving mirror models which simulate Hawking radiation from black holes and verify the above duality in these time-dependent backgrounds. In this context we analyze the time evolution of the OEE and establish the analogues of Page curves for this mixed-state correlation measure.
We develop a covariant formalism to investigate the mixed state entanglement between time-dependent boosted subsystems in T T ¯ -deformed CFT 2 s through the reflected entropy. To this end we utilize a conformal perturbation theory to obtain the Rényi reflected entropy through the partition function on the replica manifold. The correction to the reflected entropy to the first order in the deformation parameter μ is then obtained in the replica limit for finite temperature and finite-sized systems. We further perform an analytic continuation of our results to obtain the reflected entropy for timelike subsystems in such T T ¯ -deformed CFT 2 s . We verify our field theoretic computations for both spacelike and timelike subsystems by obtaining the dual EWCS in the corresponding bulk cutoff AdS 3 geometries and find perfect agreement between the two. Published by the American Physical Society 2024
A bstract We obtain the reflected entropy for bipartite mixed state configurations involving two disjoint and adjacent subsystems in a two dimensional boundary conformal field theory (BCFT 2 ) in a black hole background. The bulk dual is described by an AdS 3 black string geometry truncated by a Karch-Randall brane. The entanglement wedge cross section computed for this geometry matches with the reflected entropy obtained for the BCFT 2 verifying the holographic duality. In this context, we also obtain the analogues of the Page curves for the reflected entropy and investigate the behaviour of the Markov gap.