Covariant entanglement wedge cross-section, balanced partial entanglement and gravitational anomalies

TL;DR

This paper explores the calculation of balanced partial entanglement and entanglement wedge cross-section in covariant two-dimensional CFTs, including those with gravitational anomalies, establishing their equivalence with reflected entropy and extending to topological massive gravity.

Contribution

It provides the first prescription to evaluate EWCS corrections from Chern-Simons terms in TMG and analyzes their relation to BPE and reflected entropy in covariant scenarios.

Findings

BPE and EWCS coincide with reflected entropy in covariant setups.

Partitioning of the purifying system is determined using gravitational anomalies.

Extended analysis to topological massive gravity with Chern-Simons corrections.

Abstract

The balanced partial entanglement (BPE) was observed to give the reflected entropy and the entanglement wedge cross-section (EWCS) for various mixed states in different theories \cite{Wen:2021qgx,Camargo:2022mme}. It can be calculated in different purifications, and is conjectured to be independent from purifications. In this paper we calculate the BPE and the EWCS in generic covariant scenarios in two-dimensional CFTs with and without gravitational anomalies, and find that they coincide with the reflected entropy. In covariant configurations we determine the partition for the purifying system with the help of the gravitational anomalies, and we extend our discussion to topological massive gravity (TMG). We give the first prescription to evaluate the entropy quantity associated to the EWCS beyond Einstein gravity, i.e. the correction to the EWCS from the Chern-Simons term in TMG. Apart from the gravity theory and geometry, further input from the mixed state should be taken into account.