SciPost Phys. 13, 056 (2022) ·
published 9 September 2022
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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.
Hugo A. Camargo, Pratik Nandy, Qiang Wen, Haocheng Zhong
SciPost Phys. 12, 137 (2022) ·
published 21 April 2022
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Recently in Ref. [1], one of the authors introduced the balanced partial entanglement (BPE), which has been proposed to be dual to the entanglement wedge cross-section (EWCS). In this paper, we explicitly demonstrate that the BPE could be considered as a proper measure of the total intrinsic correlation between two subsystems in a mixed state. The total correlation includes certain crossing correlations, which are minimized by particular balance conditions. By constructing a class of purifications from Euclidean path-integrals, we find that the balanced crossing correlations show universality and can be considered as the generalization of the Markov gap for the canonical purification. We also test the relation between the BPE and the EWCS in three-dimensional asymptotically flat holography. We find that the balanced crossing correlation vanishes for the field theory invariant under BMS$_3$ symmetry (BMSFT) and dual to the Einstein gravity, indicating the possibility of a perfect Markov recovery. We further elucidate these crossing correlations as a signature of tripartite entanglement and explain their interpretation in both AdS and non-AdS holography.
