Balanced partial entanglement and mixed state correlations

Abstract

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_33 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.

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

“…In other words, the balance conditions cannot uniquely determine the partition points Q hence give different BPEs in the case of the CFT without gravitational anomalies. This contradicts with the claim of [2] that the BPE is an intrinsic measure of the mixed state correlation hence should be unique when ρ AB is settled down. In the following we will show that when the contribution from gravitational anomalies is considered, the partition point Q will be uniquely determined and give the right BP E(A, B) that reproduces the reflected entropy.…”

“…This constant is a generalization of the so-called Markov gap [70], which is defined as the difference between the reflected entropy and the mutual information. It was proposed in [2] that the crossing PEE at the balance condition is a universal tripartite correlation that is independent from purifications. This proposal has been tested in many configurations including both the holographic and non-holographic models [2].…”

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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 [1,2]. It can be calculated in different purifications, and is conjectured to be independent from purifications.

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The PEE aspects of entanglement islands from bit threads

“…The concepts of the PEE and the entanglement contour have a range of applications in studying the entanglement structures in condensed matter theory [17,18,44,45]. Moreover, they have enlightening significance in the holographic framework [14,19,[41][42][43]. However, so far the fundamental definition of the PEE based on the reduced density matrix has not been established.…”

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

“…In this paper we focus on a new quantity named the balanced partial entanglement (BPE) [1,2], which is the partial entanglement entropy (PEE) [28,29] that satisfies certain balance conditions. The PEE satisfies the key property of additivity, as well as all the properties (inequalities) satisfied by the mutual information.…”