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arXiv:2201.13362 · SciPost Physics

Balanced partial entanglement and mixed state correlations

Hugo A. Camargo, Pratik Nandy, Qiang Wen, Haocheng Zhong

Published Apr 21, 2022• 35 citations
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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 _3 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.

DOI
10.21468/scipostphys.12.4.137

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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 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.
Abstract In this article we define a new information theoretical quantity for any bipartite mixed state ρAB. We call it the balanced partial entanglement (BPE). The BPE is the partial entanglement entropy, which is an integral of the entanglement contour in a subregion, that satisfies certain balance requirements. The BPE depends on the purification hence is not intrinsic. However, the BPE could be a useful way to classify the purifications. We discuss the entropy relations satisfied by BPE and find they are quite similar to those satisfied by the entanglement of purification. We show that in holographic CFT2 the BPE equals to the area of the entanglement wedge cross section (EWCS) divided by 4G. More interestingly, when we consider the canonical purification the BPE is just half of the reflected entropy, which also directly relate to the EWCS. The BPE can be considered as an generalization of the reflected entropy for a generic purification of the mixed state ρAB. We interpret the correspondence between the BPE and EWCS using the holographic picture of the entanglement contour.
We advance a construction for the balanced partial entanglement entropy (BPE) for bipartite mixed states in a class of (1+1)(1+1)(1+1)-dimensional Galilean conformal field theories dual to Einstein gravity and topologically massive gravity in asymptotically flat spacetimes. We compare our results with the entanglement wedge cross section (EWCS) in the dual geometries and find exact agreement. Furthermore, utilizing a geometric picture, we demonstrate the holographic duality between the EWCS and the BPE in the context of flat holography.
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. 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.
We identify a non-negative and upper-bounded entanglement signal in holography which is defined as a combination of entanglement wedge cross sections (EWCS) for a tripartite mixed state ABEABEABE: EIΔ(A:B∣E)=EWCS(A:EB)+EWCS(B:EA)−EWCS(E:AB)\mathrm{EI}_Δ(A:B|E) = \mathrm{EWCS}(A:EB) + \mathrm{EWCS}(B:EA) - \mathrm{EWCS}(E:AB)EIΔ​(A:B∣E)=EWCS(A:EB)+EWCS(B:EA)−EWCS(E:AB). This quantity is an analogue of conditional mutual information (CMI) and shares similar mathematical structures in both quantum information theory and holography. We show that CMI is upper bounded by a quantum information quantity, the entanglement of assistance, which quantifies the entanglement that can be generated between two parties AAA and BBB, given assistance from a third party EEE. We prove that EIΔ\mathrm{EI}_ΔEIΔ​ is also upper bounded by the entanglement of assistance in the canonical purification state. We analyze its upper bound by maximizing EIΔ(A:B∣E)\mathrm{EI}_Δ(A:B|E)EIΔ​(A:B∣E) over all configurations of the auxiliary subsystem EEE in AdS3_33​/CFT2_22​. The maximized EIΔ\mathrm{EI}_ΔEIΔ​ displays a rich phase structure governed by the cross ratio XABX_{AB}XAB​: it vanishes below a critical threshold and, beyond a second phase transition point, saturates the bound of entanglement of assistance. We comment on the interpretation of EIΔ\mathrm{EI}_ΔEIΔ​ as characterizing the assisted bipartite quantum entanglement between AAA and BBB with the help of EEE.
Abstract We evaluate the entanglement wedge cross section (EWCS) in asymptotically AdS geometries which are dual to boundary excited states. We carry out a perturbative analysis for calculating EWCS between the vacuum and other states for a symmetric configuration consisting of two disjoint strips and obtain analytical results in the specific regimes of the parameter space. In particular, when the states described by purely gravitational excitations in the bulk we find that the leading correction to EWCS is negative and hence the correlation between the boundary subregions decreases. We also study other types of excitations upon adding the extra matter fields including current and scalar condensate. Our study reveals some generic properties of boundary information measures dual to EWCS, e.g., entanglement of purification, logarithmic negativity and reflected entropy. Finally, we discuss how these results are consistent with the behavior of other correlation measures including the holographic mutual information.
A bstract We propose a signal Δp(3) {\Delta }_p^{(3)} Δp(3)​ ∆ p 3 for tripartite correlations in finite-dimensional quantum systems and Δw(3) {\Delta }_w^{(3)} Δw(3)​ ∆ w 3 for holographic systems. We prove that Δp(3) {\Delta }_p^{(3)} Δp(3)​ ∆ p 3 is non-negative for any tripartite entangled mixed states and vanishes for mixed product states. It is a correlation signal because it is generally nonzero for a separable state containing a classical mixture. Based on the conjecture, the equality between an entanglement wedge cross section E w and entanglement of purification E p , i.e., E w = E P in the semiclassical limit, we apply the tripartite correlation signal to study the structures of tripartite entanglement in AdS 3 /CFT 2 , especially for pure AdS 3 . We comment on a generalization to n -partite correlation signals Δp(n)(A1:⋯:An) {\Delta}_p^{(n)}\left({A}_1:\cdots :{A}_n\right) Δp(n)​(A1​:⋯:An​) Δ p n A 1 : ⋯ : A n .