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arXiv:1907.12555 · Physical review. D/Physical review. D.

Quantum and classical correlations inside the entanglement wedge

Koji Umemoto

Published Dec 20, 2019• 46 citations
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Abstract

We show that the entanglement wedge cross section (EWCS) can become larger than the quantum entanglement measures such as the entanglement of formation in the AdS/CFT correspondence. We then discuss a series of holographic duals to the optimized correlation measures, finding a novel geometrical measure of correlation, the entanglement wedge mutual information (EWMI), as the dual of the $Q$-correlation. We prove that the EWMI satisfies the properties of the $Q$-correlation as well as the strong superadditivity, and that it can become larger than the entanglement measures. These results imply that both of the EWCS and the EWMI capture more than quantum entanglement in the entanglement wedge, which enlightens a potential role of classical correlations in holography.

DOI
10.1103/physrevd.100.126021

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We show that the entanglement wedge cross section (EWCS) can become larger than the quantum entanglement measures such as the entanglement of formation in the AdS/CFT correspondence. We then discuss a series of holographic duals to the optimized correlation measures, finding a novel geometrical measure of correlation, the entanglement wedge mutual information (EWMI), as the dual of the QQQ-correlation. We prove that the EWMI satisfies the properties of the QQQ-correlation as well as the strong superadditivity, and that it can become larger than the entanglement measures. These results imply that both of the EWCS and the EWMI capture more than quantum entanglement in the entanglement wedge, which enlightens a potential role of classical correlations in holography.
Information-theoretic inequalities often impose nontrivial constraints on holographic states. In this work, we study measurement-based classical and quantum correlations in holography, focusing on the proposed duals of classical correlation JWJ_WJW​, quantum discord DWD_WDW​, and one-shot distillable entanglement EDE_DED​, defined in terms of the entanglement wedge cross section (EWCS). We develop a homological framework tailored to inequalities involving multiple EWCSs and Ryu-Takayanagi surfaces, and use it to prove a family of inequalities, including monotonicity and monogamy/polygamy-type relations, as well as one-way strong superadditivity. For strong superadditivity, we additionally confirm its two-way version using Haar random states. We also examine holography-inspired boundary duals in terms of the reflected entropy and provide proofs and counterexamples for their information-theoretic inequalities. Taken together, our results provide further evidence for the duality between the EWCS and its proposed boundary counterparts -- measurement-based correlations and one-shot distillable entanglement -- while also furnishing a unified, rigorous method for proving multi-EWCS inequalities.
A bstract Information-theoretic inequalities often impose nontrivial constraints on holographic states. In this work, we study measurement-based classical and quantum correlations in holography, focusing on the proposed duals of classical correlation J W , quantum discord D W , and one-shot distillable entanglement E D , defined in terms of the entanglement wedge cross section (EWCS). We develop a homological framework tailored to inequalities involving multiple EWCSs and Ryu-Takayanagi surfaces, and use it to prove a family of inequalities, including monotonicity and monogamy/polygamy-type relations, as well as one-way strong superadditivity. For strong superadditivity, we additionally confirm its two-way version using Haar random states. We also examine holography-inspired boundary duals in terms of the reflected entropy and provide proofs and counterexamples for their information-theoretic inequalities. Taken together, our results provide further evidence for the duality between the EWCS and its proposed boundary counterparts — measurement-based correlations and one-shot distillable entanglement — while also furnishing a unified, rigorous method for proving multi-EWCS inequalities.
Recent developments have exposed close connections between quantum information and holography. In this paper, we explore the geometrical interpretations of the recently introduced QQQ-correlation and RRR-correlation, EQ{E}_{Q}EQ​ and ER{E}_{R}ER​. We find that EQ{E}_{Q}EQ​ admits a natural geometric interpretation via the surface-state correspondence: it is a minimal mutual information between a surface region AAA and a cross section of AAA's entanglement wedge with BBB. We note a strict trade-off between this minimal mutual information and the symmetric side-channel-assisted distillable entanglement from the environment EEE to AAA, Iss(E⟩A){I}^{ss}(E⟩A)Iss(E⟩A). We also show that the RRR-correlation, ER{E}_{R}ER​, coincides holographically with the entanglement wedge cross section. This further elucidates the intricate relationship between entanglement, correlations, and geometry in holographic field theories.
Abstract We study the entanglement wedge cross-section (EWCS) in holographic massive gravity theory, in which a first and second-order phase transition can occur. We find that the mixed state entanglement measures, the EWCS and mutual information (MI) can characterize the phase transitions. The EWCS and MI show exactly the opposite behavior in the critical region, which suggests that the EWCS captures distinct degrees of freedom from that of the MI. More importantly, EWCS, MI and HEE all show the same scaling behavior in the critical region. We give an analytical understanding of this phenomenon. By comparing the quantum information behavior in the thermodynamic phase transition of holographic superconductors, we analyze the relationship and difference between them and provide two mechanisms of quantum information scaling behavior in the thermodynamic phase transition.
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.
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.