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Physical review. D/Physical review. D.

Distilled density matrices of holographic partial entanglement entropy from thread-state correspondence

Yi-Yu Lin

Published Nov 13, 2023• 13 citations
Publisher

Abstract

Within the framework of holographic duality, conditional mutual information (CMI) is often understood as a correlation between ``region pairs'' and is closely related to the concept of partial entanglement entropy (PEE). The main theme of this paper is to try to understand the rigorous physical meaning of such a region-pair correlation. This relies on the idea of holographic bit threads and the recently developed thread-state correspondence. In a sense, this effort also prompted us to give a definition of PEE based on the density matrices of the holographic distilled states. Specifically, drawing from experience with the locking multiflow configuration, we first provide a bipartite entanglement explanation for the $\mathrm{PEE}=\mathrm{CMI}$ scheme, but it leads to difficulties in characterizing the entanglement entropy of disconnected regions. We then introduce multipartite entanglement through the generalized $n$-thread/perfect tensor state correspondence to solve this problem and explain the coincidence between CMI and tripartite information in the holographic quantum systems.

DOI
10.1103/physrevd.108.106010

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Within the framework of holographic duality, conditional mutual information (CMI) is often understood as a correlation between ``region pairs'' and is closely related to the concept of partial entanglement entropy (PEE). The main theme of this paper is to try to understand the rigorous physical meaning of such a region-pair correlation. This relies on the idea of holographic bit threads and the recently developed thread-state correspondence. In a sense, this effort also prompted us to give a definition of PEE based on the density matrices of the holographic distilled states. Specifically, drawing from experience with the locking multiflow configuration, we first provide a bipartite entanglement explanation for the PEE=CMI\mathrm{PEE}=\mathrm{CMI}PEE=CMI scheme, but it leads to difficulties in characterizing the entanglement entropy of disconnected regions. We then introduce multipartite entanglement through the generalized nnn-thread/perfect tensor state correspondence to solve this problem and explain the coincidence between CMI and tripartite information in the holographic quantum systems.
Within the framework of holographic duality, CMI (conditional mutual information) is often understood as a correlation between ``region pairs" and is closely related to the concept of partial entanglement entropy (PEE). The main theme of this paper is to try to understand the rigorous physical meaning of such a region-pair correlation. This relies on the idea of holographic bit threads and the recently developed thread-state correspondence. In a sense, this effort also prompted us to give a definition of PEE based on the density matrices of the holographic distilled states. Specifically, drawing from experience with the locking multiflow configuration, we first provide a bipartite entanglement explanation for the PEE=CMI scheme, but it leads to difficulties in characterizing the entanglement entropy of disconnected regions. We then introduce multipartite entanglement through the generalized nnn-thread/perfect tensor state correspondence to solve this problem and explain the coincidence between CMI and tripartite information in the holographic quantum systems.
Abstract In the holographic framework, we argue that the partial entanglement entropy (PEE) can be explicitly interpreted as the component flow flux in a locking bit thread configuration. By applying the locking theorem of bit threads, and constructing a concrete locking scheme, we obtain a set of uniquely determined component flow fluxes from this viewpoint, and successfully derive the PEE proposal and its generalized version in the multipartite cases. Moreover, from this perspective of bit threads, we also present a coherent explanation for the coincidence between the BPE (balanced partial entanglement)/EWCS (entanglement wedge cross section) duality proposed recently and the EoP (entanglement of purification)/EWCS duality. We also discuss the issues implied by this coincident between the idea of the PEE and the picture of locking thread configuration.
Abstract We study the partial entanglement entropy (PEE) aspects of the holographic BCFT setup with an entanglement island, inspired by the holographic triality of the AdS/BCFT setup developed in the recent study on the black hole information problem, and the “PEE=CFF (component flow flux)” prescription, which is proposed recently to investigate the holographic PEE in the framework of bit thread formulation. Our study provides a bit thread description of the AdS/BCFT setup, which characterizes the specific entanglement details between the different parts of the system with an entanglement island, and may provide further insight into the black hole information problem. Furthermore, we show that in the context of island, one should distinguish between the fine-grained PEE and the semi-classical PEE. Interestingly, similar to the island rule of the fine-grained entropy in the semi-classical picture, we also propose the island rules of the fine-grained PEE.
In the framework of the holographic principle, focusing on a central concept, conditional mutual information, we construct a class of coarse-grained states, which are intuitively connected to a family of thread configurations. These coarse-grained states characterize the entanglement structure of holographic systems at a coarse-grained level. Importantly, these coarse-grained states can be used to further reveal nontrivial requirements for the holographic entanglement structure. Specifically, we employ these coarse-grained states to probe the entanglement entropies of disconnected regions and the entanglement wedge cross section dual to the inherent correlation in a bipartite mixed state. The investigations demonstrate the necessity of perfect tensor state entanglement. Moreover, in a certain sense, our work establishes the equivalence between the holographic entanglement of purification and the holographic balanced partial entropy. We also construct a thread configuration with the multiscale entanglement renormalization ansatz (MERA) structure, reexamining the connection between the MERA structure and kinematic space. Published by the American Physical Society 2024
A bstract We give a scheme to geometrize the partial entanglement entropy (PEE) for holographic CFT in the context of AdS/CFT. More explicitly, given a point x we geometrize the two-point PEEs between x and any other points in terms of the bulk geodesics connecting these two points. We refer to these geodesics as the PEE threads , which can be naturally regarded as the integral curves of a divergenceless vector field Vxμ {V}_{\textbf{x}}^{\mu } Vxμ​ V x μ , which we call PEE thread flow . The norm of Vxμ {V}_{\textbf{x}}^{\mu } Vxμ​ V x μ that characterizes the density of the PEE threads can be determined by some physical requirements of the PEE. We show that, for any static interval or spherical region A , a unique bit thread configuration can be generated from the PEE thread configuration determined by the state. Hence, the non-intrinsic bit threads are emergent from the intrinsic PEE threads. For static disconnected intervals, the vector fields describing a divergenceless flow is no longer suitable to reproduce the RT formula. We weight a PEE thread with the number of times it intersects with any homologous surface. Instead, the RT formula is perfectly reformulated by the minimization of the summation of PEE threads with all possible assignment of weights.
Abstract We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured E P = E W relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.