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arXiv:2203.03111 · Journal of High Energy Physics

The PEE aspects of entanglement islands from bit threads

Yi-Yu Lin, Jia-Rui Sun, Yuan Sun, Jie-Chen Jin

Published Jul 1, 2022• 27 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep07(2022)009

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Physical review. D/Physical review. D.•Nov 13, 2023•13 citations
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.
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.
In the context of the AdS/CFT, it was recently proposed that the boundary partial entanglement entropy structure can be represented by the so-called partial entanglement entropy (PEE) threads in the AdS bulk, which are bulk geodesics with the density determined by the boundary PEE structure [. and .]. In Poincaré anti–de Sitter (AdS) space, it was shown that the PEE threads cover the AdS space uniformly, such that the number of intersections between any bulk surface and the bulk PEE threads is always given by the area of the surface divided by 4 G . In this paper, we investigate the configurations of PEE threads when the boundary state is in the island phase. The island phase was studied in the context of the holographic Weyl transformed CFT 2 , which has been shown to capture all the main features of AdS/BCFT (Boundary Conformal Field Theory). Compared with AdS 3 / CFT 2 , in the island phase instead of modifying the distribution of the bulk PEE threads, we should replace the boundary points with the corresponding cutoff spheres. Then the two-point functions and the four-point functions of twist operators can be reproduced by identifying the bulk homologous surfaces anchored on the corresponding cutoff spheres that has the minimal number of intersections with the bulk PEE threads. This gives us a better understanding about the PEE structure in the island phase and reproduces the island formula for entanglement entropy by allowing homologous surfaces to anchor on any cutoff spheres. Furthermore, it gives a demonstration for the two basic proposals and a better understanding for the entanglement contribution that makes the foundation to compute the balanced partial entanglement entropy [.], which reproduces the entanglement wedge cross section in the island phase.
Motivated by the kinematic-space description of gravitational subregions \cite{Basu:2026hbg}, we allow partial entanglement entropy (PEE) threads to be sourced not only from the asymptotic boundary, but also from boundaries of subregions. Based on this setup, we develop a framework to construct configurations for PEE threads and bit threads sourced from surfaces in the AdS bulk. The situations we have analyzed include the Poincaré AdS3_33​ and the planar BTZ black brane with an end-of-the-world (EOW) brane, and the entanglement wedges of boundary multi-intervals. We construct novel configurations of PEE threads and bit threads in these situations, which shed new light in our understanding of the AdS3_33​/BCFT2_22​ correspondence and the holography defined in the entanglement wedge motivated by surface/state correspondence. A notable feature we found is that, although the elementary PEE threads are geodesics, the integral curves of the coarse-grained bit-thread current are generically not geodesics. Across the explicit constructions, the integrated source currents reduce to endpoint distance-difference potentials, which make divergencelessness, the norm bound and bottleneck saturation geometrically transparent. Our results clarify how boundary, brane, horizon and RT-surface degrees of freedom participate in holographic entanglement and provide a concrete link between PEE, bit threads, subregion kinematic space and the surface/state correspondence.
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.
In the context of partial entanglement entropy (PEE), we study the entanglement structure of the island phases realized in several 2-dimensional holographic set-ups. From a pure quantum information perspective, the entanglement islands emerge from the self-encoding property of the system, which gives us new insights on the construction of the PEE and the physical interpretation of two-point functions of twist operators in island phases. With the contributions from the entanglement islands properly taken into account, we give a generalized prescription to construct PEE and balanced partial entanglement entropy (BPE). Here the ownerless island region, which lies inside the island \text{Is}(AB) Is ( A B ) of A\cup B A ∪ B but outside \text{Is}(A)\cup \text{Is}(B) Is ( A ) ∪ Is ( B ) , plays a crucial role. Remarkably, we find that under different assignments for the ownerless island, we get different BPEs, which exactly correspond to different saddles of the entanglement wedge cross-section (EWCS) in the entanglement wedge of A\cup B A ∪ B . The assignments can be settled by choosing the one that minimizes the BPE. Furthermore, under this assignment we study the PEE and give a geometric picture for the PEE in holography, which is consistent with the geometric picture in the no-island phases.
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.