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arXiv:1910.10978 · Physical Review Research

Formulas for partial entanglement entropy

Qiang Wen

Published May 13, 2020• 49 citations
arXivHTMLPDF

Abstract

The partial entanglement entropy (PEE) s A (A i ) characterizes how much the subset A i of A contribute to the entanglement entropy S A .We find one additional physical requirement for s A (A i ), which is the invariance under a permutation.The partial entanglement entropy proposal satisfies all the physical requirements.We show that for Poincaré invariant theories the physical requirements are enough to uniquely determine the PEE (or the entanglement contour) to satisfy a general formula.This is the first time we find the PEE can be uniquely determined.Since the solution of the requirements is unique and the PEE proposal is a solution, the PEE proposal is justified for Poincaré invariant theories.

DOI
10.1103/physrevresearch.2.023170

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The partial entanglement entropy (PEE) s A (A i ) characterizes how much the subset A i of A contribute to the entanglement entropy S A .We find one additional physical requirement for s A (A i ), which is the invariance under a permutation.The partial entanglement entropy proposal satisfies all the physical requirements.We show that for Poincaré invariant theories the physical requirements are enough to uniquely determine the PEE (or the entanglement contour) to satisfy a general formula.This is the first time we find the PEE can be uniquely determined.Since the solution of the requirements is unique and the PEE proposal is a solution, the PEE proposal is justified for Poincaré invariant theories.
We study the entanglement contour and partial entanglement entropy (PEE) in quantum field theories in 3 and higher dimensions. The entanglement entropy is evaluated from a certain limit of the PEE with a geometric regulator. In the context of the entanglement contour, we classify the geometric regulators, study their difference from the UV regulators. Furthermore, for spherical regions in conformal field theories (CFTs) we find the exact relation between the UV and geometric cutoff, which clarifies some subtle points in the previous literature. We clarify a subtle point of the additive linear combination (ALC) proposal for PEE in higher dimensions. The subset entanglement entropies in the ALC proposal should all be evaluated as a limit of the PEE while excluding a fixed class of local-short-distance correlation. Unlike the 2-dimensional configurations, naively plugging the entanglement entropy calculated with a UV cutoff will spoil the validity of the ALC proposal. We derive the entanglement contour function for spherical regions, annuli and spherical shells in the vacuum state of general-dimensional CFTs on a hyperplane.
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 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.
The partial entanglement entropy sA(Ai)s_{\mathcal{A}}(\mathcal{A}_i)sA​(Ai​) captures the contribution from the subset Ai\mathcal{A}_iAi​ of the region A\mathcal{A}A to the total entanglement entropy SAS_{\mathcal{A}}SA​ of A\mathcal{A}A. The partial entanglement entropy proposal \cite{Wen:2018whg,Wen:2019ubu} claims that sA(Ai)s_{\mathcal{A}}(\mathcal{A}_i)sA​(Ai​) equals to a linear combination of the entanglement entropies of certain relevant subsets in A\mathcal{A}A. We derive the differential version of this proposal which can directly generate the scheme independent entanglement contour function. Furthermore we derived the sufficient condition to apply the differential version of proposal. On the other way around, the proposal indicates that given the partial entanglement entropy we can calculate the entanglement entropies for the subsets. Following this idea, on the field theory side we analytically calculated the entanglement contour functions and entanglement entropies for annuli and spherical shells in holographic CFTs in arbitrary dimensions. We also comment on the phase transition of the mutual information, which is calculated holographically, across an annulus.
Entanglement contour characterizes the spatial structure of entanglement and quantifies the contribution from the degrees of freedom in any subset of the region A\mathcal{A}A to the total entanglement entropy SAS_{\mathcal{A}}SA​. Recently in \cite{Wen:2018whg} the author gave a simple proposal for the entanglement contour (or partial entanglement entropy) in two-dimensional theories, which involves the entanglement entropies of all the subsets inside A\mathcal{A}A. In this paper we explicitly study this proposal and show it satisfies many rational requirements for entanglement contour. Together with the holographic picture constructed with the modular planes \cite{Wen:2018whg}, we propose the correspondence between bulk geodesic chords and boundary partial entanglement entropies, which can be considered as a finer version of the Ryu-Takayanagi (RT) formula. As an example we calculate the fine correspondence between the points on A\mathcal{A}A and the points on the RT surface EA\mathcal{E}_{\mathcal{A}}EA​ for the BTZ black hole. We also give a strategy to extract the local modular flow from our proposal.
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.