Entanglement contour and modular flow from subset entanglement entropies

Wen, Qiang

doi:10.1007/jhep05(2020)018

Cited by 28 publications

(99 citation statements)

References 54 publications

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“…The PEE proposal [10,13] claims that for any twodimensional theories, the PEE is given by a linear combination of subset entanglement entropies. More explicitly given a connected region A, for any connected subset A 2 , which in general divide the region A into three subsets…”

Section: Partial Entanglement Entropy Proposal As a Solutionmentioning

confidence: 99%

“…Furthermore, this formula is consistent with the PEE proposal and the fine structure analysis. Though the requirement of normalization is very subtle, we show that the PEE proposal [10,13] is a solution to all the requirements.…”

Section: Introductionmentioning

confidence: 95%

“…Recently, several papers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] studied the so-called entanglement contour [5], which is a function that characterizes how much each degree of freedom in a region A contributes to the entanglement entropy S A . In other words, consider a quantum field theory in d dimensions (in this paper d means the dimension of space-time), the entanglement contour is a density function of entanglement entropy that depends on A and satisfies…”

Section: Introductionmentioning

confidence: 99%

“…The contour function constructed in this way has a clear physical meaning as the local distribution of entanglement. The third one is the partial entanglement entropy proposal [10,13,21] that claims the PEE is given by an additive linear combination of subset entanglement entropies, which we will explicitly discuss in this paper. The consistency check between the Gaussian formula and the PEE proposal for some cases of free boson and free fermion can be found in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…It has been shown to be particularly useful to characterize the evolution of the entanglement structure when studying dynamical situations [5,12,14]. The local modular flow in d = 2 can be generated from the PEE in a extremely simple way [13]. Also it has been recently demonstrated that the entanglement contour (calculated by the PEE proposal) is a useful probe of slowly scrambling and nonthermalizing dynamics for some interacting many-body systems [20].…”

Section: Introductionmentioning

confidence: 99%

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Formulas for partial entanglement entropy

Wen

2020

Phys. Rev. Research

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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.

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Section: Partial Entanglement Entropy Proposal As a Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Formulas for partial entanglement entropy

Wen

2020

Phys. Rev. Research

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Geometrizing the partial entanglement entropy: from PEE threads to bit threads

Lin,

Lu,

Wen

2024

J. High Energ. Phys.

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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 $$ {V}_{\textbf{x}}^{\mu } $$ V x μ , which we call PEE thread flow. The norm of $$ {V}_{\textbf{x}}^{\mu } $$ 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.

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Shaping contours of entanglement islands in BCFT

Ageev

2022

J. High Energ. Phys.

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In this paper, we study the fine structure of entanglement in holographic two-dimensional boundary conformal field theories (BCFT) in terms of the spatially resolved quasilocal extension of entanglement entropy — entanglement contour. We find that the boundary induces discontinuities in the contour revealing hidden localization-delocalization patterns of the entanglement degrees of freedom. Moreover, we observe the formation of “islands” where the entanglement contour vanishes identically implying that these regions do not contribute to the entanglement at all. We argue that these phenomena are the manifestation of the entanglement islands recently discussed in the literature. We apply the entanglement contour proposal to the recently discussed BCFT black hole models reproducing the Page curve — moving mirror model and the pair of BCFT in the thermofield double state. From the viewpoint of entanglement contour, the Page curve also carries the imprint of strong delocalization caused by dynamical entanglement islands.

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Entanglement contour and modular flow from subset entanglement entropies

Cited by 28 publications

References 54 publications

Formulas for partial entanglement entropy

Formulas for partial entanglement entropy

Geometrizing the partial entanglement entropy: from PEE threads to bit threads

Shaping contours of entanglement islands in BCFT

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