Geometric aspects of holographic bit threads
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Springer
Received: March 15, 2019 Accepted: May 5, 2019 Published: May 14, 2019
Geometric aspects of holographic bit threads
a
C. N. Yang Institute for Theoretical Physics, State University of New York Stony Brook, NY 11794, U.S.A. b Institute for Theoretical Physics, University of Amsterdam Amsterdam, 1090 GL, The Netherlands
E-mail: [email protected], [email protected], [email protected] Abstract: We revisit the recent reformulation of the holographic prescription to compute entanglement entropy in terms of a convex optimization problem, introduced by Freedman and Headrick. According to it, the holographic entanglement entropy associated to a boundary region is given by the maximum flux of a bounded, divergenceless vector field, through the corresponding region. Our work leads to two main results: (i) We present a general algorithm that allows the construction of explicit thread configurations in cases where the minimal surface is known. We illustrate the method with simple examples: spheres and strips in vacuum AdS, and strips in a black brane geometry. Studying more generic bulk metrics, we uncover a sufficient set of conditions on the geometry and matter fields that must hold to be able to use our prescription. (ii) Based on the nesting property of holographic entanglement entropy, we develop a method to construct bit threads that maximize the flux through a given bulk region. As a byproduct, we are able to construct more general thread configurations by combining (i) and (ii) in multiple patches. We apply our methods to study bit threads which simultaneously compute the entanglement entropy and the entanglement of purification of mixed states and comment on their interpretation in terms of entanglement distillation. We also consider the case of disjoint regions for which we can explicitly construct the so-called multi-commodity flows and show that the monogamy property of mutual information can be easily illustrated from our constructions. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 1811.08879
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)075
JHEP05(2019)075
Cesar A. Ag´ on,a Jan de Boerb and Juan F. Pedrazab
Contents 1 Introduction
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3 Constraints on geometry and matter 3.1 General geodesic foliations 3.2 Strips in a general translationally invariant background 3.3 Spheres in a general rotationally invariant background
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4 Nesting property and maximally packed flows
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5 Applications of flow constructions 5.1 Entanglement of purification 5.1.1 One interval in a BTZ black brane 5.1.2 Two intervals in pure AdS 5.2 Monogamy of mutual information 5.2.1 Quick review of the multiflow proposal 5.2.2 Max multiflow and MMI for disjoint intervals
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6 Summary and discussion
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A Geodesic integral curves for strips? A.1 A geodesic flow for strips in d = 2
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1
Introduction
An important lesson from the past decade’s research program in holography is that quantum informat
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