Bit Threads and Holographic Entanglement

Michael Freedman, Matthew Headrick

Research output: Contribution to journalArticlepeer-review

Abstract

The Ryu-Takayanagi (RT) formula relates the entanglement entropy of a region in a holographic theory to the area of a corresponding bulk minimal surface. Using the max flow-min cut principle, a theorem from network theory, we rewrite the RT formula in a way that does not make reference to the minimal surface. Instead, we invoke the notion of a “flow”, defined as a divergenceless norm-bounded vector field, or equivalently a set of Planck-thickness “bit threads”. The entanglement entropy of a boundary region is given by the maximum flux out of it of any flow, or equivalently the maximum number of bit threads that can emanate from it. The threads thus represent entanglement between points on the boundary, and naturally implement the holographic principle. As we explain, this new picture clarifies several conceptual puzzles surrounding the RT formula. We give flow-based proofs of strong subadditivity and related properties; unlike the ones based on minimal surfaces, these proofs correspond in a transparent manner to the properties’ information-theoretic meanings. We also briefly discuss certain technical advantages that the flows offer over minimal surfaces. In a mathematical appendix, we review the max flow-min cut theorem on networks and on Riemannian manifolds, and prove in the network case that the set of max flows varies Lipshitz continuously in the network parameters.

Original languageEnglish
Pages (from-to)407-438
Number of pages32
JournalCommunications in Mathematical Physics
Volume352
Issue number1
DOIs
StatePublished - 1 May 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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