Bit Threads and Holographic Entanglement

By Michael H. Freedman, M. Headrick. Communications in Mathematical Physics, 2016

230 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/af318d27d450d11c21ce514b42045d97cca3eb9f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. R. Myers
2. Dean Carmi
3. M. Pretko
4. Yizhi You
5. T. Takayanagi
6. Brian Swingle
7. Omer Ben-Ami
8. Xie Chen

In Nuclear and High Energy Physics

1. R. Myers
2. T. Takayanagi
3. Dean Carmi
4. Brian Swingle
5. Shira Chapman
6. H. Marrochio
7. Omer Ben-Ami
8. P. Nguyen

In Statistical and Nonlinear Physics

1. R. Myers
2. T. Takayanagi
3. Dean Carmi
4. Brian Swingle
5. P. Nguyen
6. Omer Ben-Ami
7. Shira Chapman
8. H. Marrochio

Citations in the corpus

• Comments on holographic complexity (2016, background): “[57] introduced the concept of ‘bit threads’, which are flow lines emanating from a boundary region A and threading the extremal RT surface, to define holographic entanglement in terms of the max flow-min cut principle.”
• On volumes of subregions in holography and complexity (2016, methodology): “The suggestion of [65] is a reformulation of the Ryu-Takayanagi formula in terms of bit threads, which loosely speaking probe the volume inside the RT surface.”
• Entanglement of purification: from spin chains to holography (2017, methodology): “On the other hand, the Ryu-Takayanagi formula has been reinterpreted recently via the min-cut/max-flow theorem as some kind of information flow [43].”
• Fracton phases of matter (2020, background): “[125] Michael Freedman and Matthew Headrick, “Bit threads and holographic entanglement,” Communications in Mathematical Physics 352, 407–438 (2016).”
• Entanglement of purification through holographic duality (2017, background): “Furthermore, as first found in [9], we can prove the following bound”