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The Ryu-Takayanagi (RT) formula relates the entanglement entropy of a regionin 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 rewritethe 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 divergencelessnorm-bounded vector field, or equivalently a set of Planck-thickness "bitthreads". The entanglement entropy of a boundary region is given by the maximumflux out of it of any flow, or equivalently the maximum number of bit threadsthat can emanate from it. The threads thus represent entanglement betweenpoints on the boundary, and naturally implement the holographic principle. Aswe explain, this new picture clarifies several conceptual puzzles surroundingthe RT formula. We give flow-based proofs of strong subadditivity and relatedproperties; unlike the ones based on minimal surfaces, these proofs correspondin a transparent manner to the properties' information-theoretic meanings. Wealso briefly discuss certain technical advantages that the flows offer overminimal surfaces. In a mathematical appendix, we review the max flow-min cuttheorem on networks and on Riemannian manifolds, and prove in the network casethat the set of max flows varies Lipshitz continuously in the networkparameters.

Bit threads and holographic entanglement