  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.
