  Quantum corrections to holographic entanglement entropy require knowledge of
the bulk quantum state. In this paper, we derive a novel dual prescription for
the generalized entropy that allows us to interpret the leading quantum
corrections in a geometric way with minimal input from the bulk state. The
equivalence is proven using tools borrowed from convex optimization. The new
prescription does not involve bulk surfaces but instead uses a generalized
notion of a flow, which allows for possible sources or sinks in the bulk
geometry. In its discrete version, our prescription can alternatively be
interpreted in terms of a set of Planck-thickness bit threads, which can be
either classical or quantum. This interpretation uncovers an aspect of the
generalized entropy that admits a neat information-theoretic description,
namely, the fact that the quantum corrections can be cast in terms of
entanglement distillation of the bulk state. We also prove some general
properties of our prescription, including nesting and a quantum version of the
max multiflow theorem. These properties are used to verify that our proposal
respects known inequalities that a von Neumann entropy must satisfy, including
subadditivity and strong subadditivity, as well as to investigate the fate of
the holographic monogamy. Finally, using the Iyer-Wald formalism we show that
for cases with a local modular Hamiltonian there is always a canonical solution
to the program that exploits the property of bulk locality. Combining with
previous results by Swingle and Van Raamsdonk, we show that the consistency of
this special solution requires the semi-classical Einstein's equations to hold
for any consistent perturbative bulk quantum state.
