  We apply the bit thread formulation of holographic entanglement entropy to
reduced states describing only the geometry contained within an entanglement
wedge. We argue that a certain optimized bit thread configuration, which we
construct, gives a purification of the reduced state to a full holographic
state obeying a precise set of conditional mutual information relations. When
this purification exists, we establish, under certain assumptions, the
conjectured $E_P = E_W$ relation equating the entanglement of purification with
the area of the minimal cross section partitioning the bulk entanglement wedge.
Along the way, we comment on minimal purifications of holographic states,
geometric purifications, and black hole geometries.

High Energy Physics - Theory;General Relativity and Quantum Cosmology;Quantum Physics