  The entanglement of purification (EoP), which measures the classical
correlations and entanglement of a given mixed state, has been conjectured to
be dual to the area of the minimal cross section of the entanglement wedge in
holography. Using the surface-state correspondence, we propose a `bit thread'
formulation of the EoP. With this formulation, proofs of some known properties
of the EoP are performed. Moreover, we show that the quantum advantage of dense
code (QAoDC), which reflects the increase in the rate of classical information
transmission through quantum channel due to entanglement, also admits a flow
interpretation. In this picture, we can prove the monogamy relation of QAoDC
with the EoP for tripartite states. We also derive a new lower bound for
$S(AB)$ in terms of QAoDC, which is tighter than the one given by the
Araki-Lieb inequality.
