  There are increasing evidences that quantum information theory has come to
play a fundamental role in quantum gravity especially the holography. In this
paper, we show some new potential connections between holography and quantum
information theory. Particularly, by utilizing the multiflow description of the
holographic entanglement of purification (HEoP) defined in relative homology,
we obtain several new inequalities of HEoP under a max multiflow configuration.
Each inequality derived for HEoP has a corresponding inequality of the
holographic entanglement entropy (HEE). This is further confirmed by geometric
analysis. In addition, we conjecture that, based on flow considerations, each
property of HEE that can be derived from bit threads may have a corresponding
property for HEoP that can be derived from bit threads defined in relative
homology.
