  We study the conjectured holographic duality between entanglement of
purification and the entanglement wedge cross-section. We generalize both
quantities and prove several information theoretic inequalities involving them.
These include upper bounds on conditional mutual information and tripartite
information, as well as a lower bound for tripartite information. These
inequalities are proven both holographically and for general quantum states. In
addition, we use the cyclic entropy inequalities to derive a new holographic
inequality for the entanglement wedge cross-section, and provide numerical
evidence that the corresponding inequality for the entanglement of purification
may be true in general. Finally, we use intuition from bit threads to extend
the conjecture to holographic duals of suboptimal purifications.
