Holographic Inequalities and Entanglement of Purification

TL;DR

This paper explores the holographic duality between entanglement of purification and the entanglement wedge cross-section, establishing new inequalities and providing evidence for their validity in holography and quantum states.

Contribution

It generalizes both quantities, proves several new information inequalities, and extends the duality conjecture to suboptimal purifications using bit thread intuition.

Findings

Proved inequalities involving entanglement of purification and entanglement wedge cross-section.

Derived a new holographic inequality for the entanglement wedge cross-section.

Provided numerical evidence supporting the inequality for entanglement of purification.

Abstract

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.