Monogamy, polygamy, and other properties of entanglement of purification

TL;DR

This paper investigates the properties of entanglement of purification, revealing its polygamous nature in contrast to quantum mutual information, and explores its additivity, bounds, and implications for quantum communication protocols.

Contribution

It demonstrates that entanglement of purification is polygamous for pure tripartite states, unlike quantum mutual information, and analyzes its additivity and bounds for various states.

Findings

Entanglement of purification is polygamous for pure tripartite states.

If not additive, entanglement of purification is subadditive.

Identifies states with additive entanglement of purification.

Abstract

For bipartite pure and mixed quantum states, in addition to the quantum mutual information, there is another measure of total correlation, namely, the entanglement of purification. We study the monogamy, polygamy, and additivity properties of the entanglement of purification for pure and mixed states. In this paper, we show that, in contrast to the quantum mutual information which is strictly monogamous for any tripartite pure states, the entanglement of purification is polygamous for the same. This shows that there can be genuinely two types of total correlation across any bipartite cross in a pure tripartite state. Furthermore, we find the lower bound and actual values of the entanglement of purification for different classes of tripartite and higher-dimensional bipartite mixed states. Thereafter, we show that if entanglement of purification is not additive on tensor product states, it is actually subadditive. Using these results, we identify some states which are additive on tensor products for entanglement of purification. The implications of these findings on the quantum advantage of dense coding are briefly discussed, whereby we show that for tripartite pure states, it is strictly monogamous and if it is nonadditive, then it is superadditive on tensor product states.