Monogamy, polygamy, and other properties of entanglement of purification

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.

Figure 1

Difference between lower bounds for state $p|W\rangle \langle W|+(1-p)\left[a\right|000\rangle \langle 000|+(1-a\left)\right|111\rangle \langle 111\left|\right]$. The difference between the new lower bound and the previous one is always positive in this case.

Figure 2

Difference between lower bounds for state $\rho =p|W\rangle \langle W|+\frac{(1-p)}{8}{I}_3$. The difference between the new and the old lower bound is always positive here. The difference in lower bounds is given by the amount of polygamy of quantum mutual information.