  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.
