Quantum communication channels differ from their classical counterparts because their capacities can be superadditive. The principle of monogamy of entanglement suggests that superadditive improvements in the transmission capacity of a channel should reduce the amount of information loss to the environment. We challenge this intuition by demonstrating that the coherent and private information of a channel and its complement can be simultaneously superadditive for arbitrarily many channel uses. To quantify the limits of this effect, we consider the notion of max (resp. total) private information of a channel, which represents the maximum (resp. sum) of the private information of the channel itself and its complement, and study its relationship with the coherent information of the individual direct and complementary channels. We show that these quantities can obey different interleaving sequences of inequalities for a varying number of channel uses.

Assume that Alice, Bob, and Charlie share a tripartite pure state$ \vert \psi _{ABC}\rangle $. We prove that if Alice cannot distill entanglement with either Bob or Charlie using$ \vert \psi _{ABC}\rangle $and local operations with any one of the following configurations for classical communication:$(A\to B, A\leftrightarrow C), (A\leftrightarrow B, A\to C)$, and$(A\leftrightarrow B, A\leftrightarrow C)$, then the same is also true for the other two configurations. Moreover, this happens precisely when the state is such that both its reductions on systems$AB$and$AC$are separable, which is further equivalent to the reductions being PPT. This, in particular, implies that any NPT bipartite state is such that either the state itself or its complement is 2-way distillable. In proving these results, we first obtain an explicit lower bound on the 2-way distillable entanglement of low rank bipartite states. Furthermore, we show that even though not all low rank states are 1-way distillable, a randomly sampled low rank state will almost surely be 1-way distillable.