Minimal entropy of states emerging from noisy quantum channels

In this paper, we consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel, which is modeled by the action of a completely positive trace-preserving (or stochastic) map. We provide strong support for the conjecture that this minimal entropy is additive, na mely, that the minimum entropy can be achieved when product states are tran smitted. Explicitly, we prove that for a tensor product of two unital stoch astic maps on qubit states, using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the mi nimum achievable using product states. We give a separate argument, based o n the geometry of the image of the set of density matrices under stochastic maps, which suggests that the minimal entropy conjecture holds for nonunit al as well, as for unital maps. We also show that the maximal norm of the o utput states is multiplicative for most product maps on n-qubit states, inc luding all those for which at least one map is unital. For the class of unital channels on C-2, We show that additivity of minimal entropy implies that the Holevo capacity of the channel is additive over t wo inputs, achievable with orthogonal states, and equal to the Shannon capa city. This implies that superadditivity of the capacity is possible only fo r nonunital channels.