Minimal entropy of states emerging from noisy quantum channels

In this paper, we consider the minimal entropy of two-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, namely that the minimum entropy can be achieved when product states are transmitted. Explicitly, we prove that for tensor products of unital stochastic maps, using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states. We give a separate argument, based on the geometry of the image of the set of density matrices under stochastic maps, which suggests that the minimal entropy conjecture holds for nonunital as well as for unital maps. We also show that the maximal norm of the output states is multiplicative for most product maps, including all those for which at least one map is unital. Partially supported by National Science Foundation Grant DMS-97-05779 Partially supported by National Science Foundation Grant DMS-97-06981 and Army Research Office Grant DAAG55-98-1-0374