Classical and Quantum Factors of Channels

Given a classical channel, a stochastic map from inputs to outputs, can we replace the input with a simple intermediate variable that still yields the correct conditional output distribution? We examine two cases: first, when the intermediate variable is classical; second, when the intermediate variable is quantum. We show that the quantum variable's size is generically smaller than the classical, according to two different measures---cardinality and entropy. We demonstrate optimality conditions for a special case. We end with several related results: a proposal for extending the special case, a demonstration of the impact of quantum phases, and a case study concerning pure versus mixed states.