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.