Countering quantum noise with supplementary classical information

We consider situations in which (i) Alice wishes to send quantum information to Bob via a noisy quantum channel, (ii) Alice has a classical description of the states she wishes to send, and (iii) Alice can make use of a finite amount of noiseless classical information. After setting up the problem in general, we focus attention on one specific scenario in which Alice sends a known qubit down a depolarizing channel along with a noiseless classical bit. We describe a protocol that we conjecture is optimal and calculate the average fidelity obtained. A surprising amount of structure is revealed even for this simple case, which suggests that relationships between quantum and classical information could in general be very intricate.