Quantum Signal Propagation in Depolarizing Channels

Let X be an unbiassed random bit, let Y be a qubit whose mixed state depends on X, and let the qubit Z be the result of passing Y through a depolarizing channel, which replaces Y with a completely random qubit with probability p. We measure the quantum mutual information between X and Y by T(X; Y) = S(X) + S(Y) - S(X,Y), where S(...) denotes von Neumann's entropy. (Since X is a classical bit, the quantity T(X; Y) agrees with Holevo's bound chi(X; Y) to the classical mutual information between X and the outcome of any measurement of Y.)