Quantum signal propagation in depolarizing channels

Let X be an unbiased 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 (1948) entropy. (Since X is a classical bit, the quantity T(X; Y) agrees with Holevo´s (1973) bound χ(X; Y) to the classical mutual information between X and the outcome of any measurement of Y.) We show that T(X; Z) ⩽ (1-p)²T(X; Y). This generalizes an analogous bound for classical mutual information due to Evans and Schulman (1993), and provides a new proof of their result