Classical information storage in an $n$-level quantum system

A game is played by a team of two — say Alice and Bob — in which the value of a random variable x is revealed to Alice only, who cannot freely communicate with Bob. Instead, she is given a quantum n-level system, respectively a classical n-state system, which she can put in possession of Bob in any state she wishes. We evaluate how successfully they managed to store and recover the value of x by requiring Bob to specify a value z and giving a reward of value f (x, z) to the team. We show that whatever the probability distribution of x and the reward function f are, when using a quantum n-level system, the maximum expected reward obtainable with the best possible team strategy is equal to that obtainable with the use of a classical n-state system. The proof relies on mixed discriminants of positive matrices and — perhaps surprisingly — an application of the Supply–Demand Theorem for bipartite graphs. As a corollary, we get an infinite set of new, dimension dependent inequalities regarding positive operator valued measures and density operators on complex n-space. As a further corollary, we see that the greatest value, with respect to a given distribution of x, of the mutual information I(x; z) that is obtainable using an n-level quantum system equals the analogous maximum for a classical n-state system.