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.

We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree.A graph parameter is said to be estimable if it converges along every Benjamini-Schramm convergent sparse graph sequence. We prove that the normalized logarithm of the number of matchings is estimable. We also show that the analogous statement for perfect matchings already fails for d-regular bipartite graphs for any fixed d ≥ 3. The latter result relies on analyzing the probability that a randomly chosen perfect matching contains a particular edge.However, for any sequence of d-regular bipartite graphs converging to the dregular tree, we prove that the normalized logarithm of the number of perfect matchings converges. This applies to random d-regular bipartite graphs. We show that the limit equals to the exponent in Schrijver's lower bound on the number of perfect matchings.Our analytic approach also yields a short proof for the Nguyen-Onak (also Elek-Lippner) theorem saying that the matching ratio is estimable. In fact, we prove the slightly stronger result that the independence ratio is estimable for claw-free graphs.