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

TL;DR

This paper proves that for a specific information storage game, quantum $n$-level systems do not outperform classical $n$-state systems in terms of maximum expected reward or mutual information, regardless of the distribution or reward function.

Contribution

It demonstrates the equivalence of quantum and classical systems in this game, introducing new inequalities and applying advanced mathematical tools like mixed discriminants and bipartite graph theorems.

Findings

Quantum and classical systems achieve equal maximum expected rewards.

Maximum mutual information is the same for quantum and classical $n$-level systems.

New dimension-dependent inequalities for positive operators are derived.

Abstract

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 $x$ in the used system 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.