Game-theoretical Comparison of Information Structures in Quantum Theory

Abstract

A family of probability distributions (i.e. a statistical model) is said to be sufficient for another, if there exists a transition matrix transforming the probability distributions in the former to the probability distributions in the latter. The so-called Blackwell-Sherman-Stein Theorem provides necessary and sufficient conditions for one statistical model to be sufficient for another, by comparing their "information values" in a game-theoretical framework. In this talk, I will extend some of these ideas to the quantum case. I will begin by considering the comparison of ensembles of quantum states in terms of their "information value" in quantum statistical decision problems. In this case, I will prove that one ensemble is "more informative" than another if and only if there exists a suitable processing of the former into the latter. I will then move on to the comparison of bipartite quantum states in terms of their "nonlocality value" in nonlocal games. In this case, I will prove that one bipartite state is "more nonlocal" than another if and only if the former can be transformed into the latter by local operations and shared randomness, arguing, moreover, that the framework provided by nonlocal games can be useful in understanding analogies and differences between the notions of quantum entanglement and nonlocality.