Quantum Minimax Theorem

TL;DR

This paper extends the classical minimax theorem to quantum systems, establishing a quantum minimax theorem and demonstrating the existence of least favorable priors in quantum statistical decision theory.

Contribution

It introduces a quantum minimax theorem applicable to general models and proves the existence of least favorable priors, advancing quantum statistical decision theory.

Findings

Quantum minimax theorem established for convex measurement sets.

Existence of least favorable priors proven in quantum settings.

Results generalize classical decision theory to quantum systems.

Abstract

Recently, many fundamental and important results in statistical decision theory have been extended to the quantum system. Quantum Hunt-Stein theorem and quantum locally asymptotic normality are typical successful examples. In the present paper, we show quantum minimax theorem, which is also an extension of a well-known result, minimax theorem in statistical decision theory, first shown by Wald and generalized by LeCam. Our assertions hold for every closed convex set of measurements and for general parametric models of density operator. On the other hand, Bayesian analysis based on least favorable priors has been widely used in classical statistics and is expected to play a crucial role in quantum statistics. According to this trend, we also show the existence of least favorable priors, which seems to be new even in classical statistics.