Statistical decision theory for quantum systems

Masao Ozawa

Wenbin Zhou

Fuyuhiko Tanaka

Fuyuhiko Tanaka

Decades ago, Holevo established quantum statistical decision theory. However, this formulation arises from theoretical interests and still is unsatisfactory compared with the classical counterpart. From the viewpoint in modern statistics and aiming at more rich framework, we reformulate quantum statistical decision theory.

Fuyuhiko Tanaka

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.