Abstract
This article gives a comprehensive account of quantum theory of statistical inference, whose necessity is recently rising, related to both of applied standpoints and fundamental standpoints. In quantum computation and quantum cryptography, etc, there is a need for extraction of precise information from quantum system, which is to be achieved by quantum statistical inference. Also, the development of the theory is expected to give a new insight to considerations of foundations of quantum mechanics. Centered on the asymptotic aspects of the theory, which is remarkably advanced by Japanese researchers in recent years, the article describe the quantum Stein's lemma in hypothesis testing, and the quantum Cramer-Rao type bound in parameter estimation, which is the minimum of sum of mean square error. For the illustration, the 2-and 3-dimensional-parameter quantum Gaussian state family are analyzed in detail. Historical overview of the theory, and brief review of other important works in recent years are also given. To supply background knowledge, we review asymptotic theory in the classical statistics and quantum measurement theory in the beginning.
