Asymptotic theory of quantum statistical inference by Masahito Hayashi

By Masahito Hayashi

Quantum statistical inference, a study box with deep roots within the foundations of either quantum physics and mathematical records, has made notable growth considering 1990. specifically, its asymptotic concept has been built in this interval. notwithstanding, there has hitherto been no publication masking this notable development after 1990; the recognized textbooks through Holevo and Helstrom deal merely with study leads to the sooner level (1960s-1970s). This e-book provides the $64000 and up to date result of quantum statistical inference. It specializes in the asymptotic concept, that is one of many crucial problems with mathematical statistics and had no longer been investigated in quantum statistical inference until eventually the early Eighties. It includes notable papers after Holevo's textbook, a few of that are of serious significance yet will not be on hand now. The reader is predicted to have purely straightforward mathematical wisdom, and hence a lot of the content material can be obtainable to graduate scholars in addition to examine staff in similar fields. Introductions to quantum statistical inference were in particular written for the publication. Asymptotic idea of Quantum Statistical Inference: chosen Papers will supply the reader a brand new perception into physics and statistical inference.

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In statistics this is often called Kullback-Leibler divergence. (1) of Chap. 5. , by combining classical Stein’s lemma, they showed that the Quantum relative entropy can be achieved by the optimal second error exponent with the constant constraint. Undoubtedly, their achievement is a great progress in Quantum hypothesis testing. However, Quantum Stein’s lemma was not completed only by the above result because an opposite inequality is required. , Hiai-Petz’s result) is called the direct part while the impossibility part is called the converse part.

2] R. E. Blahut, “Hypothesis testing and information theory,” IEEE Trans. Inform. Theory, 20, 405–417, (1974). E. Blahut, Principles and practice of information theory, Reading, Addison-Wesley, MA, 1987. [4] I. Csisz´ ar, “Information type measures of difference of probability distributions and indirect observations,” Studia Sci. Math. , 2, 299–318, 1967. [5] A. Dembo and O. Zeitouni, Large deviation techniques and applications, Jones and Bartlett, Boston, 1993. W. Helstrom, Quantum detection and estimation theory, Academic Press, New York, 1976.

When every element is randomly generated by a given probability, the maximum size satisfying the asymptotic weak orthogonality is characterized by the von Neumann entropy, but the maximum satisfying the asymptotic strong orthogonality is characterized by the quantum R´enyi entropy with degree 2. Here, we should remark that as was solved by Hausladen et al. [I-17], the von Neumann entropy equals the capacity of quantum channel coding with pure states, and that the quantum R´enyi entropy with degree 2 coincides with the smaller lower bound of the capacity by Stratonovich and Vantsjan [I-18].

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