Asymptotic theory of quantum statistical inference by Masahito Hayashi

By Masahito Hayashi

Quantum statistical inference, a learn box with deep roots within the foundations of either quantum physics and mathematical statistics, has made amazing development for the reason that 1990. particularly, its asymptotic concept has been built in this interval. even if, there has hitherto been no e-book masking this extraordinary development after 1990; the recognized textbooks by way of Holevo and Helstrom deal merely with learn leads to the sooner level (1960s-1970s). This publication offers the $64000 and up to date result of quantum statistical inference. It specializes in the asymptotic idea, that's one of many important problems with mathematical information and had now not been investigated in quantum statistical inference till the early Eighties. It comprises impressive papers after Holevo's textbook, a few of that are of significant significance yet aren't to be had now. The reader is anticipated to have simply hassle-free mathematical wisdom, and consequently a lot of the content material can be available to graduate scholars in addition to examine staff in comparable fields. Introductions to quantum statistical inference were especially written for the booklet. Asymptotic idea of Quantum Statistical Inference: chosen Papers will provide the reader a brand new perception into physics and statistical inference.

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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. S. Han and K. Kobayashi, “The strong converse theorem for hypothesis testing,” IEEE Trans.

We consider the hypothesis testing problem for the hypotheses pn and q n and denote the error probabilities by αn (An ) = pn (Acn ) and βn (An ) = q n (An ) for an acceptance region An ⊂ X n . After Blahut [3] showed that if βn (An ) ≤ e−nr with r > D(p||q) then αn (An ) necessarily tends to 1 as n → ∞ at least with a polynomial rate, Han and Kobayashi [7] proved that the actual rate is exponential and found the optimal exponent. Namely, letting α∗n (r) = min{αn (An ) | An ⊂ X n , βn (An ) ≤ e−nr }, def they derived lim inf n→∞ 1 log(1 − α∗n (r)) = −˜ u(r), n (32) where def u ˜(r) = min pˆ : D(p||q)≤r ˆ {D(ˆ p||p) + r − D(ˆ p||q)} .

40, 147–151 (1975). [11] T. Ogawa and H. Nagaoka, “Strong converse to the quantum channel coding theorem,” IEEE Trans. Inform. Theory, 45, 2486–2489 (1999). [12] T. Ogawa and H. Nagaoka, “Strong converse theorems in the quantum information theory,” In Proc. of 1999 IEEE Information Theory and Networking Workshop, (ITW99 Metsovo) 54, 1999. [13] D. Petz, “Quasi-entropies for states of a von Neumann algebra,” Publ. , 21, 787–800 (1985). [14] D. Petz, “Quasientropies for finite quantum systems,” Rep.

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