# Asymptotic Theory Of Quantum Statistical Inference: Selected by Masahito Hayashi

By Masahito Hayashi

“This ebook will provide the students new perception into physics and statistical inference.” Zentralblatt Math Quantum statistical inference, a study box with deep roots within the foundations of either quantum physics and mathematical information, has made amazing growth in view that 1990. specifically, its asymptotic idea has been built in this interval. despite the fact that, there has hitherto been no e-book masking this impressive growth after 1990; the well-known textbooks via Holevo and Helstrom deal purely with examine ends up in the sooner degree (1960s-1970s). This publication offers the $64000 and up to date result of quantum statistical inference. It specializes in the asymptotic concept, that is one of many critical problems with mathematical information and had now not been investigated in quantum statistical inference till the early Eighties. It includes extraordinary papers after Holevo's textbook, a few of that are of serious value yet aren't to be had now. The reader is anticipated to have basically simple mathematical wisdom, and as a result a lot of the content material might be obtainable to graduate scholars in addition to learn employees in comparable fields. Introductions to quantum statistical inference were especially written for the e-book. Asymptotic concept of Quantum Statistical Inference: chosen Papers will provide the reader a brand new perception into physics and statistical inference.

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Uhlmann, “Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory,” Commun. Math. , 54, 21–32 (1977). entire December 28, 2004 13:56 WSPC / Master ﬁle for review volume with part divider — 9in x 6in CHAPTER 3 The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability Fumio Hiai and D´enes Petz Abstract. Umegaki’s relative entropy S(ω, ϕ) = Tr Dω (log Dω − log Dϕ ) (of states ω and ϕ with density operators Dω and Dϕ , respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint.

N The main diﬃculty is that the probability PρM⊗n is neither identical nor independent, and has been solved with the use of the information-spectrum method, which was introduced by Han¶ . This method is a powerful tool for the analysis of classical hypothesis testing problems in such a general setting. , they asymptotically treated the simple quantum hypothesis testing problem with a general sequence of two density matrixes that are hypotheses, and obtained general formulas of this setting. In the classical setting, Hoeﬀding focused on the constraint on the decreasing rate of the second error probability, and obtained the optimal asymptotic decreasing rate of the ﬁrst error probability under this constraint.

23) n→∞ n Let An be an arbitrary test which satisﬁes αn (An ) ≤ ε. From (9), we have 1 − ε ≤ 1 − αn (An ) ≤ e−nϕ(λ) + enλ βn (An ), and hence βn (An ) ≥ e−nλ (1 − ε − e−nϕ(λ) ). By taking the minimum with respect to An , we obtain βn∗ (ε) ≥ e−nλ (1 − ε − e−nϕ(λ) ). (24) Assume now that λ > D(ρ||σ). Then ϕ(λ) > 0 and 1 − ε − e−nϕ(λ) > 0 for suﬃciently large n. Thus (24) yields 1 1 log βn∗ (ε) ≥ −λ + log(1 − ε − e−nϕ(λ) ), n n and hence 1 lim inf log βn∗ (ε) ≥ −λ. n→∞ n Since λ can be arbitrarily close to D(ρ||σ), (23) has been proved.