By Masahito Hayashi

ISBN-10: 9812560157

ISBN-13: 9789812560155

ISBN-10: 9812563075

ISBN-13: 9789812563071

Quantum statistical inference, a examine box with deep roots within the foundations of either quantum physics and mathematical records, has made impressive development seeing that 1990. specifically, its asymptotic idea has been built in this interval. even if, there has hitherto been no booklet protecting this striking growth after 1990; the recognized textbooks by way of Holevo and Helstrom deal purely with examine leads to the sooner degree (1960s-1970s). This booklet offers the real and up to date result of quantum statistical inference. It specializes in the asymptotic idea, that's one of many relevant problems with mathematical facts and had no longer been investigated in quantum statistical inference until eventually the early Eighties. It comprises impressive papers after Holevo's textbook, a few of that are of significant significance yet should not to be had now. The reader is predicted to have purely trouble-free mathematical wisdom, and as a result a lot of the content material may be obtainable to graduate scholars in addition to examine employees in comparable fields. Introductions to quantum statistical inference were specifically written for the publication. Asymptotic thought of Quantum Statistical Inference: chosen Papers will supply the reader a brand new perception into physics and statistical inference.

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**Extra resources for Asymptotic theory of quantum statistical inference**

**Sample text**

Hence lim n→∞ 1 βε (2εϕn + (1 − 2ε)ψn , ϕn ) = 0. n (2) Let A = C ⊕ C and ω 1 , ω 2 , ϕ be given with the densities (1, 0), (0, 1), 1 2 (α, 1 − α), respectively, where 0 < α < 1/2. Let ψ = 12 (ω∞ + ω∞ ). By the aﬃnity of SM (·, ϕ∞ ) (see [28]) we get SM (ψ, ϕ∞ ) = 1 1 {S(ω 1 , ϕ) + S(ω 2 , ϕ)} = − {log α + log(1 − α)}. 2 2 But we easily see that Spr (ψn , ϕn ) is the maximum of log 1 = −n log{αn + (1 − α)n }1/n αn + (1 − α)n and 1 1 1 1 1 log n + log = max − log(1 − α), − log α 2 2α 2 2(1 − α)n 2 < SM (ψ, ϕ∞ ).

N (3) Combining the above statement, we can prove Stein’s lemma. The following statement is known as a weaker converse statement. Weak Converse Part: If n → 0, then −1 log βn∗ (p q| n ) ≤ D(p q). n lim (4) Here, for reader’s convenience, we give a proof of classical Stein’s lemma. Proof: For arbitrary real number δ > 0, we choose an acceptance ren gion Tn = ω n = (ω1 , . . , ωn ) pqn (ω n ) ≥ en(D(p q)−δ) . Then, the ﬁrst error probability 1 − pn (Tn ) = pn ωn 1 n n (log p − log q)(ωi ) < D(p q) − δ i=1 n goes to 0, because the random variable n1 i=1 (log p − log q)(ωi ) goes to D(p q) in pn probability.

Kobayashi, “The strong converse theorem for hypothesis testing,” IEEE Trans. Inform. Theory, 35, 178–180 (1989). [8] F. Hiai and D. Petz, “The proper formula for relative entropy and its asymptotics in quantum probability,” Commun. Math. , 143, 99–114 (1991). (Chap. 3 of this book) [9] E. H. , 11, 267–288 (1973). [10] G. Lindblad, “Completely positive maps and entropy inequalities,” Commun. Math. , 40, 147–151 (1975). [11] T. Ogawa and H. Nagaoka, “Strong converse to the quantum channel coding theorem,” IEEE Trans.

### Asymptotic theory of quantum statistical inference by Masahito Hayashi

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