By Masahito Hayashi

Quantum statistical inference, a study box with deep roots within the foundations of either quantum physics and mathematical records, has made outstanding development on account that 1990. particularly, its asymptotic thought has been built in this interval. notwithstanding, there has hitherto been no e-book protecting this extraordinary growth after 1990; the well-known textbooks by way of Holevo and Helstrom deal simply with learn leads to the sooner level (1960s-1970s). This booklet offers the $64000 and up to date result of quantum statistical inference. It specializes in the asymptotic conception, that's one of many principal problems with mathematical records and had no longer been investigated in quantum statistical inference till the early Nineteen Eighties. It includes amazing papers after Holevo's textbook, a few of that are of significant value yet aren't on hand now. The reader is predicted to have simply common mathematical wisdom, and for this reason a lot of the content material might be available to graduate scholars in addition to learn employees in comparable fields. Introductions to quantum statistical inference were specifically 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 classical hypothesis testing the optimal exponent has been obtained by [7]. On the other hand, it is yet to be determined whether the exponent u(r) derived here for the quantum hypothesis testing is optimal or not. Appendix A. Deﬁnition of Sf (ρ||σ) Here we summarize the original deﬁnition of Sf (ρ σ) given by Petz [14], translating it into our notation, for readers’ convenience. Given arbitrary strictly positive density operators ρ, σ ∈ S(H), deﬁne the relative modular operator ∆ = ∆σ,ρ : L(H) → L(H) by ∆(A) = σAρ−1 def ∀A ∈ L(H) or, equivalently, by ∆(A), B − ρ = A, B + σ ∀A, B ∈ L(H) entire December 28, 2004 13:56 WSPC / Master ﬁle for review volume with part divider — 9in x 6in Tomohiro Ogawa and Hiroshi Nagaoka 40 where , − ρ and , A, B + σ − def ρ = denote the inner products on L(H) such that Tr ρA∗ B and A, B + def σ = Tr σBA∗ .

C 1991 Springer-Verlag. Reprinted, with permission, from Commun. Math. , 143, 99-114, 1991. 43 entire December 28, 2004 13:56 WSPC / Master ﬁle for review volume with part divider — 9in x 6in Fumio Hiai and D´ enes Petz 44 In quantum theory the states of a system correspond to positive operators of trace one. ) In particular, in the setting of matrix algebras Umegaki’s relative entropy of a state ω with respect to another state ϕ is deﬁned by S(ω, ϕ) = Tr Dω (log Dω − log Dϕ ), (2) where Tr denotes the usual trace on matrices and Dω the density of ω with respect to Tr.

Next, we show the opposite inequality lim −1 log βn∗ (p q| ) ≤ D(p q). n (5) entire December 28, 2004 13:56 WSPC / Master ﬁle for review volume with part divider — 9in x 6in Introduction to Part I 26 Let Tn be an acceptance region achieving the minimum value βn∗ (p q| ), then βn∗ (p q| ) ≥ q n ω n ∈ Tn en(D(p ≥ e−n(D(p q)+δ) n ≥ e−n(D(p q)+δ) p q (ω n ) ≥ pn (ω n ) q)+δ) n ω n ∈ Tn en(D(p q (ω n ) ≥ pn (ω n ) q)+δ) n pn (Tn ) − pn ω n en(D(p q (ω n ) ≥ pn (ω n ) q)+δ) n c , where δ is an arbitrary positive number.