By Masahito Hayashi

Quantum statistical inference, a examine box with deep roots within the foundations of either quantum physics and mathematical data, has made striking development due to the fact that 1990. specifically, its asymptotic thought has been constructed in this interval. notwithstanding, there has hitherto been no ebook masking this outstanding development after 1990; the well-known textbooks by means of Holevo and Helstrom deal basically with learn ends up in the sooner level (1960s-1970s). This booklet provides the real and up to date result of quantum statistical inference. It makes a speciality of the asymptotic concept, that's one of many relevant problems with mathematical data and had now not been investigated in quantum statistical inference till the early Nineteen Eighties. It includes impressive papers after Holevo's textbook, a few of that are of significant value yet will not be on hand now. The reader is anticipated to have merely undemanding mathematical wisdom, and accordingly a lot of the content material should be obtainable to graduate scholars in addition to study staff in similar fields. Introductions to quantum statistical inference were particularly written for the booklet. Asymptotic conception of Quantum Statistical Inference: chosen Papers will supply the reader a brand new perception into physics and statistical inference.

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Entire December 28, 2004 13:56 WSPC / Master ﬁle for review volume with part divider — 9in x 6in Fumio Hiai and D´ enes Petz 48 In the following we say that ψ is completely ergodic if it is ergodic for all γ n , n ≥ 1. We know that a weakly mixing state ψ is completely ergodic because it is weakly mixing for all γ n . Let βn (ψn , ϕn ) be deﬁned by (6) for n ≥ 1 and 0 < ε < 1. The next theorem shows that we have for large n exp 1 βn (ψn , ϕn ) n ≈ exp{−SM (ψ, ϕ∞ )} when ε is suﬃciently small and ψ is completely ergodic.

More precisely, let us suppose two hypotheses H0 and H1 so that the system A∞ has states ψ and ϕ∞ under H0 and H1 , respectively. A projection q in ⊗n1 A means a “quantum question” entire December 28, 2004 46 13:56 WSPC / Master ﬁle for review volume with part divider — 9in x 6in Fumio Hiai and D´ enes Petz of size n, whose outcomes are the eigenvalues 1 or 0. We decide that H0 (respectively H1 ) is true if the outcome of q is 1 (respectively 0). Then ϕn (q) (respectively ϕn (1 − q)) gives the “probability” of the error of accepting H0 (respectively H1 ) when H1 , (respectively H0 ) actually is true.

The relative entropies S(ψn , ϕn ) are deﬁned as (2). Then in view of the superadditivity of relative entropy [23] and the stationarity of ψ we get S(ψm+n , ϕm+n ) ≥ S(ψm , ϕm ) + S(ψn , ϕn ), m, n ≥ 1 so that limn→∞ n1 S(ψn , ϕn ) exists; in fact (cf. [10]) lim n→∞ 1 1 S(ψn , ϕn ) = sup S(ψn , ϕn ). n n≥1 n (8) We denote this limit by SM (ψ, ϕ∞ ), which is called the mean relative entropy of ψ with respect to ϕ∞ . Note that if ω is a state of A then SM (ω∞ , ϕ∞ ) = S(ω, ϕ) because S(ωn , ϕn ) = nS(ω, ϕ), n ≥ 1.