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

By Masahito Hayashi

Quantum statistical inference, a study box with deep roots within the foundations of either quantum physics and mathematical records, has made impressive development considering the fact that 1990. specifically, its asymptotic thought has been built in this interval. even if, there has hitherto been no e-book protecting this outstanding growth after 1990; the recognized textbooks by way of Holevo and Helstrom deal simply with study ends up in the sooner level (1960s-1970s). This booklet offers the $64000 and up to date result of quantum statistical inference. It makes a speciality of the asymptotic thought, that is one of many significant problems with mathematical facts and had no longer been investigated in quantum statistical inference till the early Nineteen Eighties. It comprises awesome papers after Holevo's textbook, a few of that are of significant value yet will not be to be had now. The reader is anticipated to have purely trouble-free mathematical wisdom, and accordingly a lot of the content material may be obtainable to graduate scholars in addition to learn employees in similar fields. Introductions to quantum statistical inference were especially written for the e-book. 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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**Asymptotic Theory Of Quantum Statistical Inference: Selected Papers**

Quantum statistical inference, a examine box with deep roots within the foundations of either quantum physics and mathematical information, has made striking development when you consider that 1990. particularly, its asymptotic conception has been built in this interval. even if, there has hitherto been no publication overlaying this impressive development after 1990; the well-known textbooks through Holevo and Helstrom deal in basic terms with study ends up in the sooner degree (1960s-1970s).

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N→∞ n As a special case we have for every state ω of A, 1 1 1 lim Spr (ωn , ϕn ) = lim Sco (ωn , ϕn ) = lim Scp (ωn , ϕn ) = S(ω, ϕ). n→∞ n n→∞ n n→∞ n (13) entire December 28, 2004 13:56 WSPC / Master ﬁle for review volume with part divider — 9in x 6in The Proper Formula for Relative Entropy and its Asymptotics 49 This means that Umegaki’s relative entropy comes out when we ﬁrst adopt any of the quantities Spr , Sco or Scp and then take asymptotics. The following examples show that the ergodicity assumption of ψ is essential in Theorems 2 and 3.

The next theorem shows that the asymptotic limit of Sco (ψn , ϕn ) exists and coincides with the mean relative entropy. Theorem 1: For every stationary state ψ of A∞ , lim n→∞ 1 Sco (ψn , ϕn ) = SM (ψ, ϕn ). n (9) A stationary state ψ of A is said to be ergodic if it is extremal in the set of stationary states. For ergodicity in general C ∗ -dynamical systems, see [10, 30] for instance. We say that ψ is weakly mixing if for every a, b ∈ A∞ , 1 n→∞ n n−1 |ψ(γ i (a)b) − ψ(a)ψ(b)| = 0. lim i=0 Obviously this is the case when ψ is strongly mixing (or strongly clustering), that is, limn→∞ ψ(γ n (a)b) = ψ(a)ψ(b) for every a, b ∈ A∞ .

For a test An , the error probabilities of the ﬁrst kind and the second kind are, respectively, deﬁned by αn (An ) = Tr[ ρ⊗n (In − An ) ] and βn (An ) = Tr[ σ ⊗n An ]. def def We see that αn (An ) is the probability of erroneously accepting σ ⊗n when ρ⊗n is true and βn (An ) is the error probability of the converse situation. Our interest lies in the asymptotic behavior of these error probabilities when n goes to ∞. In [8], Hiai and Petz proved an important theorem which claims that the quantum relative entropy def D(ρ||σ) = Tr[ ρ(log ρ − log σ) ] (1) is obtained as a limit of classical relative entropies (Kullback divergences).