By Masahito Hayashi

Quantum statistical inference, a learn box with deep roots within the foundations of either quantum physics and mathematical records, has made impressive development due to the fact 1990. particularly, its asymptotic conception has been constructed in this interval. even if, there has hitherto been no ebook overlaying this notable development after 1990; the well-known textbooks through Holevo and Helstrom deal in simple terms with learn leads to the sooner level (1960s-1970s). This publication provides the $64000 and up to date result of quantum statistical inference. It makes a speciality of the asymptotic concept, that's one of many crucial problems with mathematical information and had now not been investigated in quantum statistical inference till the early Eighties. It comprises striking papers after Holevo's textbook, a few of that are of significant value yet aren't to be had now. The reader is anticipated to have merely easy mathematical wisdom, and for that reason a lot of the content material may be obtainable to graduate scholars in addition to study employees in comparable fields. Introductions to quantum statistical inference were particularly written for the booklet. 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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Consequently, we obtain (29). u(r) ≥ entire December 28, 2004 13:56 WSPC / Master ﬁle for review volume with part divider — 9in x 6in Tomohiro Ogawa and Hiroshi Nagaoka 38 6. Relation with Classical Hypothesis Testing In this section, we discuss the relation between u(r) and the power exponent derived by Han and Kobayashi [7] for classical hypothesis testing. ) extensions by pn and q n . We consider the hypothesis testing problem for the hypotheses pn and q n and denote the error probabilities by αn (An ) = pn (Acn ) and βn (An ) = q n (An ) for an acceptance region An ⊂ X n .

Example 4: (1) Let 0 < ε < 1/2 and ψ be a stationary state of A∞ with ψ = ϕ∞ . Then SM (2εϕ∞ + (1 − 2ε)ψ, ϕ∞ ) > 0 because otherwise by (8) we get 2εϕ∞ + (1 − 2ε)ψ = ϕ∞ so that ψ = ϕ∞ . On the other hand, since 2εϕn (q) + (1 − 2ε)ψn (q) ≥ 1 − ε implies ϕn (q) ≥ 1/2 for a projection q in An , we have βε (2εϕn + (1 − 2ε)ψn , ϕn ) ≥ − log 2. 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.

Entire December 28, 2004 13:56 WSPC / Master ﬁle for review volume with part divider — 9in x 6in Strong Converse and Stein’s Lemma in Quantum Hypothesis Testing 31 Let λ be an arbitrary real number and let the spectral decomposition of the operator ρ⊗n − enλ σ ⊗n be written as ρ⊗n − enλ σ ⊗n = µn,j En,j , (7) j∈Jn where {µn,j }j∈Jn are the eigenvalues and {En,j }j∈Jn are the operators representing the orthogonal projections onto the eigenspaces. Deﬁne a test Xn,λ by def Xn,λ = En,j , j∈Kn def where Kn = {j | µn,j ≥ 0}.