By Cohen-Tannoudji C.

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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.