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24) Assume now that λ > D(ρ||σ). Then ϕ(λ) > 0 and 1 − ε − e−nϕ(λ) > 0 for sufficiently large n. Thus (24) yields 1 1 log βn∗ (ε) ≥ −λ + log(1 − ε − e−nϕ(λ) ), n n and hence 1 lim inf log βn∗ (ε) ≥ −λ. n→∞ n Since λ can be arbitrarily close to D(ρ||σ), (23) has been proved. 5. The Strong Converse Property For each r > 0, there uniquely exists a number λ∗ = λ∗ (r) such that ϕ(λ∗ ) = r − λ∗ (see Fig. 2), and we define u(r) = ϕ(λ∗ (r)) = r − λ∗ (r). def Theorem 6: For any test {An } and any r > 0, if 1 lim sup log βn (An ) ≤ −r n→∞ n then 1 lim sup log(1 − αn (An )) ≤ −u(r).

In particular we have limn→∞ n1 Sco (ωn , ϕn ) = S(ω, ϕ) for every state ω of A. This has an interesting corollary that SBS (ω, ϕ) ≥ S(ω, ϕ) for all states ω and ϕ of A. Moreover for n ≥ 1 and 0 < ε < 1 let us introduce the following quantities: βε (ψn , ϕn ) = inf log ϕn (q) : q is a projection in ⊗n1 A with ψn ≥ 1 − ε (6) and Spr (ψn , ϕn ) = sup ψn (q) log ψn (q) 1 − ψn (q) + (1 − ψn (q)) log : ϕn (q) 1 − ϕn (q) q is a projection in ⊗n1 A . (7) The quantity βε (ψn , ϕn ) has a natural meaning from the viewpoint of quantum hypothesis testing (cf.

23, 57–65 (1986). [15] A. Uhlmann, “Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory,” Commun. Math. , 54, 21–32 (1977). entire December 28, 2004 13:56 WSPC / Master file for review volume with part divider — 9in x 6in CHAPTER 3 The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability Fumio Hiai and D´enes Petz Abstract. Umegaki’s relative entropy S(ω, ϕ) = Tr Dω (log Dω − log Dϕ ) (of states ω and ϕ with density operators Dω and Dϕ , respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint.

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