This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. Among the applications the equality case for the strong subadditivity of the von Neumann entropy, the Imoto-Koashi theorem and exponential families are treated. The setting of the paper allows the underlying Hilbert space to be infinite dimensional.MSC: 46L53, 81R15, 62B05.
The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family ϕ n θ 0 +u/ √ n consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state φu of an algebra of canonical commutation relations. The convergence holds for all "local parameters" u ∈ R m such that θ = θ0 + u/ √ n parametrizes a neighborhood of a fixed point θ0 ∈ Θ ⊂ R m . In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and investigate the relation between the two notions of convergence. For reader's convenience and completeness we review the relevant results of the classical as well as the quantum theory.
We consider a generalization of relative entropy derived from the Wigner-Yanase-Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for Tr K * A p KB 1−p Lieb's joint concavity in (A, B) for 0 < p < 1 and Ando's joint convexity for 1 < p ≤ 2. This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them. These include the monotonicity under partial traces, and some Minkowski type matrix inequalities proved by Carlen and Lieb for Tr 1 (Tr 2 A p 12 ) 1/p . In all cases, the equality conditions are independent of p; for extensions to three spaces they are identical to the conditions for equality in the strong subadditivity of relative entropy. a In [23], only concavity of the conditional entropy was proved explicitly, but the same argument [36, Sec. V.B] yields joint convexity of the relative entropy. Independently, Lindblad ([26]) observed that this follows directly from (2) by differentiating at p = 1. Rev. Math. Phys. 2010.22:1099-1121. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 09/17/13. For personal use only. Unified Treatment of Convexity of Relative Entropy and Related Trace Functions 1101even for A = B. Although this might seem unnecessary for convexity and concavity questions, it is crucial to a unified treatment. Lieb also considered Tr K * A p KB q with p, q > 0 and 0 ≤ p + q ≤ 1 and Ando considered 1 < q ≤ p ≤ 2. In Sec. 2.2, we extend our results to this situation. However, we also show that for q = 1−p, equality holds only under trivial conditions. Therefore, we concentrate on the case q = 1 − p.Next, we introduce our notation and conventions. In Sec. 2, we first describe our generalization of relative entropy and prove its convexity; then consider the extension to q = 1 − p mentioned above; and finally prove monotonicity under partial traces including a generalization of strong subadditivity to p = 1. In Sec. 3, we consider several formulations of equality conditions. In Sec. 4, we show how to use these results to obtain equality conditions in the results of Lieb and Carlen ([7, 8]). For completeness, we include an appendix which contains the proof of a basic convexity result from [37] that is key to our results. Rev. Math. Phys. 2010.22:1099-1121. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 09/17/13. For personal use only.
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