Sufficiency and relative entropy in ∗-algebras with applications in quantum systems

Abstract

The sufficiency and weak sufficiency in * -algebras are discussed. Some properties are studied concerning the relative entropy and the sufficiency for invariant states and KMS states in W* and C*-dynami- cal systems. Introduction. The concept of sufficiency is very important in mathematical statistics. The abstract measure theoretic investigation of sufficient statistics was initiated by Halmos and Savage (13). Kullback and Leibler (19) gave the characterization of sufficiency in terms of the information (i.e., the classical relative entropy). Umegaki (33,34) studied the sufficiency and the relative entropy in the noncommutative case of semi-finite von Neumann algebras. Araki (4,5) extended the relative entropy to the case for normal positive linear functionals of general von Neumann algebras and showed its several properties. Furthermore Uhlmann (32) showed the general WYDL concavity using a quadratic inteφolation theory and defined the relative entropy of positive linear functionals of arbitrary *-algebras. In the previous paper (14), we discussed the sufficiency and the relative entropy in von Neumann algebras and gave the characterizations of invariant states and KMS states with respect to the modular automor- phism group of a faithful normal state. In this paper, we further develop the sufficiency and the relative entropy in * -algebras. In §1, we introduce besides the sufficiency another notion of weak sufficiency and establish the relation between them. In §2, we deal with the weak sufficiency of positive linear maps between ^alge- bras. In §3, we mention the Araki's and Uhlmann's relative entropies which are equal in the von Neumann algebra case. We further give a formula of relative entropy for states of C*-algebras. In §4, we establish some properties of invariant states and KMS states in WΓ*-dynamical systems and C*-dynamical systems through the relative entropy and the sufficiency. The theorems there improve or extend the results obtained in (14). Finally we give an application to the Gibbs states of quantum lattice systems.