No abstract

ABSTRACT. Some aspects of weak sufficiency of quantum statistics are investigated. In particular, we give necessary and sufficient conditions for the existence of a weakly sufficient statistic for a given family of vector states, investigate the problem of its minimality, and find the relation between weak sufficiency and other notions of sufficiency employed so far.

The paper investigates correlations in a general theory of quantum measurement based on the notion of instrument. The analysis is performed in the algebraic formalism of quantum theory in which the observables of a physical system are described by a von Neumann algebra, and the states -by normal positive normalized functionals on this algebra. The results extend and generalise those obtained for the classical case where one deals with the full algebra of operators on a Hilbert space.

Properties of Segal's entropy for semifinite and finite von Neumann algebras are investigated. In particular, its invariance with respect to a trace-preserving normal *-homomorphism is studied, as well as norm-continuity in the trace norm on the set of bounded in the operator norm density matrices.

We present some results concerning the relative modular operator in semifinite von Neumann algebras. These results allow one to prove some basic formula for trace, to obtain equivalence between Araki's relative entropy and Umegaki's information as well as to derive some formulae for quasi-entropies, and Rényi's relative entropy known in finite dimension.