Some aspects of quantum sufficiency for finite and full von Neumann algebras

Abstract

Some features of the notion of sufficiency in quantum statistics are investigated. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), strong sufficiency and Umegaki’s sufficiency. It is shown that for a finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki’s sense. Moreover, a proper version of the factorization theorem of Jenčová and Petz is obtained. The structure of the minimal sufficient subalgebra is described in the case of pure states on the full algebra of all bounded linear operators on a Hilbert space.