Weak sufficiency of quantum statistics

Weakly sufficient quantum statistics

“…The definition of weak sufficiency is closely related to the classical factorization criterion, and seems to be especially well motivated in the case when we are dealing with the full algebra of bounded operators on a Hilbert space together with vector states. In the present paper, which can be considered as a follow-up to [5], we continue the investigation of this notion. Three questions are dealt with: the problem of the existence of a weakly sufficient statistic for a given family of states, minimality of weakly sufficient statistics, and the relation of weak sufficiency to other notions of sufficiency.…”

“…Let U be an arbitrary weakly sufficient statistic affiliated with N . Then U = Φ(T ) for some real-valued Borel function Φ, thus we may assume that U has form (6), with f k given by equation (5). From assumption (7) and Proposition 4 it follows that for each k all elements in I k are equivalent, thus for each k there is an m such that I k ⊂ J m , so each J m is a sum of some I k 's.…”

“…Weak sufficiency of a quantum statistic was introduced in [1] (under the name of 'sufficiency'), and afterwards analyzed in [5]. The definition of weak sufficiency is closely related to the classical factorization criterion, and seems to be especially well motivated in the case when we are dealing with the full algebra of bounded operators on a Hilbert space together with vector states.…”

Lüders Instruments, Generalised Lüders Theorem, and Some Aspects of Sufficiency

“…Obviously, sufficiency in Umegaki's sense implies sufficiency in Petz's sense which in turn implies sufficiency. Some aspects of sufficiency, Petz's sufficiency, and Umegaki's sufficiency were investigated in [13], [6,7], and [17,18], respectively, while in [8,9] …”

State Determination and Sufficiency of Observables

“…Sufficiency of a quantum statistic in Petz's sense was introduced in [9] (under the name of 'sufficiency'), and afterwards analyzed in [7,8]. State determination and distinction was investigated in [2] and [1].…”