An operational approach to quantum probability

Abstract

In order to provide a mathmatical framework for the process of making repeated measurements on continuous observables in a statistical system we make a mathematical definition of an instrument, a concept which generalises that of an observable and that of an operation. It is then possible to develop such notions as joint and conditional probabilities without any of the commutation conditions needed in the approach via observables. One of the crucial notions is that of repeatability which we show is implicitly assumed in most of the axiomatic treatments of quantum mechanics, but whose abandonment leads to a much more flexible approach to measurement theory.

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

“…The notion of non-degeneracy was introduced in [3] for a class of instruments. A natural generalisation of this notion is as follows.…”

“…Roughly speaking, they express the celebrated von Neumann repeatability hypothesis which says: if the physical quantity is measured twice in succession in a system, then we get the same value each time (cf. [3,14]). Following [3], the measurement is said to be weakly repeatable if the instrument describing it satisfies the condition:…”

“…[3,14]). Following [3], the measurement is said to be weakly repeatable if the instrument describing it satisfies the condition:…”

On the Interpretation of Probabilities in Generalized Probabilistic Models

“…Later, it was discovered that this was strongly related to the nonexistence of joint distributions for noncommuting observables. These peculiarities and formal aspects of the probabilities involved in quantum theory have been vastly studied in the literature [5][6][7][8][9][10][11]. One of the most important axiomatizations in probability theory is due to Kolmogorov [12].…”

Representations of Baer ∗-semigroups

“…Representations of Baer »-semigroups are not only of interest in their own right, they can be important in the study of quantum logics [7], [11], [13], [18] and operational quantum mechanics [2], [3], [7], [9]. Because of the intimate connection between Baer »-semigroups and orthomodular lattices [5], [6], representations of the former can provide embeddings of the latter [8].…”