Identifying quantum logics by numerical events

Abstract

Let S be a set of states of a physical system and let p(s) be the probability of an occurrence of an event when the system is in the state s ∈ S. The function p from S to [0, 1] is called a numerical event, multidimensional probability or, alternatively, S-probability. Given a set of numerical events which has been obtained by measurements and not supposing any knowledge of the logical structure of the events that appear in the physical system, the question arises which kind of logic is inherent to the system under consideration. In particular, does one deal with a classical situation or a quantum one?In this survey several answers are presented. Starting by associating sets of numerical events to quantum logics we study structures that arise when S-probabilities are partially ordered by the order of functions and characterize those structures which indicate that one deals with a classical system. In particular, sequences of numerical events are considered that give rise to Bell-like inequalities. At the center of all studies there are so called algebras of S-probabilities, subsets of these and their generalizations. A crucial feature of these structures is that order theoretic properties can be expressed by the addition and subtraction of real functions entailing simplified algorithmic procedures.The study of numerical events and algebras of S-probabilities goes back to a cooperation of E. G. Beltrametti and M. J. Mączyński in 1991 and has since then resulted in a series of subsequent papers of physical interest the main results of which will be commented on and put in an appropriate context.

On Boolean posets of numerical events

“…There are many papers in which (arbitrary) classes of algebras of S-probabilities are characterized to be Boolean algebras by specifying some structural properties-for an overview of these papers see Dorninger (2020)-and there are numerous results on Boolean orthoposets and concrete logics which can all be applied to fathom the distance between specific sets of S-probabilities and Boolean algebras (cf. i.a.…”

On Boolean posets of numerical events

“…There are many papers in which (arbitrary) classes of algebras of S-probabilities are characterized to be Boolean algebras by specifying some structural properties -for an overview of these papers see [5] -and there are numerous results on Boolean orthoposets and concrete logics which can all be applied to fathom the distance between specific sets of S-probabilities and Boolean algebras (cf. i.a.…”