Generalized Probabilities in Statistical Theories

Abstract

We discuss different formal frameworks for the description of generalized probabilities in statistical theories. We analyze the particular cases of probabilities appearing in classical and quantum mechanics and the approach to generalized probabilities based on convex sets. We argue for considering quantum probabilities as the natural probabilistic assignments for rational agents dealing with contextual probabilistic models. In this way, the formal structure of quantum probabilities as a non-Boolean probabilistic calculus is endowed with a natural interpretation.

Contextuality, Complementarity, Signaling, and Bell Tests

“…I remark that quantum logical-approach studies generalized probabilistic theories which are contextual in the sense described in this article. In this way, the Växjö model is a particular case of the general probabilistic framework based in measures over bounded lattices [ 139 ]. However, as such, the description of quantum probabilities is more specific, because it is described in terms of orthomodular lattices.…”

Non-Kolmogorovian Probabilities and Quantum Technologies

“…If that is the case, each proposition of the form “the value of f lies in the interval ” will have a well-defined probability value, given by the formula . Quantum mechanical observable mean values and probabilities are computed using analogous formulas, which are a natural generalization of Kolmogorov’s axioms (see the comparison between classical and quantum observables discussed in [ 56 ]).…”

“…Very precise algebraic characterization can be given as the atomic and complete orthomodular lattice of orthogonal projections acting on a separable Hilbert space. For details on this algebraic notion, we refer the reader to [ 71 ] (see also [ 56 ] for a not-so-technical introduction to the subject). The empirical content of each should be very intuitive by now: the local Boolean algebras are formed by the mathematical representatives of the propositions associated with the different experiments one can perform on the quantum system, which are orthogonal projections.…”

“…Instead of using a Boolean algebra (or a sigma algebra), in quantum theory, we must use a non-distributive orthomodular lattice. It is remarkable that the problem of the axiomatization of quantum probabilities can be traced back to Hilbert’s sixth problem [ 56 , 73 ].…”

“…The diagram associated with this perspective can be described as follows: where is just the horizontal pasting of the local algebras (i.e., the possible intertwinings between contexts are not considered). We refer the reader to [ 56 ] for a review about generalized probabilistic theories in which the orthomodular lattices approach is explained in an accessible way.…”