Classification of transformations of probabilities for preparation procedures: trigonometric and hyperbolic behaviours

TL;DR

This paper classifies probabilistic transformations in physical systems, revealing both trigonometric and hyperbolic behaviors, and shows that quantum rules are just one subset of possible transformations, including experimentally observed hyperbolic polarization.

Contribution

It introduces a comprehensive classification of probability transformations, highlighting hyperbolic behaviors alongside traditional quantum trigonometric rules, and interprets experimental hyperbolic polarization.

Findings

Quantum probabilistic rule is one among many transformations.

Hyperbolic polarization has been experimentally observed.

Hyperbolic interference presents a more complex scenario.

Abstract

We provide frequency probabilistic analysis of perturbations of physical systems by preparation procedures. We obtained the classification of possible probabilistic transformations connecting input and output probabilities that can appear in physical experiments. We found that so called quantum probabilistic rule is just one of possible rules. Besides the well known trigonometric transformation (for example, for the polarization of light), there exist the hyperbolic transformation of probabilities. In fact, `hyperbolic polarization' have laready been observed in experiments with elementary particles. However, it was not interpreted in such a way. The situation is more complex with the hyperbolic interference of alternatives.