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arXiv:quant-ph/0105059 · Journal of Physics A Mathematical and General

Linear representations of probabilistic transformations induced by context transitions

Andrei Yu. Khrennikov

Published Nov 21, 2001• 73 citations
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Abstract

By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. It is shown that not only quantum preparation procedures can have trigonometric probabilistic behaviour. We propose generalizations of C -linear space probabilistic calculus to describe non-quantum (trigonometric and hyperbolic) probabilistic transformations. We also analyse the superposition principle in this framework.

DOI
10.1088/0305-4470/34/47/304

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By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. It is shown that not only quantum preparation procedures can have trigonometric probabilistic behaviour. We propose generalizations of C -linear space probabilistic calculus to describe non-quantum (trigonometric and hyperbolic) probabilistic transformations. We also analyse the superposition principle in this framework.
By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. Each class is characterized by a perturbation of the `classical probabilistic rule': (a) costheta\\cos\\thetacostheta, (b) coshtheta\\cosh \\thetacoshtheta, (c) both costheta\\cos\\thetacostheta and coshtheta\\cosh \\thetacoshtheta. Trigonometric transformations correspond to context-transitions that induce statistical deviations of relatively small magnitudes (in classical physics - negligibly small); hyperbolic - relatively large magnitudes. We found that not only preparation procedures described by conventional quantum formalism can have trigonometric probabilistic behaviour. We propose generalizations of bfC{\\bf C}bfC-linear space probabilistic calculus to describe non quantum (trigonometric and hyperbolic) probabilistic transformations.
We study transformations of conventional (`classical') probabilities induced by context transitions. It is demonstrated that the transition from one complex of conditions to another induces a perturbation of the classical rule for the addition of probabilistic alternatives. We classify such perturbations. It is shown that there are two classes of perturbations: (a) trigonometric interference; (b) hyperbolic interference. In particular, the well known `quantum interference of probabilistic alternatives' can be obtained in classical (but contextual) probabilistic framework. Therefore we need not apply to wave arguments (or consider superposition principle) to get Quantum Statistics. In particular, interference could be a feature of experiments with purely corpuscular objects.
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.
We demonstrate the quantum probabilistic rule (which differ from classical Bayes' formula by the cosinus factor) can be obtained on purely classical basis as a consequence of the perturbation effect of preparation procedures. In any case the interference can be easily simulated for macrosystems. Moreover, we obtain the classification of all possible transformations of probablities which can arise via perturbation effects. There are two main transformation classes: with trigonometric and hyperbolic ("interference") perturbations of classical Bayes' formula. Quantum probabilistic rule is just a particular case of trigonometric probablistic behaviour. Hyperbolic probablistic behaviour can be easily simulated. However, the question of its observation for "real physical phenomena" is open.
Generalized contextuality is a canonical distinguishing property of nonclassical generalized probabilistic theories, in particular quantum mechanics. Methods for certification and characterization of generalized contextuality of a given generalized probabilistic theory are well developed for prepare-measure and single-stage prepare-transform-measure scenarios. In a recent work [arXiv:2512.10000], a bottom-up, statistics-first linear-algebraic framework for contextuality in prepare-measure scenarios was introduced. We extend this approach to operational scenarios with sequential transformations with an arbitrary number of stages. We give a full decision procedure for contextuality of such scenarios within operational theories and analyze its computational complexity. In particular, our decision procedure has a complexity linearly exponential in the minimum generalized probabilistic theory (GPT) dimension, and polynomial in the number of procedures. We demonstrate our framework and approach through multiple examples, including Spekkens' toy theory and the 8-state single-qubit stabilizer theory. In particular, we construct an operational theory in which contextuality manifests itself only in the sequential structure of the transformations. Our findings thus shed new light on the significant role of compositional structures in the phenomenon of generalized contextuality.
We study the following problem: Is it possible to explain the quantum interference of probabilities in the purely corpuscular model for elementary particles? We demonstrate that (by taking into account perturbation effects of measurement and preparation procedures) we can obtain cos⁡θ\cosθcosθ-perturbation (interference term) in probabilistic rule connecting preparation procedures for purely corpuscular objects. On one hand, our investigation demonstrated that there is nothing special in so called `quantum probabilities': the right choice of statistical ensembles gives the possibility to escape all `pathologies'. On the other hand, we found that the standard trigonometric interference of alternatives (observed, in particular, in quantum mechanics) is not the unique possibility to extend (disturb) the conventional probabilistic rule for addition of alternatives. There exist two other probabilistic rules that connect three preparation procedures: hyperbolic and hyper-trigonometric interferences.