General Probabilistic Framework of Randomness

TL;DR

This paper develops a comprehensive mathematical framework for describing experiments on various systems using initial information, unifying classical and quantum probability models, and clarifying the nature of state reduction in experiments.

Contribution

It introduces a unified probabilistic framework that encompasses classical and quantum models, including non-destructive experiments and state reduction phenomena.

Findings

The framework includes both Kolmogorov's probability model and quantum theory as special cases.

It establishes a unique generalized observable for any experiment based on initial information.

Proves that mappings in quantum experiments are completely positive.

Abstract

We introduce a new mathematical framework for the probabilistic description of an experiment upon a system of any type in terms of initial information representing this system. Based on the notions of an information state, an information state space and a generalized observable, this general framework covers the description of a wide range of experimental situations including those where, with respect to a system, an experiment is perturbing. We prove that, to any experiment upon a system, there corresponds a unique generalized observable on a system initial information state space, which defines the probability distribution of outcomes under this experiment. We specify the case where initial information on a system provides "no knowledge" for the description of an experiment. Incorporating in a uniform way the basic notions of conventional probability theory and the non-commutativity aspects and the basic notions of quantum measurement theory, our framework clarifies the principle difference between Kolmogorov's model in probability theory and the statistical model of quantum theory. Both models are included into our framework as particular cases. We show that the phenomenon of "reduction" of a system initial information state is inherent, in general, to any non-destructive experiment and upon a system of any type. Based on our general framework, we introduce the probabilistic model for the description of non-destructive experiments upon a quantum system and prove that positive bounded linear mappings on the Banach space of trace class operators, arising in the description of experiments upon a quantum system, are completely positive.