Less (precision) is more (information): quantum information in fuzzy probability theory

TL;DR

This paper explores the relationship between quantum and classical theories using a common convexity framework, showing that quantum observables can be represented as fuzzy classical variables within a classical extension.

Contribution

It classifies all possible embeddings of quantum theories into classical frameworks, demonstrating the uniqueness of the canonical classical extension of quantum mechanics.

Findings

Quantum observables are equivalent to fuzzy classical variables.

The canonical classical extension is the unique effective classical representation of quantum models.

All other embeddings do not satisfy the specified constraints.

Abstract

A comparison of structural features of quantum and classical physical theories, such as the information capacity of systems subject to these theories, requires a common formal framework for the presentation of corresponding concepts (such as states, observables, probability, entropy). Such a framework is provided by the notion of statistical model developed in the convexity approach to statistical physical theories. Here we use statistical models to classify and survey all possible types of embedding and extension of quantum probabilistic theories subject to certain reasonable constraints. It will be shown that the so-called canonical classical extension of quantum mechanics is essentially the only `good' representation of the quantum statistical model in a classical framework. All quantum observables are thus identified as fuzzy classical random variables.