Quantum information as a non-Kolmogorovian generalization of Shannon's theory

TL;DR

This paper explores a generalized information theory framework extending classical probability to non-commutative structures, positioning quantum information as a specific instance within a broad family of non-commutative theories.

Contribution

It introduces a non-commutative extension of probability calculus, generalizes state spectra, and develops new entropic measures, broadening the understanding of quantum and classical information.

Findings

Quantum information as a special case of non-commutative extensions

Development of a generalized majorization relation

Introduction of new families of entropic measures

Abstract

In this article we discuss the formal structure of a generalized information theory based on the extension of the probability calculus of Kolmogorov to a (possibly) non-commutative setting. By studying this framework, we argue that quantum information can be considered as a particular case of a huge family of non-commutative extensions of its classical counterpart. In any conceivable information theory, the possibility of dealing with different kinds of information measures plays a key role. Here, we generalize a notion of state spectrum, allowing us to introduce a majorization relation and a new family of generalized entropic measures.