Vistas of Algebraic Probability: Quantum Computation and Information

TL;DR

This paper explores an algebraic framework for probability that encompasses both classical and quantum-like behaviors, emphasizing the role of non-commutativity in quantum phenomena and its relevance to quantum computing.

Contribution

It provides a finite-dimensional algebraic perspective on quantum probability, highlighting structural features and potential applications in quantum computation.

Findings

Algebraic approach unifies classical and quantum probability.

Non-commutativity distinguishes quantum-like behaviors.

Framework applicable to quantum computing models.

Abstract

Kolmogorov's foundation of probability takes measure spaces, $\sigma$-algebras, and probability measures as basic objects. It is, however, widely recognized that this classical framework is inadequate for random phenomena involving quantum effects, and more generally for \emph{quantum-like} situations. A broader formulation is provided by an algebraic viewpoint: one starts from an algebra of random variables equipped with a distinguished linear functional -- the \emph{state} -- interpreted as expectation. In this sense, the approach can also be viewed as a modern reading of ideas already implicit in early probability (e.g., the Bernoullis), while its contemporary form has been developed and used extensively in quantum physics. The algebraic framework accommodates both classical and quantum-like behaviours, yet it remains underused in classical probability and uncertainty quantification, where it can nevertheless open new perspectives and clarify structural features. Although the language carries a physics flavor, the subject is purely probabilistic. The key distinction between classical and quantum-like behaviour is \emph{commutativity}: its failure produces the characteristic effects of quantum-like situations. The rise of quantum computing is a prominent setting in which such behaviour may become relevant even for practitioners in computational science. Here we focus on the purely algebraic core of the approach. By restricting attention to finite-dimensional algebras, we avoid many analytical subtleties while retaining the main ideas, their classical limit, and their applicability to quantum-like models and quantum computation.