Connecting Commutativity and Classicality for Multi-Time Quantum Processes

TL;DR

This paper explores the relationship between classicality and quantum behavior in multi-time quantum processes by linking Kolmogorov consistency with measurement operator commutativity, revealing their distinctions and implications.

Contribution

It formalizes the connection between classicality notions in quantum processes, showing they are generally distinct and analyzing their implications for memoryless multi-time quantum systems.

Findings

Kolmogorov consistency implies classical stochastic process behavior.

Measurement operator commutativity is a structural property linked to classical physics.

The two notions of classicality are generally independent in quantum processes.

Abstract

Understanding the demarcation line between classical and quantum is an important issue in modern physics. The development of such an understanding requires a clear picture of the various concurrent notions of `classicality' in quantum theory presently in use. Here, we focus on the relationship between Kolmogorov consistency of measurement statistics -- the foundational footing of classical stochastic processes in standard probability theory -- and the commutativity (or absence thereof) of measurement operators -- a concept at the core of quantum theory. Kolmogorov consistency implies that the statistics of sequential measurements on a (possibly quantum) system could be explained entirely by means of a classical stochastic process, thereby providing an operational notion of classicality. On the other hand, commutativity of measurement operators is a structural property that holds in classical physics and its breakdown is the origin of the uncertainty principle, a fundamentally quantum phenomenon. Here, we formalise the connection between these two a priori independent notions of classicality, demonstrate that they are distinct in general and detail their implications for memoryless multi-time quantum processes.