Connecting Commutativity and Classicality for Multi-Time 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.

14.5 pages, 2 figures. Close to published version

References in corpus (9)

• Quantum Darwinism
• No-local-broadcasting theorem for quantum correlations
• Operational Markov condition for quantum processes
• Classicality in discrete Wigner functions
• Non-disturbing quantum measurements
• Lüders Theorem for Unsharp Quantum Measurements
• Quantumness Witnesses
• The collapse of a quantum state as a joint probability construction
• Statistics of projective measurement on a quantum probe as a witness of noncommutativity of algebra of a probed system