Connecting commutativity and classicality for multitime quantum processes

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. We formalize the connection between these two a priori independent notions of classicality, demonstrate that they are distinct in general and detail their implications for memoryless multitime quantum processes.