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Kolmogorov extension theorem for general (quantum) stochastic processes

Research paper by Simon Milz, Fattah Sakuldee, Felix A. Pollock, Kavan Modi

Indexed on: 07 Dec '17Published on: 07 Dec '17Published in: arXiv - Quantum Physics



Abstract

In classical physics, the Kolmogorov extension theorem provides the foundation for the definition and investigation of stochastic processes. In its original form, it does not hold in quantum mechanics. More generally, it does not hold in any theory -- classical, quantum or beyond -- of stochastic processes that does not just describe passive observations, but allows for active interventions. Thus, to date, these frameworks lack a firm theoretical underpinning. We prove a generalized extension theorem for stochastic processes that applies to all theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart, and providing the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalised one in the correct limit, and elucidate how the comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.