Abstract
In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes -- classical, quantum or beyond -- that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to all theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides 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 generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.
Highlights
The present situation implies an incompatibility between existing frameworks to describe processes with interventions and the classical theory of stochastic processes, even though they should converge to the latter in the correct limit. This suggests that the mere act of interacting with a system over time introduces a fundamental divide between the continuity of physical laws and the finite statistics that can be accessed in reality, begging the question: What do we generally mean by a ‘stochastic process’, and how can we reconcile causal modelling frameworks with the idea of an underlying process?
While the Kolmogorov extension theorem (KET) is the fundamental building block for the theory of classical stochastic processes, it does not hold in quantum mechanics, or any other theory that allows for active interventions
This breakdown goes hand in hand with the violation of Leggett-Garg inequalities: the violation of such an inequality always implies that compatibility conditions are not satisfied, and the KET does not hold
Summary
An extension theorem) implies an incompatibility between existing frameworks to describe processes with interventions (both classical and quantum) and the classical theory of stochastic processes, even though they should converge to the latter in the correct limit This suggests that the mere act of interacting with a system over time introduces a fundamental divide between the continuity of physical laws and the finite statistics that can be accessed in reality, begging the question: What do we generally mean by a (quantum) ‘stochastic process’, and how can we reconcile causal modelling frameworks with the idea of an underlying process?.
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