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In classical physics, the Kolmogorov extension theorem lays the foundationfor 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. Moregenerally, it does not hold in any theory of stochastic processes -- classical,quantum or beyond -- that does not just describe passive observations, butallows for active interventions. Such processes form the basis of the study ofcausal modelling across the sciences, including in the quantum domain. To date,these frameworks have lacked a conceptual underpinning similar to that providedby Kolmogorov's theorem for classical stochastic processes. We prove ageneralized extension theorem that applies to all theories of stochasticprocesses, putting them on equally firm mathematical ground as their classicalcounterpart. Additionally, we show that quantum causal modelling and quantumstochastic processes are equivalent. This provides the correct framework forthe description of experiments involving continuous control, which play acrucial role in the development of quantum technologies. Furthermore, we showthat the original extension theorem follows from the generalized one in thecorrect limit, and elucidate how a comprehensive understanding of generalstochastic processes allows one to unambiguously define the distinction betweenthose that are classical and those that are quantum.

Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories