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

TL;DR

This paper generalizes the Kolmogorov extension theorem to all stochastic process theories, including quantum, establishing a solid mathematical foundation for causal modeling and demonstrating the equivalence of quantum causal and stochastic processes.

Contribution

It introduces a generalized extension theorem applicable to all stochastic process theories, bridging classical and quantum causal modeling frameworks.

Findings

Proves a generalized extension theorem for all stochastic process theories.

Shows quantum causal modeling is equivalent to quantum stochastic processes.

Clarifies the classical-quantum boundary in stochastic processes.

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.