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

Simon Milz1,2, Fattah Sakuldee3,4, Felix A. Pollock2, and Kavan Modi2

1Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
2School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
3International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
4MU-NECTEC Collaborative Research Unit on Quantum Information, Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand.

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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.

While theories of general (quantum) processes with interventions have attracted considerable interest over the last decades, their axiomatic underpinnings are still opaque. In the classical case, the Kolmogorov extension theorem (KET) establishes the basic properties of stochastic processes; however, this theorem breaks down when interventions are allowed. For quantum processes, interventions are unavoidable. The present work closes this conceptual gap by providing a generalised version of the KET. Our generalised theorem lays the theoretical foundation for the description of all processes with interventions, be they classical, quantum or beyond. Two prominent and timely examples are the theories of quantum stochastic processes and quantum causal modelling. Our results have direct consequences for the characterisation and modelling of causal structure in stochastic processes throughout the quantitative sciences, and, notably, allows for a complete representation of arbitrary controlled quantum systems.

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