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

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.

Stochastic thermodynamics with arbitrary interventions

“…To close this section, we remark that causal models can be also represented in different ways [17] (see also Sec. IV E) and the picture we have given here follows closely the description in the quantum case [26][27][28]. The particular and simple description (7) is a consequence of the Markov property [26].…”

Repeated Interactions and Quantum Stochastic Thermodynamics at Strong Coupling

“…. While this step is not necessary from a thermodynamic perspective, it will allow to rigourously connect our framework to the theory of quantum stochastic processes [44][45][46], see below. It gives rise to an instantaneous unitary operation correlating the system and unit via…”

“…. , A rn to an open quantum system defines a general quantum stochastic process, which can be formally represented by a 'quantum comb' or 'process tensor' [44][45][46]. The sole difference compared to the most general case is that we do not allow for real-time feedback control, i.e., the instruments A r k are not allowed to depend on the previous results r k−1 .…”

George Sudarshan and Quantum Dynamics

“…Nevertheless, this way of looking at quantum processes naturally resolves the ambiguity of what makes a quantum processes Markovian [35] and when the memory is quantum [18]. It leads to a unifying framework for spatio-temporal correlation [11,39], where a space-time version of the Born rule appears [62]. Later we also generalised Kuah's idea [37] to fit restricted control process tensor [41].…”