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

Simon Milz , Fattah Sakuldee , Felix A. Pollock , Kavan Modi


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.
Author Simon Milz
Simon Milz,,
, Fattah Sakuldee (ICTQT)
Fattah Sakuldee,,
- International Centre for Theory of Quantum Technologies
, Felix A. Pollock
Felix A. Pollock,,
, Kavan Modi
Kavan Modi,,
Journal seriesQuantum, ISSN 2521-327X, (N/A 140 pkt)
Issue year2020
Publication size in sheets1.05
Article number255
URL https://quantum-journal.org/papers/q-2020-04-20-255/pdf/
Languageen angielski
LicenseJournal (articles only); published final; Uznanie Autorstwa (CC-BY); with publication
Score (nominal)140
Score sourcejournalList
ScoreMinisterial score = 140.0, 13-05-2020, ArticleFromJournal
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