Double or nothing: a Kolmogorov extension theorem for multitime (bi)probabilities in quantum mechanics
1Department Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
2Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, via E. Orabona 4, 70125 Bari, Italy
3Istituto Nazionale di Fisica Nucleare, Sezione di Bari, via G. Amendola 173, 70126 Bari, Italy
4Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang, 200240 Shanghai, China
5The International Centre for Theory of Quantum Technologies, University of Gdańsk, Jana Bażyńskiego 1A, 80-309 Gdańsk, Poland
6Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, PL 02-668 Warsaw, Poland
7Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń, Poland
| Published: | 2024-08-27, volume 8, page 1447 |
| Eprint: | arXiv:2402.01218v3 |
| Doi: | https://doi.org/10.22331/q-2024-08-27-1447 |
| Citation: | Quantum 8, 1447 (2024). |
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Abstract
The multitime probability distributions obtained by repeatedly probing a quantum system via the measurement of an observable generally violate Kolmogorov's consistency property. Therefore, one cannot interpret such distributions as the result of the sampling of a single trajectory. We show that, nonetheless, they do result from the sampling of one $pair$ of trajectories. In this sense, rather than give up on trajectories, quantum mechanics requires to double down on them. To this purpose, we prove a generalization of the Kolmogorov extension theorem that applies to families of complex-valued bi-probability distributions (that is, defined on pairs of elements of the original sample spaces), and we employ this result in the quantum mechanical scenario. We also discuss the relation of our results with the quantum comb formalism.
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[2] Piotr Szańkowski, "A perturbation theory for multi-time correlation functions in open quantum systems", SciPost Physics 19 3, 066 (2025).
[3] Joshua Lackman, "A Mathematical Definition of Path Integrals on Symplectic Manifolds", arXiv:2406.14547, (2024).
[4] Piotr Szańkowski and Łukasz Cywiński, "Objectivity of classical quantum stochastic processes", Quantum 8, 1390 (2024).
[5] Győző Egri, Marton Gomori, Balazs Gyenis, and Gábor Hofer-Szabó, "Trajectory of Probabilities, Probability on Trajectories, and the Stochastic-Quantum Correspondence", arXiv:2602.23491, (2026).
The above citations are from Crossref's cited-by service (last updated successfully 2026-08-10 14:08:51) and SAO/NASA ADS (last updated successfully 2026-08-10 14:08:52). The list may be incomplete as not all publishers provide suitable and complete citation data.
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