Double or nothing: a Kolmogorov extension theorem for multitime (bi)probabilities in quantum mechanics
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912003306553344 |
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| author | Lonigro, Davide Sakuldee, Fattah Cywiński, Łukasz Chruściński, Dariusz Szańkowski, Piotr |
| author_facet | Lonigro, Davide Sakuldee, Fattah Cywiński, Łukasz Chruściński, Dariusz Szańkowski, Piotr |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01218 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Double or nothing: a Kolmogorov extension theorem for multitime (bi)probabilities in quantum mechanics Lonigro, Davide Sakuldee, Fattah Cywiński, Łukasz Chruściński, Dariusz Szańkowski, Piotr Quantum Physics Mathematical Physics 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. |
| title | Double or nothing: a Kolmogorov extension theorem for multitime (bi)probabilities in quantum mechanics |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2402.01218 |