Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories
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 --...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2020-04-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2020-04-20-255/pdf/ |
| _version_ | 1817988482791899136 |
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| author | Simon Milz Fattah Sakuldee Felix A. Pollock Kavan Modi |
| author_facet | Simon Milz Fattah Sakuldee Felix A. Pollock Kavan Modi |
| author_sort | Simon Milz |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-14T00:34:40Z |
| format | Article |
| id | doaj.art-1a016b5c8c5d4d43ba9d5c0bba8c26d8 |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-04-14T00:34:40Z |
| publishDate | 2020-04-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-1a016b5c8c5d4d43ba9d5c0bba8c26d82022-12-22T02:22:25ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2020-04-01425510.22331/q-2020-04-20-25510.22331/q-2020-04-20-255Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theoriesSimon MilzFattah SakuldeeFelix A. PollockKavan ModiIn 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.https://quantum-journal.org/papers/q-2020-04-20-255/pdf/ |
| spellingShingle | Simon Milz Fattah Sakuldee Felix A. Pollock Kavan Modi Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories Quantum |
| title | Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories |
| title_full | Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories |
| title_fullStr | Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories |
| title_full_unstemmed | Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories |
| title_short | Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories |
| title_sort | kolmogorov extension theorem for quantum causal modelling and general probabilistic theories |
| url | https://quantum-journal.org/papers/q-2020-04-20-255/pdf/ |
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