Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario
It is well-known that the law of total probability does not generally hold in quantum theory. However, recent arguments on some of the fundamental assumptions in quantum theory based on the extended Wigner’s friend scenario show a need to clarify how the law of total probability should be formulated...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/7/903 |
| _version_ | 1797433584077766656 |
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| author | Jianhao M. Yang |
| author_facet | Jianhao M. Yang |
| author_sort | Jianhao M. Yang |
| collection | DOAJ |
| description | It is well-known that the law of total probability does not generally hold in quantum theory. However, recent arguments on some of the fundamental assumptions in quantum theory based on the extended Wigner’s friend scenario show a need to clarify how the law of total probability should be formulated in quantum theory and under what conditions it still holds. In this work, the definition of conditional probability in quantum theory is extended to POVM measurements. A rule to assign two-time conditional probability is proposed for incompatible POVM operators, which leads to a more general and precise formulation of the law of total probability. Sufficient conditions under which the law of total probability holds are identified. Applying the theory developed here to analyze several quantum no-go theorems related to the extended Wigner’s friend scenario reveals logical loopholes in these no-go theorems. The loopholes exist as a consequence of taking for granted the validity of the law of total probability without verifying the sufficient conditions. Consequently, the contradictions in these no-go theorems only reconfirm the invalidity of the law of total probability in quantum theory rather than invalidating the physical statements that the no-go theorems attempt to refute. |
| first_indexed | 2024-03-09T10:19:03Z |
| format | Article |
| id | doaj.art-2b70c0d80d9846088cbc4cc6ad0162d8 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T10:19:03Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-2b70c0d80d9846088cbc4cc6ad0162d82023-12-01T22:06:38ZengMDPI AGEntropy1099-43002022-06-0124790310.3390/e24070903Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend ScenarioJianhao M. Yang0Qualcomm, San Diego, CA 92121, USAIt is well-known that the law of total probability does not generally hold in quantum theory. However, recent arguments on some of the fundamental assumptions in quantum theory based on the extended Wigner’s friend scenario show a need to clarify how the law of total probability should be formulated in quantum theory and under what conditions it still holds. In this work, the definition of conditional probability in quantum theory is extended to POVM measurements. A rule to assign two-time conditional probability is proposed for incompatible POVM operators, which leads to a more general and precise formulation of the law of total probability. Sufficient conditions under which the law of total probability holds are identified. Applying the theory developed here to analyze several quantum no-go theorems related to the extended Wigner’s friend scenario reveals logical loopholes in these no-go theorems. The loopholes exist as a consequence of taking for granted the validity of the law of total probability without verifying the sufficient conditions. Consequently, the contradictions in these no-go theorems only reconfirm the invalidity of the law of total probability in quantum theory rather than invalidating the physical statements that the no-go theorems attempt to refute.https://www.mdpi.com/1099-4300/24/7/903the law of total probabilityextended Wigner’s friend scenarioPVOM measurement |
| spellingShingle | Jianhao M. Yang Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario Entropy the law of total probability extended Wigner’s friend scenario PVOM measurement |
| title | Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario |
| title_full | Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario |
| title_fullStr | Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario |
| title_full_unstemmed | Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario |
| title_short | Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario |
| title_sort | law of total probability in quantum theory and its application in wigner s friend scenario |
| topic | the law of total probability extended Wigner’s friend scenario PVOM measurement |
| url | https://www.mdpi.com/1099-4300/24/7/903 |
| work_keys_str_mv | AT jianhaomyang lawoftotalprobabilityinquantumtheoryanditsapplicationinwignersfriendscenario |