Operational meanings of a generalized conditional expectation in quantum metrology
A unifying formalism of generalized conditional expectations (GCEs) for quantum mechanics has recently emerged, but its physical implications regarding the retrodiction of a quantum observable remain controversial. To address the controversy, here I offer operational meanings for a version of the GC...
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2023-11-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2023-11-03-1162/pdf/ |
| _version_ | 1797639306511122432 |
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| author | Mankei Tsang |
| author_facet | Mankei Tsang |
| author_sort | Mankei Tsang |
| collection | DOAJ |
| description | A unifying formalism of generalized conditional expectations (GCEs) for quantum mechanics has recently emerged, but its physical implications regarding the retrodiction of a quantum observable remain controversial. To address the controversy, here I offer operational meanings for a version of the GCEs in the context of quantum parameter estimation. When a quantum sensor is corrupted by decoherence, the GCE is found to relate the operator-valued optimal estimators before and after the decoherence. Furthermore, the error increase, or regret, caused by the decoherence is shown to be equal to a divergence between the two estimators. The real weak value as a special case of the GCE plays the same role in suboptimal estimation – its divergence from the optimal estimator is precisely the regret for not using the optimal measurement. For an application of the GCE, I show that it enables the use of dynamic programming for designing a controller that minimizes the estimation error. For the frequentist setting, I show that the GCE leads to a quantum Rao-Blackwell theorem, which offers significant implications for quantum metrology and thermal-light sensing in particular. These results give the GCE and the associated divergence a natural, useful, and incontrovertible role in quantum decision and control theory. |
| first_indexed | 2024-03-11T13:15:52Z |
| format | Article |
| id | doaj.art-a9e82030a845409687c93fc73bb8b250 |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-03-11T13:15:52Z |
| publishDate | 2023-11-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-a9e82030a845409687c93fc73bb8b2502023-11-03T13:16:17ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2023-11-017116210.22331/q-2023-11-03-116210.22331/q-2023-11-03-1162Operational meanings of a generalized conditional expectation in quantum metrologyMankei TsangA unifying formalism of generalized conditional expectations (GCEs) for quantum mechanics has recently emerged, but its physical implications regarding the retrodiction of a quantum observable remain controversial. To address the controversy, here I offer operational meanings for a version of the GCEs in the context of quantum parameter estimation. When a quantum sensor is corrupted by decoherence, the GCE is found to relate the operator-valued optimal estimators before and after the decoherence. Furthermore, the error increase, or regret, caused by the decoherence is shown to be equal to a divergence between the two estimators. The real weak value as a special case of the GCE plays the same role in suboptimal estimation – its divergence from the optimal estimator is precisely the regret for not using the optimal measurement. For an application of the GCE, I show that it enables the use of dynamic programming for designing a controller that minimizes the estimation error. For the frequentist setting, I show that the GCE leads to a quantum Rao-Blackwell theorem, which offers significant implications for quantum metrology and thermal-light sensing in particular. These results give the GCE and the associated divergence a natural, useful, and incontrovertible role in quantum decision and control theory.https://quantum-journal.org/papers/q-2023-11-03-1162/pdf/ |
| spellingShingle | Mankei Tsang Operational meanings of a generalized conditional expectation in quantum metrology Quantum |
| title | Operational meanings of a generalized conditional expectation in quantum metrology |
| title_full | Operational meanings of a generalized conditional expectation in quantum metrology |
| title_fullStr | Operational meanings of a generalized conditional expectation in quantum metrology |
| title_full_unstemmed | Operational meanings of a generalized conditional expectation in quantum metrology |
| title_short | Operational meanings of a generalized conditional expectation in quantum metrology |
| title_sort | operational meanings of a generalized conditional expectation in quantum metrology |
| url | https://quantum-journal.org/papers/q-2023-11-03-1162/pdf/ |
| work_keys_str_mv | AT mankeitsang operationalmeaningsofageneralizedconditionalexpectationinquantummetrology |