Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117583
Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551
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Abstract
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.
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Cited by
[1] Arthur J. Parzygnat and James Fullwood, “From Time-Reversal Symmetry to Quantum Bayes’ Rules”, PRX Quantum 4 2, 020334 (2023).
[2] James Fullwood and Arthur J. Parzygnat, “On dynamical measures of quantum information”, arXiv:2306.01831, (2023).
[3] Xiao-Jie Tan and Mankei Tsang, “Quantum limit to subdiffraction incoherent optical imaging. III. Numerical analysis”, arXiv:2308.04317, (2023).
[4] Arthur J. Parzygnat, James Fullwood, Francesco Buscemi, and Giulio Chiribella, “Virtual quantum broadcasting”, arXiv:2310.13049, (2023).
The above citations are from SAO/NASA ADS (last updated successfully 2023-11-03 13:10:02). The list may be incomplete as not all publishers provide suitable and complete citation data.
Could not fetch Crossref cited-by data during last attempt 2023-11-03 13:10:00: Could not fetch cited-by data for 10.22331/q-2023-11-03-1162 from Crossref. This is normal if the DOI was registered recently.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
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- Source: https://quantum-journal.org/papers/q-2023-11-03-1162/
