Operational meanings of a generalized conditional expectation in quantum metrology
arXiv:2212.13162 · doi:10.22331/q-2023-11-03-1162
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.
arXiv comment25 pages, 3 figures. v6: extended and accepted by Quantum
References in corpus (40)
- From Classical to Quantum Shannon Theory
- Quantum Theory of Superresolution for Two Incoherent Optical Point Sources
- Technical advantages for weak value amplification: When less is more
- Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
- Fundamental Quantum Limit to Waveform Estimation
- Weak value amplification is suboptimal for estimation and detection
- Weak Values, Quantum Trajectories, and the Stony-Brook Cavity QED experiment
- Far-field Superresolution of Thermal Electromagnetic Sources at the Quantum Limit
- Ultimate precision bound of quantum and sub-wavelength imaging
- Prior information: how to circumvent the standard joint-measurement uncertainty relation
- How the result of a single coin toss can turn out to be 100 heads
- Weak Values are Interference Phenomena
- Time-Symmetric Quantum Theory of Smoothing
- Resolving starlight: a quantum perspective
- Bayesian quantum frequency estimation in presence of collective dephasing
- Nondemolition Principle of Quantum Measurement Theory
- Bayes' theorem and quantum retrodiction
- Quantum metrology in the presence of limited data
- Quantum Nonlocality in Weak-Thermal-Light Interferometry
- Quantum State Smoothing
- Can a quantum state over time resemble a quantum state at a single time?
- From time-reversal symmetry to quantum Bayes' rules
- Optimal waveform estimation for classical and quantum systems via time-symmetric smoothing. II. Applications to atomic magnetometry and Hardy's paradox
- Spectrum analysis with quantum dynamical systems
- Ultimate precision limit of noise sensing and dark matter search
- Axioms for retrodiction: achieving time-reversal symmetry with a prior
- Quantum metrology of noisy spreading channels
- Comparison of Information Structures and Completely Positive Maps
- Unifying theory of quantum state estimation using past and future information
- Beyond the diffraction limit via optical amplification
- The Quantum Blackwell Theorem and Minimum Error State Discrimination
- Quantum noise spectroscopy as an incoherent imaging problem
- Comment on "How the result of a single coin toss can turn out to be 100 heads"
- Quantum optimality of photon counting for temperature measurement of thermal astronomical sources
- A smoothing theory for open quantum systems: The least mean square approach
- How to estimate past measurement interventions on a quantum system undergoing continuous monitoring
- Quantum U-statistics
- Mismatched Quantum Filtering and Entropic Information
- Quantum smoothing
- Optimal waveform estimation for classical and quantum systems via time-symmetric smoothing
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- Virtual quantum broadcasting
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