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Operational meaning of a generalized conditional expectation in quantum metrology

Mankei Tsang
Dec 2022
摘要
A unifying formalism of generalized conditional expectations (GCEs) forquantum mechanics has recently emerged, but its physical implications regardingthe retrodiction of a quantum observable remain controversial. To address thecontroversy, here I offer an operational meaning of a version of the GCEs -- ofwhich the real weak value is a special case -- in the context of Bayesianquantum parameter estimation. When a quantum sensor is corrupted bydecoherence, the GCE is found to relate the operator-valued optimal estimatorsbefore and after the decoherence. Furthermore, the error increase, or regret,caused by the decoherence is shown to be equal to a divergence between the twoestimators. The real weak value, in particular, plays the same role insuboptimal estimation -- its divergence from the optimal estimator is preciselythe regret for not using the optimal measurement. As an application of theformalism, I show that it enables the use of dynamic programming for designinga controller that minimizes the estimation error. These results give the GCEand the associated divergence a natural, useful, and incontrovertible role inquantum decision and control theory.
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