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European Quantum Electronics Conference (EQEC)

The Estimation of Signal Force Parameters in Quantum Variation Measurement

Sergey P. Vyatchanin

Published Aug 24, 2005• 1 citation
Publisher

Abstract

Abstract It is shown by examples that the procedure of a quantum variation measurement allows one to estimate simultaneously several parameters of the signal force (amplitude, arrival time, frequency) with errors much less than the standard quantum limit.
DOI
10.1109/eqec.1998.714775

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Multiparameter estimation remains a fundamental challenge in quantum estimation theory because the optimal measurements associated with different parameters are generally incompatible. In this work, we develop a Bayesian framework for the simultaneous estimation of multiple parameters. Our approach builds on Personick's method, in which the estimation problem is reduced to a set of Lyapunov equations whose solutions define optimal observables for the individual parameters. Since these observables generally do not commute, we construct a convex combination of the solutions, parameterized by a set of variational parameters. The spectral decomposition of the resulting operator defines a parametrized projection-valued measure. The resulting projective measurement determines the likelihood function, from which the minimum mean-square error estimators and the corresponding Bayesian mean-square errors are obtained, following standard Bayesian procedures, as functions of the variational parameters. To determine their optimal values, we formulate a minimax optimization problem based on the normalized Bayesian mean-square errors, introducing a min-max normalization procedure that enables a meaningful comparison of estimation errors associated with different parameters. This optimization yields a variationally optimized projective measurement for simultaneous Bayesian estimation. Two qubit examples, involving phase estimation and the estimation of parameters defining convex combinations of unitary operations, demonstrate the construction of optimized projective measurements and the corresponding minimum mean-square error estimators within the proposed framework for finite data sets.
Precise measurements of tiny forces and displacements play an important role in science and technology. The precision of recent experiments, while beginning to reach the limits imposed by quantum mechanics, is necessarily spoiled by the unavoidable influence of noise. Here we obtain a quantum limit for the uncertainty in the estimation of the amplitude of a classical force with arbitrary time dependence, acting on a noisy quantum-mechanical oscillator. We determine the best initial state of the oscillator and the best measurement procedure, thus getting a rigorous and useful benchmark for experiments aiming to detect extremely small forces.