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arXiv:0802.1181

Estimation of n non-identical unitary channels

Caleb J O'Loan

Published Feb 8, 2008• 0 citations
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Abstract

We investigate the simultaneous estimation of $n$ not necessarily identical unitary channels using multi-partite entanglement. We examine whether it is possible for the rate at which the mean square error decreases to be greater than that using the channels individually. For a reasonably general situation, in which there is no functional dependence between the channels, we show that this is not possible. We look at a case in which the channels are not necessarily identical but depend on a common variable. In this case, the mean square error decreases more rapidly using multi-partite entanglement.

DOI
10.48550/arxiv.0802.1181

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We investigate the simultaneous estimation of nnn not necessarily identical unitary channels using multi-partite entanglement. We examine whether it is possible for the rate at which the mean square error decreases to be greater than that using the channels individually. For a reasonably general situation, in which there is no functional dependence between the channels, we show that this is not possible. We look at a case in which the channels are not necessarily identical but depend on a common variable. In this case, the mean square error decreases more rapidly using multi-partite entanglement.
The quantum Cramer-Rao bound places a limit on the mean square error of a parameter estimation procedure, and its numerical value is determined by the quantum Fisher information. For single parameters, this leads to the well- known Heisenberg limit that surpasses the classical shot-noise limit. When estimating multiple parameters, the situation is more complicated and the quantum Cramer-Rao bound is generally not attainable. In such cases, the use of entanglement typically still offers an enhancement in precision. Here, we demonstrate that entanglement is detrimental when estimating some nuisance parameters. In general, we find that the estimation of coupled parameters does not benefit from either classical or quantum correlations. We illustrate this effect in a practical application for optical gyroscopes.
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So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. As for channels, however, only a little had been known. In this paper, we impose monotonicity by concatenation of channels before and after the given channel families, and invariance by tensoring identity channels. Under these axioms, we identify the largest and the smallest metrics. Also, we studied asymptotic theory of metric in parallel and adaptive repetition settings, and applied them to the study of channel estimation. First we express the achievable lower bound of the mean square error (MSE) of an estimate by a monotone channel metric, and show this equals O(1/n) for noisy channels, where n is the number of times of channel use. This result shows Heisenberg rate, or O(1/n^2)-rate of the MSE observed in case of estimation of unitary, collapses with very small arbitrary noise.
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.