Abstract
The quantum Cramér-Rao bound is a cornerstone of modern quantum metrology, as it provides the ultimate precision in parameter estimation. In the multiparameter scenario, this bound becomes a matrix inequality, which can be cast to a scalar form with a properly chosen weight matrix. Multiparameter estimation thus elicits trade-offs in the precision with which each parameter can be estimated. We show that, if the information is encoded in a unitary transformation, we can naturally choose the weight matrix as the metric tensor linked to the geometry of the underlying algebra su(n), with applications in numerous fields. This ensures an intrinsic bound that is independent of the choice of parametrization.
Highlights
Introduction.—A central challenge in quantum metrology is to develop strategies for enhancing the precision of parameter estimation
The quantum Cramer-Rao bound is a cornerstone of modern quantum metrology, as it provides the ultimate precision in parameter estimation
If the information is encoded in a unitary transformation, we can naturally choose the weight matrix as the metric tensor linked to the geometry of the underlying algebra suðnÞ, with applications in numerous fields
Summary
The quantum Cramer-Rao bound is a cornerstone of modern quantum metrology, as it provides the ultimate precision in parameter estimation. Multiparameter quantum metrology finds many important applications in fields as diverse as imaging [4,5,6], field sensing [7,8,9], sensor networks [10,11,12], and remote sensing [13] to cite but a few examples In this case, the QCRB is a matrix inequality, and the ultimate bound is generally not saturable for all parameters. The Holevo Cramer-Rao bound (HCRB) [2] epitomizes the conundrums associated with incompatible observables: its main tenet is to map the matrix QCRB onto a scalar inequality by using a positive-definite weight matrix and optimize this scalar bound over all physically viable measurement procedures for a given probe state In this manner, one obtains a weighted mean square error that has to be minimized. When this metric is used as our weight matrix, we obtain a QCRB with intrinsic properties, 0031-9007=21=127(11)=110501(7)
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