New Cramér-Rao bounds for quantum metrology

Luigi Seveso, Matteo A. C. Rossi, Matteo G. A. Paris

Published 2016-05-27Version 1

A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it does not influence the sample space and measurements aimed at extracting information on the parameter itself. This assumption is crucial to obtaining the quantum generalization of the Cram\'er-Rao theorem of classical statistics, i.e. the introduction of the quantum Fisher information as an upper bound of the Fisher information of any possible measurement. However, there are relevant estimation problems where this assumption does not hold and an alternative approach should be developed to find a proper bound to the ultimate precision allowed by quantum mechanics. Here we derive generalized Cram\'er-Rao bounds where those additional sources of information on the parameter are taken into account. The long-standing problem of estimating the direction of a magnetic field is revisited in order to illustrate our point.