Entropic Heisenberg limits and uncertainty relations from the Holevo information bound

Abstract

Strong and general entropic and geometric Heisenberg limits are obtained, for estimates of multiparameter unitary displacements in quantum metrology, such as the estimation of a magnetic field from the induced rotation of a probe state in three dimensions. A key ingredient is the Holevo bound on the Shannon mutual information of a quantum communication channel. This leads to a Bayesian bound on performance, in terms of the prior distribution of the displacement and the asymmetry of the input probe state with respect to the displacement group. A geometric measure of performance related to entropy is proposed for general parameter estimation. It is also shown how strong entropic uncertainty relations for mutually unbiased observables, such as number and phase, position and momentum, energy and time, and orthogonal spin-1/2 directions, can be obtained from elementary applications of Holevo’s bound. A geometric interpretation of results is emphasised, in terms of the ‘volumes’ of quantum and classical statistical ensembles.