  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.
