  We consider two variants of a quantum-statistical generalization of the
Cramer-Rao inequality that establishes an invariant lower bound on the mean
square error of a generalized quantum measurement. The proposed complex variant
of this inequality leads to a precise formulation of a generalized uncertainty
principle for arbitrary states, in contrast to Helstrom's symmetric variant in
which these relations are obtained only for pure states. A notion of canonical
states is introduced and the lower mean square error bound is found for
estimating of the parameters of canonical states, in particular, the canonical
parameters of a Lie group. It is shown that these bounds are globally
attainable only for canonical states for which there exist efficient
measurements or quasimeasurements.
