Generalized uncertainty relations and efficient measurements in quantum systems

We consider two variants of a quantum-statistical generalization of the Cramer–Rao inequality that establish an invariant lower bound on the mean square error of a generalized quantum measurement. In contrast to Helstrom's variant [1], the proposed complex variant of this inequality leads to a precise formulation of a generalized uncertainty principle for arbitrary states. A bound is found for the accuracy of estimating the parameters of canonical states and, 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 effficient measurements and quasimeasurements.