Physical symmetries and Fisher's information measure

We investigate the physical consequences of imposing symmetry requirements to the Cramer-Rao inequality, and investigate in particular translation, inversion, and rotation and show the above relation remains invariant under these transformations, which adds additional flavor to the adjective shift-invariant attached to the concept of Fisher measure. In particular, if the inequality is saturated, it remains so under any transformation represented by a square matrix, representative of a (physical) unitary operator, as is the case in the all important instance of the classical harmonic oscillator.