Physical symmetries and Fisher's information measure

Abstract

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.