Sharp and Unsharp Descriptions of Physical Magnitudes in Finite Dimensional Quantum Mechanics

Abstract

Generalized (i.e., positive operator valued) observables generated by a standard (i.e., projection valued) observable via suitable confidence functions are introduced in the context of finite dimensional Hilbert quantum mechanics. These generalized observables are considered as unsharp realizations of the unique physical magnitude associated to the standard realization.We show a theorem about the “physical equivalence” (in the sense of equal probability distributions) of the two situations “sharp state — unsharp observable” and “unsharp state — sharp observable”; in particular, this equivalence is obtained by a partially defined nonlinear, isometric (nonunitary) operator.