Workshop on

Quantum Mechanics: Axiomatics of Measurements and connections with Computing and Information Retrieval

Satisfaction of Bell inequalities in the unsharp approach to quantum mechanics

aula 101 - Venerdì 7 Aprile, ore 11.00
  GIANPIERO CATTANEO, University of Milano Bicocca

Full Professor in "Dynamical System Theory" at the University of Milano -Bicocca. From 1994 to 1997 he has been Regular Visiting Professor at the London School of Economics, where, since 1998, he has a position of Research Associate. From 1997 to 1999 he has been Maitre de Conferences at the Nancy-Metz Academy, and Maitre de Conferences at la Ecole Normale Superieure in Lyon. Coordinator of several grants on Fluid dynamics and Cellular Automata. His research activity has been centered on Topological Chaos, Cellular Automata, Algebraic Approach to Fuzzy Logic and Rough Sets, Axiomatic Foundations of Quantum Mechanics, Realization of reversible gates by quantum computing techniques.

Gianpiero Cattaneo

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 nonlinear, isometric (nonunitary) partially defined operator. A similar result for the case of the localization observable in the case of the infinite dimensional Hilbert space L2(R3) is given.
An application to the Bohm version of EPR experiment is given, showing that the Clauser-Shimony description can be formulated as a "sharp singlet state -- unsharp spin correlation observation", which turns out to be physically equivalent to the Santos "unsharp singlet state--sharp spin correlation observation" description. In particular we characterize, according to P. Busch, the range of "unsharpness parametres" in which Bell's inequalities are verified.