Local Realistic Theories and Quantum Mechanics for the two-neutral-kaon system

The predictions of local realistic theories for the observables concerning the evolution of a $K^0\bar{K}^0$ quantum entangled pair (created in the decay of the $\phi$-meson) are discussed. It is shown, in agreement with Bell's theorem, that the most general local hidden-variable model fails in reproducing the whole set of quantum-mechanical joint probabilities. We achieve these conclusion by employing two different approaches. In a first one the local realistic observables are deduced from the most general premises concerning locality and realism, and Bell-like inequalities are not employed. The other approach makes use of Bell's inequalities. Within the former scheme, under particular conditions for the detection times, the discrepancy between quantum mechanics and local realism for the time-dependent asymmetry turns out to be not less than 20%. The same incompatibility can be made evident by means of a Bell-type test by employing both Wigner's and (once properly normalized probabilities are used) Clauser-Holt-Shimony-Holt's inequalities. Because of the relatively low experimental accuracy, the data obtained by the CPLEAR collaboration for the asymmetry parameter do not allow for a decisive test of local realism. Such a test, both with and without the use of Bell's inequalities, should be feasible in the future at the Frascati $\Phi$-factory.