Bell inequalities for K⁰K⁰ pairs from Φ-resonance decays

We analyze the premises of recent propositions to test local realism via the Bell inequalities using neutral kaons from Φ resonance decays as entangled Einstein-Podolsky-Rosen pairs. We pay special attention to the derivation of the Bell inequalities, or related expressions, for unstable and oscillating kaon “quasispin” states and to the possibility of the actual identification of these states through their associated decay modes. We discuss an indirect method to extract probabilities to find these states by combining experimental information with theoretical input. However, we still find inconsistencies in previous derivations of the Bell inequalities. We show that the identification of the quasispin states via their associated decay mode does not allow the free choice to perform different tests on them, a property which is crucial to establish the validity of any Bell inequality in the context of local realism. In view of this we propose a different kind of Bell inequality in which the free choice or adjustability of the experimental setup is guaranteed. We also show that the proposed inequalities are violated by quantum mechanics.