A Bigger Contradiction Between Quantum Theory and Locality for Two Particles Without Inequalities

D. A. Slavnov

Franck Laloë

The Bell theorem can take the form of several inequalities, as we have seen in §4.2. Moreover, since it was discovered, the theorem has stimulated the discovery of several other mathematical contradictions between the predictions of quantum mechanics and those of local realism. We review a few of them in this chapter: GHZ contradictions (§5.1) and their generalization (§5.2), Cabello's inequality (§5.3), and Hardy's impossibilities (§5.4). Finally, in §5.5, we discuss the notion of contextuality and introduce the BKS theorem. GHZ contradiction For many years, everyone thought that Bell had basically exhausted the subject by considering all really interesting situations, and that two-spin systems provided the most spectacular quantum violations of local realism. It therefore came as a surprise to many when in 1989 Greenberger, Horne, and Zeilinger (GHZ) showed that systems containing more than two correlated particles may actually exhibit even more dramatic violations of local realism [188, 189]. They involve a sign contradiction (100% violation) for perfect correlations, while the BCHSH inequalities are violated by about 40% (Cirelson bound) and deal with situations where the results of measurements are not completely correlated. In this section, we discuss three-particle systems, but generalizations to N particles are possible (§5.2). Derivation GHZ contradictions may occur in various systems, not necessarily involving spins. Initially, they were introduced in the context of entanglement swapping (§6.3.2) for four particles [188] or entanglement of three spinless particles [189].

David Böhm, Basil J. Hiley

Lucién Hardy

First, we demonstrate Bell's theorem, without using inequalities, for an experiment with two particles. Then we show that, if we assume realism and we assume that the ``elements of reality'' corresponding to Lorentz-invariant observables are themselves Lorentz invariant, we can derive a contradiction with quantum mechanics.

Mohammad LAMEHI-RACHTI, W. Mittig

Bell showed (''Bell's inequality) that local hidden variable theories lead to predictions in contradiction with quantum mechanics. This was tested in low energy proton-proton scattering by the simultaneous measurement of the polarization of the two protons. The results are in agreement with quantum mechanics and thus in contradiction with the inequality of Bell. (FR)

Travis Norsen