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].
Special relativity demands a locality principle (no instantaneous action at a distance); locality implies Bell’s theorem; quantum mechanics violates Bell’s inequality, therefore, quantum mechanics contradicts relativity! Or so it would seem. It is shown, however, that the locality principle needed for Bell’s theorem is stronger than the simple locality that is needed to satisfy the demands of relativity and that quantum mechanics satisfies the latter. The stronger locality principle is equivalent to the conjunction of simple locality and predictive completeness, and it is the latter principle that fails. The notion of predictive completeness is weaker than, and is implied by, the completeness criterion of Einstein, Podolsky, and Rosen. But the quantum mechanical state description is not only incomplete but incompletable, for any local complete state description would satisfy Bell’s inequality and disagree with experiment.
Bell’s inequality is derived by resorting to a hidden-variable theory devised for resolving the Einstein-Bohr debate on the conceptual foundations of quantum mechanics. The legitimacy of quantum superposition for describing the physical world is the essence of the debate. Einstein argued against the legitimacy of quantum superposition. Testing Bell’s inequality by experiment with the experimental result explained by Bell’s theorem opened the door to so-called quantum information technologies. In quantum information theory, “quantum bit” (or “qubit” for short) in a form of quantum superposition is supposed to carry quantum information. Although most physicists believe that Einstein’s vision of the physical world contradicts the experimental result of testing Bell’s inequality, actually neither the experimental result nor Bell’s theorem is relevant to Einstein’s viewpoint. In the present paper, a new principle, the general principle of measurements, is proved as a mathematical theorem. Based on this principle, the experiment for testing Bell’s inequality and so-called experimental evidence for physically realizable “qubit” are scrutinized. The findings are as follows. The experimental result of testing Bell’s inequality and the measurement outcomes of experiments involving “qubit” are all erroneously explained. Bell’s inequality failed to capture the essence of the Einstein-Bohr debate, and Bell’s theorem is irrelevant to Einstein’s vision of the physical world. Quantum mechanics can be completed by using disjunction (“or”) as the logical relation between the orthonormal vectors that span an arbitrarily given Hilbert space without resorting to any hidden-variable theory, while the mathematical setting will remain essentially unchanged. All kinds of “qubit” violate the general principle of measurements and can only describe imaginary objects that do not exist in the physical world. The findings inevitably lead to a very regrettable conclusion: Quantum information has no physical carers and all quantum information technologies are not physically realizable.