On the statistical meaning of complex numbers in quantum mechanics

Bell’s inequality is a necessary condition for the existence of a classical probabilistic model for a given set of correlation functions. This condition is not satisfied by the quantum-mechanical correlations of two-spin systems in a singlet state. We give necessary and sufficient conditions, on the transition probabilities, for the existence of a classical probabilities model. We also give necessary and sufficient conditions for the existence of a complex (respectively real) Hilbert space model. Our results apply to individual-spin systems hence they need no «locality» assumption. When applied to the quantum-mechanical transition probabilities, they prove not only the necessity of a nonclassical probabilities model, but also the necessity of using complex rather than real Hilbert spaces.