Unexpected reemergence of von Neumann theorem

TL;DR

This paper reveals that a recent test of quantum properties for a single system is fundamentally related to von Neumann's theorem, challenging assumptions in hidden variable theories and impacting classical models of quantum phenomena.

Contribution

It demonstrates the equivalence between a recent quantum test and von Neumann's theorem, highlighting the invalidity of certain hidden variable assumptions.

Findings

The test relates to von Neumann's theorem on hidden variables.

The assumption about sums of hidden variable values is unjustified.

The criterion can reject classical stochastic models.

Abstract

Is is shown here that the "simple test of quantumness for a single system" of arXiv:0704.1962 (for a recent experimental realization see arXiv:0804.1646) has exactly the same relation to the discussion of to the problem of describing the quantum system via a classical probabilistic scheme (that is in terms of hidden variables, or within a realistic theory) as the von Neumann theorem (1932). The latter one was shown by Bell (1966) to stem from an assumption that the hidden variable values for a sum of two non-commuting observables (which is an observable too) have to be, for each individual system, equal to sums of eigenvalues of the two operators. One cannot find a physical justification for such an assumption to hold for non-commeasurable variables. On the positive side. the criterion may be useful in rejecting models which are based on stochastic classical fields. Nevertheless the example used by the Authors has a classical optical realization.