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Is is shown here that the "simple test of quantumness for a single system" ofarXiv:0704.1962 (for a recent experimental realization see arXiv:0804.1646) hasexactly the same relation to the discussion of to the problem of describing thequantum system via a classical probabilistic scheme (that is in terms of hiddenvariables, or within a realistic theory) as the von Neumann theorem (1932). Thelatter one was shown by Bell (1966) to stem from an assumption that the hiddenvariable values for a sum of two non-commuting observables (which is anobservable too) have to be, for each individual system, equal to sums ofeigenvalues of the two operators. One cannot find a physical justification forsuch an assumption to hold for non-commeasurable variables. On the positiveside. the criterion may be useful in rejecting models which are based onstochastic classical fields. Nevertheless the example used by the Authors has aclassical optical realization.

Unexpected reemergence of von Neumann theorem