  The determination of a quantum observable from the first and second moments
of its measurement outcome statistics is investigated. Operational conditions
for the moments of a probability measure are given which suffice to determine
the probability measure. Differential operators are shown to lead to physically
relevant cases where the expectation values of large classes of noncommuting
observables do not distinguish superpositions of states and, in particular,
where the full moment information does not determine the probability measure.
