Two questions on quantum statistics
  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. 
Pekka Lahti, Juha-Pekka Pellonpaa, Kari Ylinen