Simultaneous Measurement, Phase‐Space Distributions, and Quantum State Determination

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Abstract
Noting that a classical phase‐space probability distribution w(q, p) may be calculated from moment expectation values {qmpn}, we inquire as to whether similar data in quantum mechanics would be adequate to determine the statistical operator ϱ. For the family of simultaneous (q, p) measurement schemes investigated, it turns out that such moments do not suffice to fix ϱ. Comparison of the empirical information that is adequate to determine ϱ with that required to find w(q, p) reveals that in a sense more data are needed for state determination in quantum statistics than are needed in the classical case.