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We introduce a concept of a minimal sufficient positive-operator valuedmeasure (POVM), which is the least redundant POVM among the POVMs that have theequivalent information about the measured quantum system. Assuming the systemHilbert space to be separable, we show that for a given POVM a sufficientstatistic called a Lehmann-Scheff\'{e}-Bahadur statistic induces a minimalsufficient POVM. We also show that every POVM has an equivalent minimalsufficient POVM and that such a minimal sufficient POVM is unique up torelabeling neglecting null sets. We apply these results to discrete POVMs andinformation conservation conditions proposed by the author.

Minimal sufficient positive-operator valued measure on a separable Hilbert space