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We introduce a reliable compressive procedure to uniquely characterize anygiven low-rank quantum measurement using a minimal set of probe states that isbased solely on data collected from the unknown measurement itself. Theprocedure is most compressive when the measurement constitutes pure detectionoutcomes, requiring only an informationally complete number of probe statesthat scales linearly with the system dimension. We argue and provide numericalevidence showing that the minimal number of probe states needed is evengenerally below the numbers known in the closely-related classicalphase-retrieval problem because of the quantum constraint. We also presentaffirmative results with polarization experiments that illustrate significantcompressive behaviors for both two- and four-qubit detectors just by usingrandom product probe states.

Compressively certifying quantum measurements