  We introduce a reliable compressive procedure to uniquely characterize any
given low-rank quantum measurement using a minimal set of probe states that is
based solely on data collected from the unknown measurement itself. The
procedure is most compressive when the measurement constitutes pure detection
outcomes, requiring only an informationally complete number of probe states
that scales linearly with the system dimension. We argue and provide numerical
evidence showing that the minimal number of probe states needed is even
generally below the numbers known in the closely-related classical
phase-retrieval problem because of the quantum constraint. We also present
affirmative results with polarization experiments that illustrate significant
compressive behaviors for both two- and four-qubit detectors just by using
random product probe states.
