Compressively certifying quantum measurements

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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.
© 2021 Published by the American Physical Society. We thank Emanuele Roccia for useful discussion. This work is supported in part by the National Research Foundation of Korea (NRF) (Grant Nos. NRF-2019R1A6A1A10073437, No. NRF-2019M3E4A1080074, and No. NRF-2020R1A2C1008609), the Spanish MINECO (Grant Nos. FIS201567963-P and No. PGC2018-099183-B-I00), European Union’s Horizon 2020 research and innovation program (Project Quan-tERA ApresSF), and a Mega-grant of the Ministry of Education and Science of the Russian Federation (Contract No.14.W03.31.0032). I.G. is supported by Ministero dell’Istruzione, dell’Università e della Ricerca Grant of Excellence Departments (ARTICOLO 1, COMMI 314-337 LEGGE 232/2016).