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Quantum Probability Estimation for Randomness with Quantum Side Information



Emanuel H. Knill, yanbao zhang, Honghao Fu


We develop a quantum version of the probability estimation framework [arXiv:1709.06159] for randomness generation with quantum side information. We show that most of the properties of probability estimation hold for quantum probability estimation (QPE). This includes asymptotic optimality at constant error and randomness expansion with logarithmic input entropy. QPE is implemented by constructing model-dependent quantum estimation factors (\QEFs) which yield statistical confidence upper bounds on data-conditional normalized R\'enyi powers. These upper bounds then lead to conditional min-entropy estimates ready for applying quantum-proof randomness extractors. The bounds are valid for relevant models of sequences of experimental trials without requiring independent and identical or stationary behavior. QEFs may be adapted to changing conditions during the sequence and trials can be stopped any time, such as when the results so far are satisfactory. QEFs can be constructed from entropy estimators to improve the bounds for conditional min-entropy of classical-quantum states from the entropy accumulation framework [Dupuis, Fawzi and Renner, arXiv:1607.01796]. QEFs are applicable to a larger class of models, including models permitting measurement devices with super-quantum but non-signaling behaviors and semi-device dependent models. The improved bounds are relevant for finite data or error bounds of the form $e^{-\kappa s}$ where $s$ is the number of random bits produced. We give a general construction of entropy estimators based on maximum probability estimators, which exist for many experimental configurations. For the important class of $(k,2,2)$ Bell-test configurations we provide schemas for directly optimizing \QEFs to overcome the limitations of entropy-estimator-based constructions...
Cornell Library University


Bell tests, quantum randomness.
Created June 12, 2018, Updated May 19, 2020