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Pseudo-Bayesian quantum tomography with rank-adaptation. (English) Zbl 1395.62379

Summary: Quantum state tomography, an important task in quantum information processing, aims at reconstructing a state from prepared measurement data. Bayesian methods are recognized to be one of the good and reliable choices in estimating quantum states [R. Blume-Kohout, New J. Phys. 12, No. 4, Article ID 043034, 25 p. (2010; Zbl 1375.81065)]. Several numerical works showed that Bayesian estimations are comparable to, and even better than other methods in the problem of \(1\)-qubit state recovery. However, the problem of choosing prior distribution in the general case of \(n\) qubits is not straightforward. More importantly, the statistical performance of Bayesian type estimators has not been studied from a theoretical perspective yet. In this paper, we propose a novel prior for quantum states (density matrices), and we define pseudo-Bayesian estimators of the density matrix. Then, using PAC-Bayesian theorems [O. Catoni, PAC-Bayesian supervised classification: the thermodynamics of statistical learning. Beachwood, OH: IMS, Institute of Mathematical Statistics (2007; Zbl 1277.62015)], we derive rates of convergence for the posterior mean. The numerical performance of these estimators is tested on simulated and real datasets.

MSC:

62P35 Applications of statistics to physics
62F15 Bayesian inference
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
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