Pseudo-Bayesian quantum tomography with rank-adaptation

ALQUIER Pierre, MAI The Tien
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 (Blume-Kohout, 2010). 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 (Catoni, 2007), we derive rates of convergence for the posterior mean. The numerical performance of these estimators is tested on simulated and real datasets.
MAI, T.T. et ALQUIER, P. (2017). Pseudo-Bayesian quantum tomography with rank-adaptation. Journal of Statistical Planning and Inference, 184, pp. 62-76.