Pseudo-Bayesian quantum tomography with rank-adaptation

- Definition of a prior for Bayesian quantum tomography. This prior can be easily extended to any dimension.
- Best up-to-date estimation rates in the case of a low-rank density matrix.
- Good experimental results; clear improvement over the direct inversion method.

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.