estimation has been considered in this context. The papers [@buvzek1998reconst; @Baier_comparison] compare Bayesian methods to other methods on simulated data. More recently, [@kravtsov2013experimental; @ferrie2014quantum; @ferrie2015have; @schmied2014quantum] discuss efficient algorithms for computing Bayesian estimators. Importantly, [@blume2010optimal] showed that Bayesian method comes with natural error bars and is the most accurate scheme w.r.t. the expected error (operational divergence) (even) with finite samples. However, there is no theoretical guarantee on the convergence of these estimators.

More works on quantum state tomography in various settings include [@audenaert2009quantum; @carlen2010trace; @RauGibbsstate; @rau2014Gibbstate; @ferrie2014likelihood_free].

In this paper, we consider a pseudo-Bayesian estimation, where the likelihood is replaced by pseudo-likelihoods based on various moments (two estimators, corresponding to two different pseudo-likelihood, are actually proposed). Using PAC-Bayesian theory [@STW; @McA; @catoni2004statistical; @catonibook; @dalalyan2008aggregation; @suzuki2012pac], we derive oracle inequalities for the pseudo-posterior mean. We obtain rates of convergence for these estimators in the complete measurement setting. One of them has a rate as good as the best known rate up to date $\mathcal{O}( {\rm rank}(\rho^0) 3^n / N)$ (still, the other one is interesting for computationnal reasons that are discussed in the paper).

The rest of the paper is organized as follow. We recall the standard notations and basics about 