2.11 (2e-3) & 2.06 (2e-3) & 2.09 (1e-3)\

Real data tests
---------------

The experiments performed to produce the data is explained in [@barreiro2010experimental]. The data was kindly provided by M. Gu[ţ]{}[ă]{} and T. Monz. It had been used in [@alquier2013rank; @guctua2012rank]. We apply two proposed estimators to the real data set of a system of 4 ions which is Smolin state further manipulated. In Figure \[realdata\] we plot the eigenvalues of the inversion estimator and our ones.

![eigenvalues plots for real data test with $ n=4 $[]{data-label="realdata"}](real_data.png)

Note that the distribution of the eigenvalues of the three estimators are rather different. Still, it seems that all estimators return results compatible with a rank-2 state.

Discussion and conclusion {#conclusion}
=========================

We propose a novel prior and introduce two pseudo-Bayesian estimators for the density matrix: the dens-estimator and the prob-estimator. The prob-estimator reaches the best up-to-date rate of convergence in the low-rank case. On the other hand, computation of the dens-estimator is an easier task. In practice, we recommend the prob-estimator. However, in cases where the MCMC shows activities of lacking of convergence, the dens-estimator can be used as a reasonable alternative.

Note also that the prob-estimator can be extended to the 