\rho_{rank-2} = \frac{1}{2}\psi_1 \psi_1^{\dagger}
    +  \frac{1}{2}\psi_2 \psi_2^{\dagger}$ with $ \psi_1,\psi_2  $ being two normalized orthogonal vectors in $  \mathbb{C}^{d\times 1} $,

-   an “approximate rank-$2$" density matrix: $ \rho = w \rho_{rank-2} + (1-w)\frac{\mathbb{I}_d}{d}, w=0.98 $. Note that by “approximate rank-$2$", we mean that $\rho $ is very well approximated by a rank-$2$ matrix $\rho_{rank-2} $ (in the sense that $\|\rho- \rho_{rank-2} \|_F^2 $ is small), but in general $\rho $ itself is full rank,

-   a maximal mixed state (rank-$d$).

The experiments are done for $ m=20;200;1000;2000 $. The parameter for $\mathcal{D}ir(\alpha,\dots,\alpha) $ is $ \alpha=0.5 $. We repeat each experiment $10$ times, and compute the mean of the square error, MSE, $\|\hat{\rho}-\rho\|_F^2$ for each estimator, together with the associated standard deviation (between brackets in Tables \[tablen\_4\],\[tablen\_3\],\[tablen\_2\]).

Results
-------

We compare the prob- and dens-estimator to the simple inversion procedure and to the thresholding estimator of [@butucea2015spectral]. The results are given in Tables \[tablen\_4\],\[tablen\_3\],\[tablen\_2\] (outputs from the **R** software). The conclusions are:

-   The prob-estimator seems to be the most accurate but also comes with a larger standard deviation. This might be due to slow convergence of the MCMC procedure. Indeed each step is computationally highly expensive.

-   The 