specifically, when the density matrix of the system is $\rho^0$ with $ r = {\rm rank}(\rho^0) $, the authors of [@alquier2013rank] proved that the Frobenius norm of the estimation error satisfies $\|\hat{\rho}_{{\rm rank-pen}} - \rho^0 \|_F^2
=\mathcal{O}(r 4^n /N)$ where $N$ is the number of quantum measurements. The rate was improved to $\mathcal{O}(r3^n/N)$ by [@butucea2015spectral], using a thresholding method. Note that the rate $\mathcal{O}(r2^n/N)$ was first claimed in the paper, but in the Corrigendum [@butucea2016corrigendum], the authors acknowledge that this is not the case. The paper however contains a proof that no method can reach a rate smaller than $r2^n / N$. So, the minimax-optimal rate is somewhere in between $r 2^n / N$ and $r3^n / N$.

Note that all the aforementioned papers only cover the complete measurement case (the definition is given in Section \[set-up\], basically it means that we have observations for all the observables given by the Pauli basis). The statistical relationship between matrix completion and quantum tomography with incomplete measurements (in the Le Cam paradigm) has been investigated in [@wang2013]. Thus compressed sensing ideas have been successfully proposed in estimating a density state from incomplete measurements [@gross2010quantum; @gross2011recovering; @flammia2012quantum; @koltchinskii2011].

On the other hand, Bayesian 