proper subset of $\mathcal{E}^n$ ($A \varsubsetneq \mathcal{E}^n$) is referred to as the incomplete measurement case. In this paper, we focus on the complete measurement case, but the extension to the incomplete case is discussed in Section \[conclusion\].

Popular estimation methods
--------------------------

A natural idea is to define the empirical frequencies $$\hat{p}_{\mathbf{a},\mathbf{s}}
= \frac{1}{m}\sum_{i=1}^m \mathbf{1}_{\{R_i^\mathbf{a}=\mathbf{s}\}}.$$ Note that $ \hat{p}_{\mathbf{a},\mathbf{s}}$ is an unbiased estimator of the probability $ p_{\mathbf{a},\mathbf{s}} $. The inversion method is based on solving the linear system of equations $$\label{rhohat}
\left\{ \begin{array}{l}
\hat p_{\mathbf{a},\mathbf{s}} = {\rm Trace}
\left(\hat {\rho} \cdot
 P_{\mathbf{s}}^{\mathbf{a}} \right),
 \\
 \mathbf{a}\in\mathbb{E}^n,
 \\
 \mathbf{s} \in \mathcal{R}^n.
 \end{array}
 \right.$$ As mentioned above, the computation of $\hat{\rho}$ is quite straighforward. Explicit formulas are classical, see e.g. [@alquier2013rank].

Another commonly used method is maximum likelihood (ML) estimation, where the likelihood is $$\mathcal{L}(\rho;\mathcal{D})  \propto \prod_{\mathbf{a}\in\mathbb{E}^n}
 \prod_{\mathbf{s} \in \mathcal{R}^n}  [{\rm Trace} \left(\rho \cdot
 P_{\mathbf{s}}^{\mathbf{a}} \right)]^{ n_{\mathbf{a},\mathbf{s}} },$$ where $n_{\mathbf{a},\mathbf{s}} = m \hat{p}_{\mathbf{a},\mathbf{s}}$ is the number of times we observed output $\mathbf{s}$ in experiment $\mathbf{a}$ (obviously, $\sum_{\mathbf{s}} n_{\mathbf{a},\mathbf{s}} = m$). As mentioned in the introduction, both methods suffer many drawbacks. The inversion method returns a matrix $\hat{\rho}$ that usually does not satisfy the axioms of a density matrix. ML becomes expensive (inpractical) for 