$ n \geq 10 $. Moreover, these two methods can not take advantage of a prior knowledge (e.x. low-rank state).

Considering the expansion of the density matrix $ \rho $ in the $n$–Pauli basis, i.e. $ \mathcal{B}= \{ \sigma_b =
\sigma_{{b}_1} \otimes \ldots \otimes \sigma_{{b}_n}, b
\in \{ I,x,y,z \}^n \}, \sigma_I = I ,$ $$\begin{aligned}
\label{Pauli expansion}
\rho = \sum_{b\in \{ I,x,y,z \}^n }\rho_b \sigma_b.\end{aligned}$$ One can also estimate the density matrix via estimating the coefficients in the Pauli expansion. This was studied in [@cai2015Pauli] where the authors also make a sparsity assumption: that is, most of $\rho_b $ are small or very close to $0$. Note that, this is not related to the setting we explore (low-rank assumption).

We now turn to the definition of a prior distribution on density matrices that will allow to perform (pseudo-)Bayesian estimation.

Pseudo-Bayesian estimation and prior distribution on density matrices {#Def Prior}
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Peudo-Bayesian estimation
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We remind that the idea of Bayesian statistics is to encode the prior information on density matrices through a prior distribution $\pi({\rm d}\rho)$. Inference is then done through the posterior distribution $\pi({\rm d}\rho|\mathcal{D})
\propto \mathcal{L}(\rho) \pi({\rm d}\rho)$. Here, for computational reasons, we replace the likelihood by 