quantum theory in Section \[set-up\]. Then the definition of the prior and of the estimators are presented in Section \[Def Prior\]. The statistical analysis of the estimators are given in Section \[MAIN\], while all the proofs are delayed to the Appendix \[Appendix\]. Some numerical experiments on simulated and real datasets are given in Section \[num\].

Preliminaries {#set-up}
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Notations
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A very good introduction to the notations and problems of quantum statistics is given in [@artiles]. Here, we only provide the basic definitions required for the paper.

In quantum physics, all the information on the physical state of a system can be encoded in its [*density matrix*]{} $\rho$. Depending on the system in hand, this matrix can have a finite or infinite number of entries. A two-level system of $n$-qubits is represented by a $2^{n}\times 2^{n}$ density matrix $\rho$, with coefficients in $\mathbb{C}$. For the sake of simplicity, the notation $d=2^n$ is used in [@butucea2015spectral], so note that $\rho$ is a $d\times d$ matrix. This matrix is Hermitian $\rho^\dagger=\rho$ (i.e. self-adjoint), semidefinite positive $\rho\geq 0$ and has ${\rm Trace}(\rho)=1$. Additionally, it often makes sense to assume that the rank of $\rho$ is small [@gross2010quantum; @gross2011recovering]. In theory, the rank can 