the single qubit state estimation $ n=1$, the representation of the quantum constraints is explicit [@Baier_comparison; @schmied2014quantum]. Thus, one can place a prior distribution on the polar reparametrization of the density. Up to our knowledge, this has not been extended to the case $n>1$, and this extension seems not straightforward. For general n-qubit densities, uninformative priors (e.g the Haar measure) are put on $ \psi_{d\times K} $ matrices ($K\geq d$) and the density state is built by $ \rho = \psi_{d\times K}
\psi^{\dagger}_{d\times K} $ [@struchalin2016experimental; @granade2015practical; @huszar2012adaptive; @ferrie2015have; @zyczkowski2011generating]. One could also define a prior on the coefficients $\{\rho_b\}$ of $\rho$ on the Pauli basis. Nevertheless, none of these approaches seem helpful for rank adaptation.

The idea for our prior is inspired by the priors used for low-rank matrix estimation in machine learning, e.g. [@mai2015; @cottet20161] and the references therein. Hereafter, we describe in details the prior construction.

Let $V$ be a vector in $ \mathbb{C}^{d\times 1}\setminus \{\mathbf{0}\}$ ($ d=2^n $ in our model), then $VV^{\dagger}$ is a Hermitian, semi-definite positive matrix in $ \mathbb{C}^{d\times d}$ with ${\rm rank}(VV^{\dagger}) = 1$. Additionally, we can normalize $V$ (that is replace $V$ by $V/\|V\|$), this lead to ${\rm Trace}
(VV^{\dagger}) = 