reconstructing the (unknown) state of a physical quantum system [@MR2181445bookParis], usually represented by the so-called density matrix $\rho$ (the exact definition of a density matrix is given in Section \[set-up\]). This task is done by using outcomes of measurements performed on many independent systems identically prepared in the same state.

The ’tomographic’ method, also named as linear/direct inversion [@vogel1989determination; @vrehavcek2010operational], is the simplest and oldest estimation procedure. It is actually the analogous of the least-square estimator in the quantum setting. Although easy in computation and providing unbiased estimate [@PhysRevLett.114.080403], it does not generate a physical density matrix as an output [@shang2014quantum]. Maximum likelihood estimation [@hradil20043] is the current procedure of choice. Unfortunately, it has some critical flaws detailed in [@blume2010optimal], including a huge computational complexity. Furthermore, both these methods are not adaptive to the case where a system is in a state $\rho$ for which some additional information is available. Note especially that, physicists focus on so-called pure states, for which ${\rm rank}(\rho)=1$.

The problem of rank-adaptivity was tackled thanks to adequate penalization. Rank-penalized maximum likelihood (BIC) was introduced in [@guctua2012rank] while a rank-penalized least-square estimator $\hat{\rho}_{{\rm rank-pen}}$ was proposed in [@alquier2013rank], together with a proof of its consistency. More 