  Quantum compressed sensing is the fundamental tool for low-rank density
matrix tomographic reconstruction in the informationally incomplete case. We
examine situations where the acquired information is not enough to allow one to
obtain a precise compressed sensing reconstruction. In this scenario, we
propose a Deep Neural Network-based post-processing to improve the initial
reconstruction provided by compressed sensing. The idea is to treat the
estimated state as a noisy input for the network and perform a deep-supervised
denoising task. After the network is applied, a projection onto the space of
feasible density matrices is performed to obtain an improved final state
estimation. We demonstrate through numerical experiments the improvement
obtained by the denoising process and exploit the possibility of looping the
inference scheme to obtain further advantages. Finally, we test the resilience
of the approach to out-of-distribution data.
