Fluctuations of quantum coherences in the Anderson model are described by the quantum symmetric simple exclusion process

Using the transfer matrix method, we numerically investigate the structure of spatial coherences and their fluctuations in the three-dimensional (3D) Anderson model in the metallic phase when driven out of equilibrium by external leads at zero temperature and in linear response. We find that the stationary state entails nonlocal non-Gaussian correlations in the longitudinal direction, which are characteristic of diffusive nonequilibrium steady states. These correlations are quantitatively matched, at least up to third order, by those analytically derived in the quantum symmetric simple exclusion process (QSSEP) which describes diffusive fermions in 1D subject to dynamical disorder. Furthermore, the large deviation scaling and $U(1)$ invariance of these correlations imply a link between the Anderson model and free probability theory. Our findings suggest the existence of a universal structure of correlations in noninteracting diffusive quantum systems that might be captured by QSSEP.