  In non-interacting isolated quantum systems out of equilibrium, local
subsystems typically relax to non-thermal stationary states. In the standard
framework, information on the rest of the system is discarded, and such states
are described by a Generalized Gibbs Ensemble (GGE), maximizing the entropy
while respecting the constraints imposed by the local conservation laws. Here
we show that the latter also completely characterize a recently introduced
projected ensemble (PE), constructed by performing projective measurements on
the rest of the system and recording the outcomes. By focusing on the time
evolution of fermionic Gaussian states in a tight-binding chain, we put forward
a random ensemble constructed out of the local conservation laws, which we call
deep GGE (dGGE). For infinite-temperature initial states, we show that the dGGE
coincides with a universal Haar random ensemble on the manifold of Gaussian
states. For both infinite and finite temperatures, we use a Monte Carlo
approach to test numerically the predictions of the dGGE against the PE. We
study in particular the $k$-moments of the state covariance matrix and the
entanglement entropy, finding excellent agreement. Our work provides a first
step towards a systematic characterization of projected ensembles beyond the
case of chaotic systems and infinite temperatures.
