  Motivated by recent quantum gas microscope experiments for fermions in
optical lattices, we present proof of principle calculations showing that it is
possible to obtain the complete information about the quantum state on a small
subsystem from equilibrium determinantal quantum Monte Carlo simulations. Both
diagonal (in the occupation number basis) and off-diagonal elements of the
reduced density matrix are calculated for a square plaquette, which is embedded
in a much larger system of the two-dimensional Hubbard model, both at half
filling and in the doped case. The diagonalization of the reduced density
matrix is done by exploiting the point group symmetry and particle number
conservation, which allows to attach symmetry labels to its eigenvalues.
Knowledge of the probabilities of plaquette occupation number configurations is
useful for meticulous benchmarking of quantum gas microscope experiments. As
the quantum state on the plaquette is exact and self-consistently embedded in
an exact, correlated bath, the present approach connects to various cluster
approximation techniques.
