  We introduce 1-RDMFT in the canonical ensemble and then proceed to
approximate the interacting ensemble by a non-interacting ensemble that
maximizes the entropy, independently of temperature. Bosonic and Fermionic
Sinkhorn algorithms are derived and used to invert the relationship between the
Natural Orbital Occupation Numbers (NOONs) and the effective orbital energies
of the non-interacting ensemble. Both the Bosonic and Fermionic Sinkhorn
algorithms are shown to perform well in reproducing the NOONs of simulated
distributions and the ground-state NOONs of H$_2$O and H$_2$. In the case of
H$_2$ we also study the resulting non-interacting entropy and non-interacting
approximation to the interaction energy within several wavefunction subspaces
as the bond length varies. This provides several new starting points for
approximations of the interaction energy, also at zero-temperature. Connections
to entropically-regularized Multi-Marginal Optimal Transport (MMOT) are
highlighted that may prove interesting for future research.
