  In a recent work, Bindini and De Pascale have introduced a regularization of
$N$-particle symmetric probabilities which preserves their one-particle
marginals. In this short note, we extend their construction to mixed quantum
fermionic states. This enables us to prove the convergence of the Levy-Lieb
functional in Density Functional Theory , to the corresponding multi-marginal
optimal transport in the semi-classical limit. Our result holds for mixed
states of any particle number $N$, with or without spin.
