  This paper establishes new connections between many-body quantum systems,
One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal
Transport (OT), by interpreting the problem of computing the ground-state
energy of a finite dimensional composite quantum system at positive temperature
as a non-commutative entropy regularized Optimal Transport problem. We develop
a new approach to fully characterize the dual-primal solutions in such
non-commutative setting. The mathematical formalism is particularly relevant in
quantum chemistry: numerical realizations of the many-electron ground state
energy can be computed via a non-commutative version of Sinkhorn algorithm. Our
approach allows to prove convergence and robustness of this algorithm, which,
to our best knowledge, were unknown even in the two marginal case. Our methods
are based on careful a priori estimates in the dual problem, which we believe
to be of independent interest. Finally, the above results are extended in
1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on
the problem.
