  This survey has been written in occasion of the School and Workshop about
Optimal Transport on Quantum Structures at Erd\"os Center in September 2022. We
discuss some recent results on noncommutative entropic optimal transport
problems and their relation to the study of the ground-state energy of a
finite-dimensional composite quantum system at positive temperature, following
the work [FGP23]. In the first part, we review some of the classical
primal-dual formulations of optimal transport in the commutative setting,
including extensions to multimarginal problems and entropic regularisation. We
discuss the main features of the entropic problem and show how optimisers can
be efficiently computed via the so-called Sinkhorn algorithm. In the second
part, we discuss how to apply these ideas to a noncommutative setting, in
particular on the space of density matrices over finite dimensional Hilbert
spaces. In this framework, we present equivalences between primal and dual
formulations, and use them to characterise the optimisers. Despite the lack of
explicit formulas due to the noncommutative nature of the problem, one can also
show that a suitable quantum version of the Sinkhorn algorithm converges to the
minimiser of the entropic problem. In the final part of this work, we discuss
similar results for bosonic and fermionic systems.
