The Schrödinger problem consists in finding the most likely evolution of a (random) Brownian particles which evolves from a given initial law to a given target one. It is in fact possible to equivalently describe this problem in terms of entropy regularisation of an optimal transport problem, by perturbing the classical quadratic Wasserstein transport problem with an entropy contribution, weighted by a factor that plays the role of the inverse of the temperature. In this talk, we study a multimarginal, non-commutative analogue of this problem, meaning an entropic regularised optimal transport problem between density matrices on finite dimensional Hilbert spaces. Physically, this describes a composite quantum system at positive temperature, under the knowledge of all its subsystems. Our main contributions are the proof of a duality formula for the multimarginal entropic problem
and the introduction and proof of convergence to the optimal states of a non-commutative analogue of the Sinkhorn algorithm. The settings of Bosonic and Fermionic systems are also discussed, with a new variational interpretation of the Pauli's principle. This is a joint work with Dario Feliciangeli and Augusto Gerolin, arXiv:2106.11217.