Fermionic optimal transport

Quadratic Wasserstein distances are obtained between dynamical systems (with states as special case), on ${\mathbb{Z}}_2$-graded von Neumann algebras. This is achieved through a systematic translation from non-graded to ${\mathbb{Z}}_2$-graded transport plans, on usual and fermionic (or ${\mathbb{Z}}_2$-graded) tensor products respectively. The metric properties of these fermionic Wasserstein distances are shown, and their symmetries relevant to deviation of a system from quantum detailed balance are investigated. The latter is done in conjunction with the development of a complete mathematical framework for detailed balance in systems involving indistinguishable fermions.