We give an overview of a theory of Wasserstein distance between quantum dynamical systems with stationary faithful states, through transport plans. Even for the case of Wasserstein distance between states themselves, the dynamics given by the modular group associated to each state plays an integral role to ensure symmetry. The role of the commutant of the observable algebra of a system, as well as a related bimodule structure in the Hilbert space representation theory, leading in particular to the triangle inequality, will be emphasized. In the general setup the structure and a number of tools for this approach are provided by Tomita-Takesaki modular theory of von Neumann algebras.
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