We present a non-commutative transportation framework for quantum channels acting on von Neumann algebras. Our central object is the Lipschitz cost measure, a transportation-based quantity that evaluates the minimal cost required to move between quantum states via a given channel. Accompanying this is the Lipschitz contraction coefficient, which captures how much the channel contracts the Wasserstein-type distance between states. We establish foundational properties of these quantities, including continuity, dual formulations, and behavior under composition and tensorization. We discuss various applications of our framework. First, the framework yields lower bounds on quantum-circuit volume and depth, revealing intrinsic limitations on how efficiently certain channels can be implemented. Second, whenever the contraction coefficient is strictly below one, it provides quantitative bounds on entropy decay and mixing times for quantum systems. By unifying metric, probabilistic, and operator-algebraic perspectives, our approach offers a systematic tool for studying convergence phenomena and computational complexity in quantum information theory.

This is joint work of Roy Araiza and Marius Junge.