Non-commutative analog of the Wasserstein W2 metric

Develop a non-commutative version of the L2-Wasserstein distance suitable for quantum metric-measure spaces, enabling formulation of N-Ricci curvature lower bounds and optimal transport methods in noncommutative settings.

Background

In the probabilistic approach to lower Ricci curvature on metric-measure spaces, convexity of entropy along W2-geodesics plays a central role and yields precompactness results.

To transfer these techniques to quantum settings, a meaningful noncommutative W2 metric compatible with states and semigroup dynamics is needed; existing free probability analogs do not directly apply to the tensor-product-based quantum geometry considered here.