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.

References

In order to use the above ideas of optimal transport on needs a non-commutative analog of the Wasserstein metric $W_2$. This is an interesting open problem.

— Moduli space of Conformal Field Theories and non-commutative Riemannian geometry  (2506.00896 - Soibelman, 1 Jun 2025) in Section 5 (Ricci curvature, diameter and dimension), overview of approaches to precompactness

Accordingly, an intrinsic characterization of noncommutative quantum optimal transport would require an additional structural condition beyond the four axioms. A natural candidate is a noncommutative convex-duality requirement: the distance should arise from a cotangent Dirichlet form $Q_\rho(A)$, built from commutators with distinguished quantum gradients, whose Legendre transform defines a tangent action and whose induced length distance is the transport metric. We do not attempt to formulate such an axiom here, since it is not presently clear how to do so without building a particular metric into the definition.

— Quantum Chaos and Quantum Optimal Transport  (2608.27350 - Cotler et al., 27 Aug 2026) in Appendix, subsection “Some examples of quantum optimal transport distances and their relation to the axioms,” immediately after the classical analogue proposition