Optimal Transport by Quantum Channels: Non-quadratic Problems, Metric Properties, and Isometries

Explore a 41-minute lecture by Dániel Virosztek from the Alfréd Rényi Institute titled "Optimal transport by quantum channels: non-quadratic problems, metric properties, and isometries," recorded during IPAM's Dynamics of Density Operators Workshop on April 28, 2025. The presentation begins with a brief review of quantum optimal transport approaches, contrasting their recent emergence with classical optimal transport theory's established role in mathematical physics and probability since the 1980s. Delve into a non-quadratic generalization of quantum mechanical optimal transport problems where quantum channels facilitate transport, as introduced by De Palma and Trevisan. Discover how this framework enables the introduction of p-Wasserstein distances and divergences, and examine their fundamental geometric properties. The lecture concludes by demonstrating that quadratic quantum Wasserstein divergences function as genuine metrics, with a summary of recent findings on isometries of the qubit state space with respect to Wasserstein distances induced by distinguished transport cost operators.