Completely Positive Maps and the V-transform

Explore a 45-minute lecture by Melchior Wirth from the Institute of Science and Technology Austria on "Completely positive maps and the V-transform" presented at IPAM's Dynamics of Density Operators Workshop. Recorded on April 29, 2025, this talk introduces the V-transform as a powerful tool for studying completely positive maps with two key connections to quantum optimal transport: first, establishing a representation of generators of KMS-symmetric quantum Markov semigroups in terms of derivations, which forms the foundation for dynamical quantum optimal transport distances pioneered by Carlen and Maas; and second, demonstrating how the V-transform proves subadditivity of cost in certain static quantum optimal transport distances introduced by De Palma and Trevisan. The presentation is based on joint work with Matthijs Vernooij and is part of IPAM UCLA's workshop series on density operators.