In this talk I will present the V-transform as a new powerful tool in the study of completely positive maps. This relates to quantum optimal transport in two ways: First, the V-transform was introduced to establish a representation of generators of KMS-symmetric quantum Markov semigroups in terms of derivations. Such generation results in terms of derivations (for GNS-symmetric QMS) have been the basis for the construction of dynamical quantum optimal transport distances as pioneered by Carlen and Maas, and they have also been central for studying geodesic convexity of the relative entropy with respect to this distance. Second, I will discuss how the V-transform can be used to show that in certain cases, the cost in the definition of a static quantum optimal transport distance introduced by De Palma and Trevisan is subadditive. (This talk is based on joint work with Matthijs Vernooij.)
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