The quantum 2-Wasserstein (semi-)distances

The optimal transport problem, established by Monge and refined by Kantorovich and Wasserstein, has ubiquitous applications in statistics, machine learning, computer vision and early Universe reconstruction. Recently, several approaches towards its quantum version have been proposed. In my talk I will present a direct generalisation of the 2-Wasserstein distance to the quantum realm, based on a specific class of quantum cost matrices. In the simplest case of a projector matrix, the induced quantity is a unitarily invariant semi-distance. Furthermore, it does enjoy the triangle inequality on the space of qubits and numerical evidence strongly suggest that this feature holds in any dimension. More general quantum cost matrices yield a weak distance structure and admit a formulation of the dual optimisation problem. Such quantum analogue of the 2-Wasserstein distance offers a new measure of proximity between quantum states and may prove useful in quantum Generative Adversarial Networks machine learning schemes. The talk is based on joint works with Rafal Bistron, Sam Cole, Shmuel Friedland and Karol Zyczkowski, arXiv: 2102.07787, 2105.06922, 2204.07405.