Comment on 'Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices'

Friedland et al. [PRL 129, 110402 (2022)] proposed and studied a quantum analogue of the $p$-Wasserstein distance based on quantum cost matrices and quantum couplings.

They conjectured that, despite being only a semidistance in general, this quantity is a true distance for a particular quantum cost matrix and for cost matrices in a small neighborhood of it.

We disprove these conjectures by exhibiting an explicit family of triples of states for which the triangle inequality fails.