Quantum wasserstein distance of order 1 between channels

Abstract

We analyse a formulation of the quantum Wasserstein distance of order $1$ and set up a general theory leading to a Wasserstein distance of order $1$ between the unital maps from one specific algebra to another specified algebra. This gives us a metric on the set of unital maps from one composite system to another, which is deeply connected to the reductions of the unital maps. We use the fact that channels are unital maps with extra structure, to systematically define a quantum Wasserstein distance of order $1$ between channels, i.e., a metric on the set of channels. Lastly, the additivity and stability properties of this metric are studied.