On the monotonicity of a quantum optimal transport cost

TL;DR

This paper investigates the properties of a quantum optimal transport cost, disproves a recent conjecture about its monotonicity, and proposes a modified version that maintains monotonicity under quantum channels.

Contribution

It demonstrates that the existing quantum Wasserstein distance is not monotone under partial traces and introduces a stabilized, monotone version of the cost.

Findings

Original quantum Wasserstein distance is not monotone under partial traces.

Proposed a stabilized version that is monotone under quantum channels.

Disproved a recent conjecture on quantum optimal transport cost.

Abstract

We show that the quantum generalization of the $2$-Wasserstein distance proposed by Chakrabarti et al. is not monotone under partial traces. This disproves a recent conjecture by Friedland et al. Finally, we propose a stabilized version of the original definition, which we show to be monotone under the application of general quantum channels.