Quantum Wasserstein distance for Gaussian states

TL;DR

This paper derives a general formula for the quantum Wasserstein distance of order 2 between any two one-mode Gaussian states, facilitating comparison with classical distances and advancing quantum state analysis.

Contribution

It provides the first closed-form expression for the quantum Wasserstein distance between one-mode Gaussian states, extending previous formalism and enabling direct comparison with classical distances.

Findings

Derived a general formula for quantum Wasserstein distance between Gaussian states

Showed how classical Gaussian and thermal quantum states distances are recovered

Facilitated comparison of quantum Wasserstein with other distance measures

Abstract

Optimal transport between classical probability distributions has been proven useful in areas such as machine learning and random combinatorial optimization. Quantum optimal transport, and the quantum Wasserstein distance as the minimal cost associated with transforming one quantum state to another, is expected to have implications in quantum state discrimination and quantum metrology. In this work, following the formalism introduced in [De Palma, G. and Trevisan, D. Ann. Henri Poincar\'e, {\bf 22} (2021), 3199-3234] to compute the optimal transport plan between two quantum states, we give a general formula for the Wasserstein distance of order 2 between any two one-mode Gaussian states. We discuss how the Wasserstein distance between classical Gaussian distributions and the quantum Wasserstein distance by De Palma and Trevisan for thermal states can be recovered from our general formula for Gaussian states. This opens the path to directly compare various known distance measures with the Wasserstein distance through their closed-form solutions.