Quantum Wasserstein distance for Gaussian states

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 Poincaré, {\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.

💡 Research Summary

This paper extends the quantum optimal transport framework introduced by De Palma and Trevisan to the full class of one‑mode Gaussian states and derives a closed‑form expression for the order‑2 quantum Wasserstein distance between any two such states. The authors begin by reviewing the classical optimal transport problem, its Kantorovich formulation, and several proposals for a quantum analogue. They focus on the De Palma‑Trevisan approach, which identifies a quantum transport plan with a completely positive, trace‑preserving (CPTP) map (a quantum channel). The cost functional is chosen to be quadratic in the canonical quadrature operators, leading to the definition of the squared Wasserstein‑2 distance as the minimal average cost over all admissible couplings.

Gaussian states are introduced via their displacement vector (d) and covariance matrix (\gamma). For a single mode, the symplectic eigenvalue (\nu = \sqrt{\det\gamma}) fully characterises the state’s second moments. The authors exploit the fact that, because the cost is quadratic, the optimal coupling can be restricted to Gaussian couplings. Consequently, the joint covariance matrix of the coupling takes a block form \