Rate-Limited Optimal Transport for Quantum Gaussian Observables

The rate-limited optimal transport problem is introduced for the continuous-variable quantum measurement systems in the form of output-constrained rate-distortion coding. The main coding theorem provides a single-letter characterization of the achievable rate region for lossy quantum-to-classical source coding that transforms a sufficiently large tensor product of IID continuous-variable quantum states from a quantum source to a sequence of IID samples from a classical continuous destination distribution with a prescribed distortion level. The evaluation of rate region is performed for the systems with quantum Gaussian source and Gaussian destination distribution. We establish a Gaussian observable optimality theorem for such systems and provide an analytical formulation of the rate-limited quantum-classical Wasserstein distance in the case of isotropic and one-mode Gaussian quantum systems.11The proofs of the theorems and the details of the results are provided in the extended online version [1] available at https://arxiv.org/abs/2305.10004 for further reference. This work was supported in part by NSF grants CCF 2007878 and CCF 2132815.