Rate-Limited Quantum-To-Classical Optimal Transport

Abstract

The goal of optimal transport is to map a source probability measure to a destination one with the minimum possible cost. However, the optimal mapping might not be feasible under some practical constraints. One such example is to realize a transport mapping through an information bottleneck. As the optimal mapping may induce infinite mutual information between the source and the destination, the existence of an information bottleneck forces one to resort to some suboptimal mappings. Investigating this type of constrained optimal transport problems is clearly of both theoretical significance and practical interest.

In this work, we substantiate a particular form of constrained optimal transport in the context of quantum-to-classical systems by establishing an Output-Constrained Rate-Distortion Theorem similar to the classical case introduced by Yuksel et al. This theorem develops a noiseless communication channel and finds the least required transmission rate R and common randomness Rc to transport a sufficiently large block of n i.i.d. source quantum states, to samples forming a perfectly i.i.d. classical destination distribution, while maintaining the distortion between them. The coding theorem provides operational meanings to the problem of Rate-Limited Optimal Transport, which finds the optimal transportation from source to destination subject to the rate constraints on transmission and common randomness.

We further provide an analytical evaluation of the quantum-to-classical rate-limited optimal transportation cost for the case of qubit source state and Bernoulli output distributions with unlimited common randomness. The evaluation results in a transcendental system of equations whose solution provides the rate-distortion curve of the transportation protocol.

We further extend this theorem to continuous-variable quantum systems by employing a clipping and quantization argument and using our discrete coding theorem. Moreover, we derive an analytical solution for rate-limited Wasserstein distance of 2nd order for Gaussian quantum systems with Gaussian output distribution. We also provide a Gaussian optimality theorem for the case of unlimited common randomness, showing that Gaussian measurement optimizes the rate in a system with Gaussian source and destination.