A Coding Theorem for Rate-Limited Quantum-Classical Optimal Transport

We establish a coding theorem for rate-limited quantum-classical optimal transport systems with limited classical common randomness. This theorem characterizes the rate region of measurement protocols on a product source state for faithful construction of a given destination state while maintaining the source-destination distortion below a prescribed threshold with respect to a general distortion observable. It also provides a solution to the problem of rate-limited optimal transport, which aims to find the optimal cost of transforming a source quantum state to a destination state via an entanglement-breaking channel with a limited communication rate. The coding theorem is further extended to cover Bosonic continuous-variable quantum systems. The analytical evaluation is performed for the case of a qubit measurement system with unlimited common randomness.