Rate-limited Quantum-to-Classical Optimal Transport: A Lossy Source Coding Perspective

We consider the rate-limited quantum-to-classical optimal transport in terms of output-constrained rate-distortion coding for discrete quantum measurement systems with limited classical common randomness. The main coding theorem provides the achievable rate region of a lossy measurement source coding for an exact construction of the destination distribution (or the equivalent quantum state) while maintaining a threshold of distortion from the source state according to a generally defined distortion observable. The constraint on the output space fixes the output distribution to an i.i.d. predefined probability mass function. Therefore, this problem can also be viewed as information-constrained optimal transport which finds the optimal cost of transporting the source quantum state to the destination state via an entanglement-breaking channel with limited communication rate and common randomness.1