  We consider the rate-limited quantum-to-classical optimal transport in terms
of output-constrained rate-distortion coding for both finite-dimensional and
continuous-variable quantum-to-classical systems with limited classical common
randomness. The main coding theorem provides a single-letter characterization
of the achievable rate region of a lossy quantum 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 IID 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 classical distribution
via a quantum measurement with limited communication rate and common
randomness.
  We develop a coding framework for continuous-variable quantum systems by
employing a clipping projection and a dequantization block and using our
finite-dimensional coding theorem. Moreover, for the Gaussian quantum systems,
we derive an analytical solution for rate-limited Wasserstein distance of order
2, along with a Gaussian optimality theorem, showing that Gaussian measurement
optimizes the rate in a system with Gaussian quantum source and Gaussian
destination distribution. The results further show that in contrast to the
classical Wasserstein distance of Gaussian distributions, which corresponds to
an infinite transmission rate, in the Quantum Gaussian measurement system, the
optimal transport is achieved with a finite transmission rate due to the
inherent noise of the quantum measurement imposed by Heisenberg's uncertainty
principle.
