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.

翻译：我们考虑在有限经典公共随机性条件下，针对有限维和连续变量量子到经典系统的输出受限率失真编码，研究有限速率量子到经典最优传输问题。主要编码定理给出了有损量子测量源编码的可达速率区域的单字母刻画，该编码通过一般定义的失真可观测量在维持源状态失真阈值的同时，精确重构目标分布（或等价量子态）。输出空间约束将输出分布固定为独立同分布预定义概率质量函数。因此，该问题也可视为信息约束下的最优传输，即通过有限通信速率和公共随机性的量子测量，寻找将源量子态传输到目标经典分布的最优代价。我们通过采用截断投影和去量子化模块，并利用有限维编码定理，构建了连续变量量子系统的编码框架。此外，针对高斯量子系统，我们推导了二阶有限速率Wasserstein距离的解析解，并提出了高斯最优性定理，表明在高斯量子源和高斯目标分布的系统中，高斯测量能优化传输速率。结果进一步表明，与高斯分布的经典Wasserstein距离（对应无穷传输速率）不同，在高斯量子测量系统中，由于海森堡不确定性原理带来的量子测量固有噪声，最优传输可在有限传输速率下实现。