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—In this work, we address the lossy quantum-classical (QC) source coding problem, where the task is to compress the classical information about a quantum source, obtained after performing a measurement, below the Shannon entropy of the measurement outcomes, while incurring a bounded reconstruction error. We propose a new formulation, namely, "rate-channel theory", for the lossy QC source coding problem based on the notion of a backward (posterior) channel. We employ a singleletter posterior channel to capture the reconstruction error in place of the single-letter distortion observable. The formulation requires the reconstruction of the compressed quantum source to satisfy a block error constraint as opposed to the average singleletter distortion criterion in the rate-distortion setting. We also develop an analogous formulation for the classical variant with respect to a corresponding posterior channel. Furthermore, we characterize the asymptotic performance limit of the lossy QC and classical source coding problems in terms of single-letter quantum mutual information and mutual information quantities of the given posterior channel, respectively. We provide examples for the above formulations.