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We consider the lossy quantum source coding problem where the task is tocompress a given quantum source below its von Neumann entropy. Inspired by theduality connections between the rate-distortion and channel coding problems inthe classical setting, we propose a new formulation for the lossy quantumsource coding problem. This formulation differs from the existing quantumrate-distortion theory in two aspects. Firstly, we require that thereconstruction of the compressed quantum source fulfill a global errorconstraint as opposed to the sample-wise local error criterion used in thestandard rate-distortion setting. Secondly, instead of a distortion observable,we employ the notion of a backward quantum channel, which we refer to as a"posterior reference map", to measure the reconstruction error. Using these, wecharacterize the asymptotic performance limit of the lossy quantum sourcecoding problem in terms of single-letter coherent information of the givenposterior reference map. We demonstrate a protocol to encode (at the specifiedrate) and decode, with the reconstruction satisfying the provided global errorcriterion, and therefore achieving the asymptotic performance limit. Theprotocol is constructed by decomposing coherent information as a difference oftwo Holevo information quantities, inspired from prior works in quantumcommunication problems. To further support the findings, we develop analogousformulations for the quantum-classical and classical variants and express theasymptotic performance limit in terms of single-letter mutual informationquantities with respect to appropriately defined channels analogous toposterior reference maps. We also provide various examples for the threeformulations, and shed light on their connection to the standardrate-distortion formulation wherever possible.

Lossy Quantum Source Coding with a Global Error Criterion based on a Posterior Reference Map