We consider the lossy quantum source coding problem where the task is to compress a given quantum source below its von Neumann entropy. Inspired by the duality connections between the rate-distortion and channel coding problems in the classical setting, we propose a new formulation for the lossy quantum source coding problem. This formulation differs from the existing quantum rate-distortion theory in two aspects. Firstly, we require that the reconstruction of the compressed quantum source fulfill a global error constraint as opposed to the sample-wise local error criterion used in the standard 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, we characterize the asymptotic performance limit of the lossy quantum source coding problem in terms of single-letter coherent information of the given posterior reference map. We demonstrate a protocol to encode (at the specified rate) and decode, with the reconstruction satisfying the provided global error criterion, and therefore achieving the asymptotic performance limit. The protocol is constructed by decomposing coherent information as a difference of two Holevo information quantities, inspired from prior works in quantum communication problems. To further support the findings, we develop analogous formulations for the quantum-classical and classical variants and express the asymptotic performance limit in terms of single-letter mutual information quantities with respect to appropriately defined channels analogous to posterior reference maps. We also provide various examples for the three formulations, and shed light on their connection to the standard rate-distortion formulation wherever possible.

翻译：本文研究有损量子信源编码问题，其任务是将给定量子信源压缩至其冯·诺依曼熵以下。受经典信源编码中率失真与信道编码之间对偶关系的启发，我们提出了一种有损量子信源编码问题的新表述。该表述与现有量子率失真理论存在两点差异：首先，我们要求压缩量子信源的重构满足全局误差约束，而非标准率失真框架中采用的逐样本局部误差准则；其次，我们采用"后验参考映射"这一反向量子信道概念来度量重构误差，而非使用失真可观测量。基于此，我们通过给定后验参考映射的单字母相干信息表征了有损量子信源编码问题的渐近性能极限。我们演示了一种编码（按指定速率）与解码协议，其重构满足所提全局误差准则，从而达到了该渐近性能极限。该协议通过将相干信息分解为两个霍列沃信息量之差构建，这一思路受量子通信问题的前期工作启发。为进一步支撑研究结论，我们发展了量子-经典及经典变体的类似表述，并通过适当定义的后验参考映射类比信道，以单字母互信息量表示其渐近性能极限。最后，我们为三种表述提供了多种示例，并尽可能阐明其与标准率失真表述之间的关联。