Lossy Quantum Source Coding With a Global Error Criterion Based on a Posterior Reference Map

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, to measure the reconstruction error, instead of a distortion observable, we employ the notion of a backward quantum channel which we refer to as a “posterior reference map”. Using these, we characterize the asymptotic performance limit in terms of single-letter coherent information of the given posterior reference map. We also develop analogous formulations for the quantum-classical and classical variants and characterize their asymptotic performance limits in terms of single-letter mutual information quantities with respect to appropriately defined channels analogous to the posterior reference map. We also provide various examples for the three formulations, and shed light on their connection to the standard rate-distortion formulation wherever possible.