A Rate-Distortion Perspective on Quantum State Redistribution

We consider a rate-distortion version of the quantum state redistribution task, where the error of the decoded state is judged via an additive distortion measure; it thus constitutes a quantum generalisation of the classical Wyner-Ziv problem. The quantum source is described by a tripartite pure state shared between Alice (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i>, encoder), Bob (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i>, decoder) and a reference (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i>). Both Alice and Bob are required to output a system (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ã</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B̃</i>, respectively), and the distortion measure is encoded in an observable on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ÃB̃R</i>. It includes as special cases most quantum rate-distortion problems considered in the past, and in particular quantum data compression with the fidelity measured per copy; furthermore, it generalises the well-known state merging and quantum state redistribution tasks for a pure state source, with per-copy fidelity, and a variant recently considered by us, where the source is an ensemble of pure states [ZBK & AW, Proc. ISIT 2020, pp. 1858-1863 and ZBK, PhD thesis, UAB 2020, arXiv:2012.14143]. We derive a single-letter formula for the rate-distortion function of compression schemes assisted by free entanglement. A peculiarity of the formula is that in general it requires optimisation over an unbounded auxiliary register, so the rate-distortion function is not readily computable from our result, and there is a continuity issue at zero distortion. However, we show how to overcome these difficulties in certain situations.