  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 ($A$, encoder), Bob ($B$, decoder) and a reference ($R$).
Both Alice and Bob are required to output a system ($\widetilde{A}$ and
$\widetilde{B}$, respectively), and the distortion measure is encoded in an
observable on $\widetilde{A}\widetilde{B}R$.
  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 [1], [2].
  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.
