  We extend quantum rate distortion theory by considering auxiliary resources
that might be available to a sender and receiver performing lossy quantum data
compression. The first setting we consider is that of quantum rate distortion
coding with the help of a classical side channel. Our result here is that the
regularized entanglement of formation characterizes the quantum rate distortion
function, extending earlier work of Devetak and Berger. We also combine this
bound with the entanglement-assisted bound from our prior work to obtain the
best known bounds on the quantum rate distortion function for an isotropic
qubit source. The second setting we consider is that of quantum rate distortion
coding with quantum side information (QSI) available to the receiver. In order
to prove results in this setting, we first state and prove a quantum reverse
Shannon theorem with QSI (for tensor-power states), which extends the known
tensor-power quantum reverse Shannon theorem. The achievability part of this
theorem relies on the quantum state redistribution protocol, while the converse
relies on the fact that the protocol can cause only a negligible disturbance to
the joint state of the reference and the receiver's QSI. This quantum reverse
Shannon theorem with QSI naturally leads to quantum rate-distortion theorems
with QSI, with or without entanglement assistance.
