Quantum expanders and the quantum entropy difference problem

We define quantum expanders in a natural way. We show that under certain conditions classical expander constructions generalize to the quantum setting, and in particular so does the Lubotzky, Philips and Sarnak construction of Ramanujan expanders from Cayley graphs of the group PGL. We show that this definition is exactly what is needed for characterizing the complexity of estimating quantum entropies.