Quantum expanders and the quantum entropy difference problem

TL;DR

This paper introduces quantum expanders, generalizes classical expander constructions to the quantum domain, and links these to the complexity of estimating quantum entropies, providing a new framework for quantum information theory.

Contribution

It defines quantum expanders, extends classical constructions like Ramanujan expanders to quantum settings, and connects these to quantum entropy estimation complexity.

Findings

Quantum expanders are naturally defined and applicable.

Classical expander constructions can be generalized to quantum expanders.

The definition of quantum expanders characterizes the complexity of quantum entropy estimation.

Abstract

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.