Quantum Margulis expanders

TL;DR

This paper introduces a quantum version of the Margulis expander, which operates on discrete Wigner functions and retains key properties of the classical expander, offering a novel approach without non-Abelian harmonic analysis.

Contribution

It presents a simple method to quantize the Margulis expander, creating a quantum expander with properties mirroring the classical one, applicable to discrete and continuous phase space systems.

Findings

Quantum Margulis expander acts on discrete Wigner functions.

Shares degree and spectrum with classical Margulis expander.

Method avoids non-Abelian harmonic analysis.

Abstract

We present a simple way to quantize the well-known Margulis expander map. The result is a quantum expander which acts on discrete Wigner functions in the same way the classical Margulis expander acts on probability distributions. The quantum version shares all essential properties of the classical counterpart, e.g., it has the same degree and spectrum. Unlike previous constructions of quantum expanders, our method does not rely on non-Abelian harmonic analysis. Analogues for continuous variable systems are mentioned. Indeed, the construction seems one of the few instances where applications based on discrete and continuous phase space methods can be developed in complete analogy.