Cayley graphs that have a quantum ergodic eigenbasis

TL;DR

This paper explores conditions under which finite Cayley graphs possess a quantum ergodic eigenbasis, showing it holds for certain groups but not universally, thus advancing understanding of quantum ergodicity in algebraic structures.

Contribution

It establishes a criterion based on the sum of irreducible representation dimensions for the existence of a quantum ergodic eigenbasis in Cayley graphs.

Findings

Cayley graphs on groups with sum of irreducible representation dimensions being o(n) have quantum ergodic eigenbases.

Some Cayley graphs do not admit any quantum ergodic eigenbasis.

The paper delineates conditions affecting quantum ergodicity in algebraic graph structures.

Abstract

We investigate which finite Cayley graphs admit a quantum ergodic eigenbasis, proving that this holds for any Cayley graph on a group of size $n$ for which the sum of the dimensions of its irreducible representations is $o(n)$, yet there exist Cayley graphs that do not have any quantum ergodic eigenbasis.