Fractional Revival and Association Schemes

TL;DR

This paper investigates fractional revival in graphs within association schemes, especially Hamming schemes, linking algebraic combinatorics and orthogonal polynomials to characterize entanglement phenomena in quantum networks.

Contribution

It provides a characterization of fractional revival in association scheme graphs, focusing on balanced cases related to maximal entanglement, using algebraic combinatorics techniques.

Findings

Characterization of fractional revival in association scheme graphs.

Identification of conditions for balanced fractional revival in Hamming schemes.

Connection established between algebraic combinatorics and quantum entanglement phenomena.

Abstract

Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose-Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming scheme. Our proofs exploit the intimate connections between algebraic combinatorics and orthogonal polynomials.