Weakly Hadamard diagonalizable graphs and Quantum State Transfer

TL;DR

This paper explores the properties and constructions of weak Hadamard matrices and weakly Hadamard diagonalizable graphs, extending the study of quantum state transfer beyond traditional Hadamard diagonalizable graphs.

Contribution

It introduces the concept of weak Hadamard matrices and weakly Hadamard diagonalizable graphs, providing foundational properties and constructions for future quantum information research.

Findings

Introduced weak Hadamard matrices with $PP^T$ tridiagonal

Characterized properties of weakly Hadamard diagonalizable graphs

Provided new constructions for these matrices and graphs

Abstract

Hadamard diagonalizable graphs are undirected graphs for which the corresponding Laplacian is diagonalizable by a Hadamard matrix. Such graphs have been studied in the context of quantum state transfer. Recently, the concept of a weak Hadamard matrix was introduced: a $\{-1,0, 1\}$-matrix $P$ such that $PP^T$ is tridiagonal, as well as the concept of weakly Hadamard diagonalizable graphs. We therefore naturally explore quantum state transfer in these generalized Hadamards. Given the infancy of the topic, we provide numerous properties and constructions of weak Hadamard matrices and weakly Hadamard diagonalizable graphs in order to better understand them.