A Theorem of Joseph-Alfred Serret and its Relation to Perfect Quantum State Transfer

TL;DR

This paper explores the mathematical foundations of perfect quantum state transfer by connecting Serret's theorem on palindromic continued fractions with polynomial continued fractions and symmetric tridiagonal matrices.

Contribution

It introduces a novel application of Serret's theorem to quantum information theory, linking classical continued fraction theory with quantum state transfer.

Findings

Established a connection between palindromic continued fractions and polynomial continued fractions.

Linked symmetric tridiagonal matrices to the problem of quantum state transfer.

Provided a new mathematical perspective on perfect quantum state transfer.

Abstract

In this paper we recast the Serret theorem about a characterization of palindromic continued fractions in the context of polynomial continued fractions. Then, using the relation between symmetric tridiagonal matrices and polynomial continued fractions we give a quick exposition of the mathematical aspect of the perfect quantum state transfer problem.