Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials

TL;DR

This paper investigates perfect quantum state transfer in weighted paths with potentials, demonstrating limitations for PST with Laplacian matrices and proposing a conjecture linking weight rationality to PST capability.

Contribution

It introduces a method using orthogonal polynomials to analyze PST in weighted paths and proves PST cannot occur for certain weighted structures, suggesting a link between weight rationality and PST.

Findings

PST with respect to the Laplacian cannot occur for weighted paths with three or more vertices.

PST cannot occur for certain symmetric weighted trees.

A conjecture links the rationality of weights to the capacity for PST in weighted paths.

Abstract

A simple method for transmitting quantum states within a quantum computer is via a quantum spin chain---that is, a path on $n$ vertices. Unweighted paths are of limited use, and so a natural generalization is to consider weighted paths; this has been further generalized to allow for loops (\emph{potentials} in the physics literature). We study the particularly important situation of perfect state transfer with respect to the corresponding adjacency matrix or Laplacian through the use of orthogonal polynomials. Low-dimensional examples are given in detail. Our main result is that PST with respect to the Laplacian matrix cannot occur for weighted paths on $n\geq 3$ vertices nor can it occur for certain symmetric weighted trees. The methods used lead us to a conjecture directly linking the rationality of the weights of weighted paths on $n>3$ vertices, with or without loops, with the capacity for PST between the end vertices with respect to the adjacency matrix.