Birth and death processes and quantum spin chains

TL;DR

This paper explores the connection between quantum spin chains and classical birth and death processes, revealing how transition amplitudes relate to orthogonal polynomials and characterizing systems with certain return probabilities.

Contribution

It establishes a novel link between quantum walks in spin chains and birth-death processes using orthogonal polynomials, providing new insights into their probabilistic behavior.

Findings

Transition amplitudes expressed via orthogonal polynomials

Characterization of spin systems with certain return probabilities

Connection to Karlin-McGregor representation formula

Abstract

This papers underscores the intimate connection between the quantum walks generated by certain spin chain Hamiltonians and classical birth and death processes. It is observed that transition amplitudes between single excitation states of the spin chains have an expression in terms of orthogonal polynomials which is analogous to the Karlin-McGregor representation formula of the transition probability functions for classes of birth and death processes. As an application, we present a characterization of spin systems for which the probability to return to the point of origin at some time is 1 or almost 1.