Quantum state transfer in spin chains with q-deformed interaction terms

TL;DR

This paper explores quantum state transfer in spin chains using q-deformed orthogonal polynomials, discovering a new model with perfect transfer linked to q-Krawtchouk polynomials and analyzing correlation functions through complex q-series calculations.

Contribution

It introduces a novel quantum spin chain model with perfect state transfer based on q-Krawtchouk polynomials and extends the analytical framework for correlation functions involving q-series.

Findings

Perfect state transfer achieved with q-Krawtchouk polynomials

Other q-polynomial cases do not exhibit perfect transfer

Correlation functions involve advanced q-series manipulations

Abstract

We study the time evolution of a single spin excitation state in certain linear spin chains, as a model for quantum communication. Some years ago it was discovered that when the spin chain data (the nearest neighbour interaction strengths and the magnetic field strengths) are related to the Jacobi matrix entries of Krawtchouk polynomials or dual Hahn polynomials, so-called perfect state transfer takes place. The extension of these ideas to other types of discrete orthogonal polynomials did not lead to new models with perfect state transfer, but did allow more insight in the general computation of the correlation function. In the present paper, we extend the study to discrete orthogonal polynomials of q-hypergeometric type. A remarkable result is a new analytic model where perfect state transfer is achieved: this is when the spin chain data are related to the Jacobi matrix of q-Krawtchouk polynomials. The other cases studied here (affine q-Krawtchouk polynomials, quantum q-Krawtchouk polynomials, dual q-Krawtchouk polynomials, q-Hahn polynomials, dual q-Hahn polynomials and q-Racah polynomials) do not give rise to models with perfect state transfer. However, the computation of the correlation function itself is quite interesting, leading to advanced q-series manipulations.