Krawtchouk matrices from the Feynman path integral and from the split quaternions

TL;DR

This paper explores the mathematical properties of Krawtchouk matrices through their connections to the Feynman path integral and split quaternions, clarifying their role in quantum information and providing algebraic insights.

Contribution

It introduces a novel interpretation of Krawtchouk matrices via the Feynman path integral and characterizes them algebraically using split quaternions, enhancing understanding of their spectral properties.

Findings

Krawtchouk matrices linked to the Feynman path integral.

Algebraic characterization via split quaternions.

Simplified derivation of spectral decomposition.

Abstract

An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integral is given. Also, an algebraic characterization in terms of the algebra of split quaternions is provided. The resulting properties include an easy inference of the spectral decomposition. It is also an occasion for an expository clarification of the role of Krawtchouk matrices in different areas, including quantum information.