Encoding discrete quantum algebras in a hierarchy of binary words

TL;DR

This paper demonstrates how to encode discrete quantum algebra structures using hierarchical binary patterns, connecting to Dyck languages and Temperley-Lieb algebras, with potential applications in signal processing and waveform engineering.

Contribution

It introduces a method to represent quantum algebraic structures within binary hierarchies, linking abstract algebra, formal languages, and computational models.

Findings

Hierarchical binary patterns can form a normed C*-algebra structure.

Connections established between quantum algebras, Dyck languages, and Temperley-Lieb algebras.

Effective arithmetic coding is possible within these structures, despite increased complexity.

Abstract

It is shown how to endow a hierarchy of sets of binary patterns with the structure of an abstract,normed C*-algebra. In the course we also recover an intermediate connection with the words of a Dyck language and Tempereley-Lieb algebras for which we also find that an effective arithmetic code is possible albeit of greater complexity. We also discuss possible applications associated with signal theory and waveform engineering on possible ways to embed discrete computational structures in an analog continuum substrate.