Qudit Isotopy

TL;DR

This paper introduces a diagrammatic framework for understanding qudits and their braiding, linking algebraic and topological methods to quantum entanglement and information protocols.

Contribution

It develops a novel diagrammatic approach to qudits using planar para algebras, connecting topological and algebraic perspectives in quantum information.

Findings

Defined qudit Pauli operators $X,Y,Z$ from algebraic and diagrammatic views

Introduced an entanglement-relay protocol for long-distance quantum communication

Established connections between planar para algebras and quantum entanglement

Abstract

We explore a general diagrammatic framework to understand qudits and their braiding, especially in its relation to entanglement. This involves understanding the role of isotopy in interpreting diagrams that implement entangling gates as well as some standard quantum information protocols. We give qudit Pauli operators $X,Y,Z$ and comment on their structure, both from an algebraic and from a diagrammatic point of view. We explain alternative models for diagrammatic interpretations of qudits and their transformations. We use our diagrammatic approach to define an entanglement-relay protocol for long-distance entanglement. Our approach rests on algebraic and topological relations discovered in the study of planar para algebras. In summary, this work provides bridges between the new theory of planar para algebras and quantum information, especially in questions involving entanglement.