Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators

TL;DR

This paper explores the complex geometric and graph-theoretic structures of generalized Pauli operators in qubit-qutrit and higher-dimensional systems, revealing multi-line properties and their algebraic representations.

Contribution

It introduces a novel geometric framework for understanding the commutation relations of higher-order Pauli operators, highlighting multi-line features in their associated graphs.

Findings

Dual Pauli graph is isomorphic to the projective line over Z2xZ3.

Multi-line property observed in two-qutrit and three-qubit systems.

Higher-dimensional systems likely exhibit similar multi-line geometric features.

Abstract

The commutation relations of the generalized Pauli operators of a qubit-qutrit system are discussed in the newly established graph-theoretic and finite-geometrical settings. The dual of the Pauli graph of this system is found to be isomorphic to the projective line over the product ring Z2xZ3. A "peculiar" feature in comparison with two-qubits is that two distinct points/operators can be joined by more than one line. The multi-line property is shown to be also present in the graphs/geometries characterizing two-qutrit and three-qubit Pauli operators' space and surmised to be exhibited by any other higher-level quantum system.