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

Michel R. P. Planat (1) , Anne-Céline Baboin (1) , Metod Saniga (1, 2)

1 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)
2 Astronomical Institute of the Slovak Academy of Sciences

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. Remarkably, all such "multi-line" substructures are found to satisfy the second (antiflag) axiom of a generalized quadrangle. This multi-line property is then 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.

Mots clés: Generalized Pauli Operators; Pauli Graphs; Finite Projective Geometries

Domaines: Physique Quantique [quant-ph]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Michel R. P. Planat, Anne-Céline Baboin, Metod Saniga. Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators. 2007. ⟨hal-00147435v1⟩