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