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

Abstract : The commutation relations between the generalized Pauli operators of N-qudits i. e., N p-level quantum systems, and the structure of their maximal sets of commuting bases, follow a nice graph theoretical-geometrical pattern. One may identify vertices-points with the operators so that edges-lines join commuting pairs of them to form the so-called Pauli graph P {p^N} . As per two-qubits p = 2, N = 2 all basic properties and partitionings of this graph are embodied in the geometry of the symplectic generalized quadrangle of order two, W2. The structure of the two-qutrit p = 3, N = 2 graph is more involved; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the geometry of generalized quadrangle Q4, 3, the dual of W3. Finally, the generalized adjacency graph for multiple N > 3 qubits-qutrits is shown to follow from symplectic polar spaces of order two-three. The relevance of these mathematical concepts to mutually unbiased bases and to quantum entanglement is also highlighted in some detail.

Keywords : Generalized Pauli operators Pauli graph Quantum entanglement Mutually unbiased bases Generalized quadrangles Symplectic polar spaces

Author: Michel Planat - Metod Saniga -



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