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Proceedings Paper

Pauli graph and finite projective lines/geometries
Author(s): Michel Planat; Metod Saniga
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Paper 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 PpN . 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, W(2). 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 Q(4, 3), the dual of W(3). 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.

Paper Details

Date Published: 11 May 2007
PDF: 12 pages
Proc. SPIE 6583, Photon Counting Applications, Quantum Optics, and Quantum Cryptography, 65830W (11 May 2007); doi: 10.1117/12.721687
Show Author Affiliations
Michel Planat, Institut FEMTO-ST, CNRS (France)
Metod Saniga, Slovak Academy of Sciences (Slovak Republic)

Published in SPIE Proceedings Vol. 6583:
Photon Counting Applications, Quantum Optics, and Quantum Cryptography
Miloslav Dusek; Ivan Prochazka; Mark S. Hillery; Alan L. Migdall; Wolfgang P. Schleich; Alexandre Pauchard, Editor(s)

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