SU₂ Nonstandard Bases: Case of Mutually Unbiased Bases

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2007
• ISSN: 1815-0659
• Hauptverfasser: Albouy, O., Kibler, M.R.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2007
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/147375
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: SU₂ Nonstandard Bases: Case of Mutually Unbiased Bases / O. Albouy, M.R. Kibler // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 78 назв. — англ.

This paper deals with bases in a finite-dimensional Hilbert space. Such a space can be realized as a subspace of the representation space of SU₂ corresponding to an irreducible representation of SU₂. The representation theory of SU₂ is reconsidered via the use of two truncated deformed oscillators. This leads to replacement of the familiar scheme {j²,jz} by a scheme {j²,vra}, where the two-parameter operator vra is defined in the universal enveloping algebra of the Lie algebra su₂. The eigenvectors of the commuting set of operators {j²,vra} are adapted to a tower of chains SO₃⊃C₂j₊₁ (2j∈N∗), where C₂j₊₁ is the cyclic group of order 2j+1. In the case where 2j+1 is prime, the corresponding eigenvectors generate a complete set of mutually unbiased bases. Some useful relations on generalized quadratic Gauss sums are exposed in three appendices