SU₂ Nonstandard Bases: Case of Mutually Unbiased Bases

• Main Authors: Albouy, O., Kibler, M.R.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2007
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Online Access: http://dspace.nbuv.gov.ua/handle/123456789/147375

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