Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 076, 22 pages      quant-ph/0701230      https://doi.org/10.3842/SIGMA.2007.076

SU2 Nonstandard Bases: Case of Mutually Unbiased Bases

Olivier Albouy and Maurice R. Kibler
Université de Lyon, Institut de Physique Nucléaire, Université Lyon 1 and CNRS/IN2P3, 43 bd du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

Received April 07, 2007, in final form June 16, 2007; Published online July 08, 2007

Abstract
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 SU$_2$ corresponding to an irreducible representation of SU$_2$. The representation theory of SU$_2$ is reconsidered via the use of two truncated deformed oscillators. This leads to replacement of the familiar scheme $\{ j^2 , j_z \}$ by a scheme $\{ j^2 , v_{ra} \}$, where the two-parameter operator $v_{ra}$ is defined in the universal enveloping algebra of the Lie algebra su$_2$. The eigenvectors of the commuting set of operators $\{ j^2 , v_{ra} \}$ are adapted to a tower of chains SO$_3 \supset C_{2j+1}$ ($2j \in \mathbb{N}^{\ast}$), where $C_{2j+1}$ 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.

Key words: symmetry adapted bases; truncated deformed oscillators; angular momentum; polar decomposition of su$_2$; finite quantum mechanics; cyclic systems; mutually unbiased bases; Gauss sums.

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