SU2 Nonstandard Bases: Case of Mutually Unbiased Bases

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 É C2j+1 (2j Î 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.