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SIGMA 3 (2007), 076, 22 pages quant-ph/0701230
doi: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 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.
Key words:
symmetry adapted bases; truncated deformed oscillators;
angular momentum; polar decomposition of su2; finite quantum
mechanics; cyclic systems; mutually unbiased bases; Gauss sums.
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