in2p3-00128008, version 3
SU(2) nonstandard bases: the case of mutually unbiased bases
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) 3 (2007) 076
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 replace 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 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) > C(2j+1), 2j integer, 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.
- 1:
- CNRS : UMR5822 – IN2P3 – Université Claude Bernard - Lyon I
- Domain : Mathematics/Mathematical Physics
Physics/Mathematical PhysicsPhysics/Quantum Physics - Comment : 33 pages – version2: rescaling of generalized Hadamard matrices – acknowledgment and references added – misprints corrected – version 3: published in SIGMA (Symmetry – Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/ (22 pages)
- Available versions : v1 (2007-01-30) v2 (2007-02-10) v3 (2007-07-09)
- in2p3-00128008, version 3
- http://hal.in2p3.fr/in2p3-00128008
- oai:hal.in2p3.fr:in2p3-00128008
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- Submitted on: Monday, 9 July 2007 10:52:27
- Updated on: Monday, 9 July 2007 11:46:47
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