SU_2 Nonstandard Bases: Case of Mutually Unbiased Bases

• Main Authors: Olivier Albouy, Maurice R. Kibler
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2007-07-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: symmetry adapted bases; truncated deformed oscillators; angular momentum; polar decomposition of su$_2$; finite quantum mechanics; cyclic systems; mutually unbiased bases; Gauss sums
• Online Access: http://www.emis.de/journals/SIGMA/2007/076/

This paper deals with bases in a finite-dimensionalHilbert space. Such a~space can be realized as a subspace of therepresentation space of SU$_2$ corresponding to an irreduciblerepresentation of SU$_2$. The representation theory of SU$_2$ isreconsidered 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 theLie algebra su$_2$. The eigenvectors of the commuting set ofoperators ${ j^2 , v_{ra} }$ are adapted to a tower of chainsSO$_3 supset C_{2j+1}$ ($2j in mathbb{N}^{ast}$), where$C_{2j+1}$ is the cyclic group of order $2j+1$. In the case where$2j+1$ is prime, the corresponding eigenvectors generate acomplete set of mutually unbiased bases. Some useful relations ongeneralized quadratic Gauss sums are exposed in three appendices.