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\(\text{SU}_2\) nonstandard bases: case of mutually unbiased bases. (English) Zbl 1139.81357

Summary: 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 \(\text{SU}_2\) corresponding to an irreducible representation of \(\text{SU}_2\). The representation theory of \(\text{SU}_2\) 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 \(\text{su}_2\). The eigenvectors of the commuting set of operators \(\{j^2,v_{ra}\}\) are adapted to a tower of chains \(\text{SO}_2\supset C_{2j+1}\) \((2j\in\mathbb N^*)\), 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.

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
47L90 Applications of operator algebras to the sciences
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