\(\text{SU}_2\) 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.


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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