$\mathrm{SU}_2$ Nonstandard Bases: Case of Mutually Unbiased Bases

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 $\mathrm{SU}_2$ corresponding to an irreducible representation of $\mathrm{SU}_2$. The representation theory of $\mathrm{SU}_2$ 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 $\mathrm{su}_2$. The eigenvectors of the commuting set of operators $\{j^2,v_{ra}\}$ are adapted to a tower of chains $\mathrm{SO}_3\supset C_{2j+1}$ ($2j\in\mathbb N^{\ast}$), where $C_{2j+1}$ is the cyclic group of order $2j+1$. In the case where $2j+1$ is prime, the corresponding eigenvectors generate a complete set of mutually unbiased bases. Some useful relations on generalized quadratic Gauss sums are exposed in three appendices.