SU(2) nonstandard bases: the case of mutually unbiased bases

TL;DR

This paper explores nonstandard bases in finite-dimensional Hilbert spaces related to SU(2), introducing a new scheme that generates mutually unbiased bases when the dimension is prime, with implications for quantum information theory.

Contribution

It introduces a novel scheme {j^2, v(ra)} for SU(2) representations that produces mutually unbiased bases in prime-dimensional spaces.

Findings

Eigenvectors of {j^2, v(ra)} form mutually unbiased bases when 2j+1 is prime.

Reformulation of SU(2) representation theory using truncated deformed oscillators.

Provides relations on generalized quadratic Gauss sums.

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.