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Symmetry Integrability and Geometry Methods and Applications

SU 2 Nonstandard Bases: Case of Mutually Unbiased Bases

Olivier Albouy

Published Jul 8, 2007• 7 citations
Publisher

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 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 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 ∈ 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.

DOI
10.3842/sigma.2007.076

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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 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 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 ∈ 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.
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.
A simple recipe for generating a complete set of mutually unbiased bases in dimension 2j + 1, with 2j integer and 2j + 1 prime, is developed from a single matrix Va acting on a space of constant angular momentum j and defined in terms of the irreducible characters of the cyclic group C2j + 1. This recipe yields an (apparently new) compact formula for the vectors spanning the various mutually unbiased bases. In dimension (2j + 1)e, with 2j integer, 2j + 1 prime and e positive integer, the use of direct products of matrices of type Va makes it possible to generate mutually unbiased bases. As two pending results, the matrix Va is used in the derivation of a polar decomposition of SU (2) and of a FFZ algebra.
Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for quantum information and quantum computation are constructed from angular momentum theory and su(2) Lie algebraic methods. We report on a formula for deriving in one step the (1 + p)p qupits (i.e., qudits with d = p a prime integer) of a complete set of 1 + p mutually unbiased bases in C p . Repeated application of the formula can be used for generating mutually unbiased bases in C d with d = p e (e ≥ 2) a power of a prime integer. A connection between mutually unbiased bases and the unitary group SU(d) is briefly discussed in the case d = p e .
The Lie algebra of the group SU 2 is constructed from two deformed oscillator algebras for which the deformation parameter is a root of unity. This leads to an unusual quantization scheme, the {J2, Ur} scheme, an alternative to the familiar {J2, Jz} quantization scheme corresponding to common eigenvectors of the Casimir operator J2 and the Cartan operator Jz. A connection is established between the eigenvectors of the complete set of commuting operators {J2, Ur} and mutually unbiased bases in spaces of constant angular momentum.
Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for quantum information and quantum computation are constructed from angular momentum theory and su(2) Lie algebraic methods. We report on a formula for deriving in one step the (1+p)p qupits (i.e., qudits with d = p a prime integer) of a complete set of 1+p mutually unbiased bases in C^p. Repeated application of the formula can be used for generating mutually unbiased bases in C^d with d = p^e (e > or = 2) a power of a prime integer. A connection between mutually unbiased bases and the unitary group SU(d) is briefly discussed in the case d = p^e.
The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of SU (2). A single formula for the bases is obtained from a polar decomposition of SU (2) and is analyzed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.