An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group

Abstract

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining 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 analysed 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.

Topical review (40 pages). Dedicated to the memory of Yurii Fedorovich Smirnov

References in corpus (15)

• Classicality in discrete Wigner functions
• Maximal Sets of Mutually Unbiased Quantum States in Dimension Six
• Mutually Unbiased Bases for Continuous Variables
• Projective Ring Line of an Arbitrary Single Qudit
• Projective Ring Line Encompassing Two-Qubits
• Variations on a theme of Heisenberg, Pauli and Weyl
• Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions
• Feynman's path integral and mutually unbiased bases
• Projective Ring Line of a Specific Qudit
• The isotropic lines of Z_{d}^{2}
• Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators
• Non-Abelian Discrete Flavor Symmetries
• Qudits of composite dimension, mutually unbiased bases and projective ring geometry
• Quasiprobability distribution functions for finite-dimensional discrete phase spaces: Spin-tunneling effects in a toy model
• Feynman's Integral is About Mutually Unbiased Bases