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 (45)

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• Security of quantum key distribution using d-level systems
• A new proof for the existence of mutually unbiased bases
• Discrete phase space based on finite fields
• SIC-POVMs and the Extended Clifford Group
• Quantum Encodings in Spin Systems and Harmonic Oscillators
• Mutually unbiased binary observable sets on N qubits
• On SIC-POVMs and MUBs in Dimension 6
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• Classicality in discrete Wigner functions
• Constructing Mutually Unbiased Bases in Dimension Six
• Mubs and Hadamards of Order Six
• Maximal Sets of Mutually Unbiased Quantum States in Dimension Six
• On the structure of the sets of mutually unbiased bases for N qubits
• The mean king's problem: Prime degrees of freedom
• Multicomplementary operators via finite Fourier transform
• Qubits in phase space: Wigner function approach to quantum error correction and the mean king problem
• Aspects of mutually unbiased bases in odd prime power dimensions
• Separability and Fourier representations of density matrices
• Wigner Functions and Separability for Finite Systems
• On SIC-POVMs in Prime Dimensions
• A SU(2) recipe for mutually unbiased bases
• Mean king's problem with mutually unbiased bases and orthogonal Latin squares
• Mutually Unbiased Bases and Trinary Operator Sets for N Qutrits
• The limitations of nice mutually unbiased bases
• Mutually Unbiased Bases for Continuous Variables
• The mean king's problem: Spin 1
• Projective Ring Line Encompassing Two-Qubits
• Projective Ring Line of an Arbitrary Single Qudit
• Variations on a theme of Heisenberg, Pauli and Weyl
• Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions
• Angular Momentum and Mutually Unbiased Bases
• Feynman's path integral and mutually unbiased bases
• Projective Ring Line of a Specific Qudit
• Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators
• The isotropic lines of Z_{d}^{2}
• 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
• A complementarity-based approach to phase in finite-dimensional quantum systems
• Talbot effect for dispersion in linear optical fibers and a wavelet approach
• Feynman's Integral is About Mutually Unbiased Bases
• Compilation of relations for the antisymmetric tensors defined by the Lie algebra cocycles of $su(n)$
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Cited by in corpus (30)

• On mutually unbiased bases
• Topological properties of linear circuit lattices
• Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems
• Superintegrability of Geodesic Motion on the Sausage Model
• General phase spaces: from discrete variables to rotor and continuum limits
• Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
• Phase operators, phase states and vector phase states for SU(3) and SU(2,1)
• The complete Heyting algebra of subsystems and contextuality
• Optimal reconstruction of states in qutrits system
• Weak mutually unbiased bases
• Quantum mechanics on profinite groups and partial order
• $SO(3)$-invariant phase of the $O(N{)}^3$ tensor model
• Non-Abelian Monopole in the Parameter Space of Point-like Interactions
• SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms
• On the geometry and invariants of qubits, quartits and octits
• Three dimensional quantum key distribution in the presence of several eavesdroppers
• Dressed coherent states in finite quantum systems: a cooperative game theory approach
• Partial order and a $T_0$-topology in a set of finite quantum systems
• An analytic function approach to weak mutually unbiased bases
• Tensor Commutation Matrices and Some Generalizations of the Pauli Matrices
• Extended Weyl-Heisenberg algebra, phase operator, unitary depolarizers and generalized Bell states
• On Two Ways to Look for Mutually Unbiased Bases
• Finite oscillator obtained through finite frame quantization
• Equilibration properties of small quantum systems: further examples
• Matrices de commutation tensorielle: de l'équation de Dirac vers une application en physique des particules
• Polyadic sigma matrices
• On mutually unbiased bases: Passing from d to d**2
• Paths of zeros of analytic functions describing finite quantum systems
• Balanced Tripartite Entanglement, the Alternating Group A4 and the Lie Algebra $sl(3,C)\oplus u(1)$
• Bases for qudits from a nonstandard approach to SU(2)