Projective Ring Line of an Arbitrary Single Qudit

Abstract

As a continuation of our previous work (arXiv:0708.4333) an algebraic geometrical study of a single $d$-dimensional qudit is made, with $d$ being {\it any} positive integer. The study is based on an intricate relation between the symplectic module of the generalized Pauli group of the qudit and the fine structure of the projective line over the (modular) ring $\bZ_{d}$. Explicit formulae are given for both the number of generalized Pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of $\bZ^{2}_{d}$. We find, remarkably, that a perp-set is not a set-theoretic union of the corresponding points of the associated projective line unless $d$ is a product of distinct primes. The operators are also seen to be structured into disjoint `layers' according to the degree of their representing vectors. A brief comparison with some multiple-qudit cases is made.

References in corpus (9)

• Projective Representations I. Projective lines over rings
• Multiple Qubits as Symplectic Polar Spaces of Order Two
• Projective Ring Line Encompassing Two-Qubits
• Projective Ring Line of a Specific Qudit
• Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators
• Divisible Designs, Laguerre Geometry, and Beyond
• A Classification of the Projective Lines over Small Rings
• Qudits of composite dimension, mutually unbiased bases and projective ring geometry
• Pauli graph and finite projective lines/geometries

Cited by in corpus (23)

• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Variations on a theme of Heisenberg, Pauli and Weyl
• Factor-Group-Generated Polar Spaces and (Multi-)Qudits
• Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
• The isotropic lines of Z_{d}^{2}
• Weak mutually unbiased bases
• Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions
• Twin "Fano-Snowflakes" Over the Smallest Ring of Ternions
• On the geometry and invariants of qubits, quartits and octits
• The Veldkamp Space of the Smallest Slim Dense Near Hexagon
• SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms
• A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies
• An analytic function approach to weak mutually unbiased bases
• Space versus Time: Unimodular versus Non-Unimodular Projective Ring Geometries?
• Generalized spin bases for quantum chemistry and quantum information
• Twelve-dimensional Pauli group contextuality
• Matrix reduction and Lagrangian submodules
• Affine and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices
• Hexagons govern three-qubit contextuality
• Bases for spin systems and qudits from angular momentum theory
• Free Cyclic Submodules and Non-Unimodular Vectors
• Pauli graphs, Riemann hypothesis, Goldbach pairs
• Bases for qudits from a nonstandard approach to SU(2)