Projective Ring Line of a Specific Qudit

Abstract

A very particular connection between the commutation relations of the elements of the generalized Pauli group of a $d$-dimensional qudit, $d$ being a product of distinct primes, and the structure of the projective line over the (modular) ring $\bZ_{d}$ is established, where the integer exponents of the generating shift ($X$) and clock ($Z$) operators are associated with submodules of $\bZ^{2}_{d}$. Under this correspondence, the set of operators commuting with a given one -- a perp-set -- represents a $\bZ_{d}$-submodule of $\bZ^{2}_{d}$. A crucial novel feature here is that the operators are also represented by {\it non}-admissible pairs of $\bZ^{2}_{d}$. This additional degree of freedom makes it possible to view any perp-set as a {\it set-theoretic} union of the corresponding points of the associated projective line.

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Cited by in corpus (16)

• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Projective Ring Line of an Arbitrary Single Qudit
• 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}
• Qudits of composite dimension, mutually unbiased bases and projective ring geometry
• Weak mutually unbiased bases
• Twin "Fano-Snowflakes" Over the Smallest Ring of Ternions
• The Veldkamp Space of the Smallest Slim Dense Near Hexagon
• The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems
• A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies
• An analytic function approach to weak mutually unbiased bases
• Generalized spin bases for quantum chemistry and quantum information
• Matrix reduction and Lagrangian submodules
• Bases for spin systems and qudits from angular momentum theory