Qudits of composite dimension, mutually unbiased bases and projective ring geometry

Abstract

The $d^2$ Pauli operators attached to a composite qudit in dimension $d$ may be mapped to the vectors of the symplectic module ${\mathcal{Z}}_d^2$ (${\mathcal{Z}}_d$ the modular ring). As a result, perpendicular vectors correspond to commuting operators, a free cyclic submodule to a maximal commuting set, and disjoint such sets to mutually unbiased bases. For dimensions $d=6,10,15,12$, and 18, the fine structure and the incidence between maximal commuting sets is found to reproduce the projective line over the rings ${\mathcal{Z}}_6$, ${\mathcal{Z}}_{10}$, ${\mathcal{Z}}_{15}$, ${\mathcal{Z}}_6\times {\mathbf{F}}_4$ and ${\mathcal{Z}}_6\times {\mathcal{Z}}_3$, respectively.

10 pages (Fast Track communication). Journal of Physics A Mathematical and Theoretical (2008) accepted

References in corpus (3)

• Pauli Diagonal Channels Constant on Axes
• Multiple Qubits as Symplectic Polar Spaces of Order Two
• Projective Ring Line of a Specific Qudit

Cited by in corpus (4)

• 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
• Generalized spin bases for quantum chemistry and quantum information