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 (8)

• A new proof for the existence of mutually unbiased bases
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• Mutually unbiased phase states, phase uncertainties, and Gauss sums
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• Variations on a theme of Heisenberg, Pauli and Weyl
• Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
• Experimental observation of impossible-to-beat quantum advantage on a hybrid photonic system
• 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
• Generalized spin bases for quantum chemistry and quantum information
• Space versus Time: Unimodular versus Non-Unimodular Projective Ring Geometries?