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arXiv · 0709.2623

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

Michel Planat·Anne-Céline Baboin
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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.

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Michel Planat, Anne-Céline Baboin. 2007-10-10. Qudits of composite dimension, mutually unbiased bases and projective ring geometry. https://doi.org/10.1088/1751-8113/40/46/f04

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