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

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

2007-09-17

arxiv.org/abs/0709.2623v2

Journal of Physics A Mathematical and Theoretical 40 (2007) F1-F8

Physics

Quantum Physics

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

Scientific paper

10.1088/1751-8113/40/46/F04

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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