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Journal Articles Journal of Physics A: Mathematical and Theoretical Year : 2007

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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Dates and versions

hal-00169103 , version 1 (31-08-2007)
hal-00169103 , version 2 (27-12-2007)

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Hans Havlicek, Metod Saniga. Projective Ring Line of a Specific Qudit. Journal of Physics A: Mathematical and Theoretical, 2007, 40, pp.F943-F952. ⟨10.1088/1751-8113/40/43/F03⟩. ⟨hal-00169103v2⟩

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