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 TMF, 2007, Volume 151, Number 1, Pages 44–53 (Mi tmf6010)

Projective line over the finite quotient ring $GF(2)[x]/\langle x^3-x\rangle$ and quantum entanglement: Theoretical background

M. Sanigaa, M. Planatb

a Astronomical Institute, Slovak Academy of Sciences
b CNRS — Institut FEMTO-ST, Département LPMO

Abstract: We consider the projective line over the finite quotient ring $R_{\diamondsuit}\equiv{GF}(2)[x]/\langle x^3-x\rangle$. The line is endowed with 18 points, spanning the neighborhoods of three pairwise distant points. Because $R_{\diamondsuit}$ is not a local ring, the neighbor (or parallel) relation is not an equivalence relation, and the sets of neighbors for two distant points hence overlap. There are nine neighbors of any point on the line, forming three disjoint families under the reduction modulo either of the two maximal ideals of the ring. Two of the families contain four points each, and they swap their roles when switching from one ideal to the other, the points in one family merging with (the image of) the point in question and the points in the other family passing in pairs into the remaining two points of the associated ordinary projective line of order two. The single point in the remaining family passes to the reference point under both maps, and its existence stems from a nontrivial character of the Jacobson radical $\mathcal J_{\diamondsuit}$ of the ring. The quotient ring $\widetilde R_{\diamondsuit} \equiv R_{\diamondsuit}/\mathcal J_{\diamondsuit}$ is isomorphic to ${GF}(2)\otimes{GF}(2)$. The projective line over $\widetilde R_{\diamondsuit}$ features nine points, each of them surrounded by four neighbors and four distant points, and any two distant points share two neighbors. We surmise that these remarkable ring geometries are relevant for modeling entangled qubit states, which we will discuss in detail in Part II of this paper.

Keywords: projective ring line, finite quotient ring, neighbor/distant relation, quantum entanglement

DOI: https://doi.org/10.4213/tmf6010

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English version:
Theoretical and Mathematical Physics, 2007, 151:1, 474–481

Bibliographic databases:

Citation: M. Saniga, M. Planat, “Projective line over the finite quotient ring $GF(2)[x]/\langle x^3-x\rangle$ and quantum entanglement: Theoretical background”, TMF, 151:1 (2007), 44–53; Theoret. and Math. Phys., 151:1 (2007), 474–481

Citation in format AMSBIB
\Bibitem{SanPla07} \by M.~Saniga, M.~Planat \paper Projective line over the~finite quotient ring $GF(2)[x]/\langle x^3-x\rangle$ and quantum entanglement: Theoretical background \jour TMF \yr 2007 \vol 151 \issue 1 \pages 44--53 \mathnet{http://mi.mathnet.ru/tmf6010} \crossref{https://doi.org/10.4213/tmf6010} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2347301} \zmath{https://zbmath.org/?q=an:1119.81016} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2007TMP...151..474S} \elib{http://elibrary.ru/item.asp?id=9521570} \transl \jour Theoret. and Math. Phys. \yr 2007 \vol 151 \issue 1 \pages 474--481 \crossref{https://doi.org/10.1007/s11232-007-0035-y} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000245809000003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34247259473} 

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• https://doi.org/10.4213/tmf6010
• http://mi.mathnet.ru/eng/tmf/v151/i1/p44

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Michel Planat, Metod Saniga, Maurice R. Kibler, “Quantum Entanglement and Projective Ring Geometry”, SIGMA, 2 (2006), 066, 14 pp.
2. M. Saniga, M. Planat, M. Minarovjech, “Projective line over the finite quotient ring $GF(2)[x]/\langle x^3-x\rangle$ and quantum entanglement: The Mermin “magic” square/pentagram”, Theoret. and Math. Phys., 151:2 (2007), 625–631
3. Havlicek, H, “Projective ring line of a specific qudit”, Journal of Physics A-Mathematical and Theoretical, 40:43 (2007), F943
4. M. Saniga, M. Planat, P. Pracna, “Projective ring line encompassing two-qubits”, Theoret. and Math. Phys., 155:3 (2008), 905–913
5. Planat M, Saniga M, “On the Pauli graphs on N-qudits”, Quantum Information & Computation, 8:1–2 (2008), 127–146
6. Hans Havlicek, Boris Odehnal, Metod Saniga, “Factor-Group-Generated Polar Spaces and (Multi-)Qudits”, SIGMA, 5 (2009), 096, 15 pp.
7. Green R.M. Saniga M., “The Veldkamp Space of the Smallest Slim Dense Near Hexagon”, Int. J. Geom. Methods Mod. Phys., 10:2 (2013), 1250082
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