Abstract algebra, projective geometry and time encoding of quantum information

Michel R. P. Planat; Metod Saniga

hal-00004513, version 2

Abstract algebra, projective geometry and time encoding of quantum information

Michel R. P. Planat () ¹, Metod Saniga ²

Endophysics, Time, Quantum and the Subjective World Scientific (Ed.) (2005) pp. 409-426

Résumé : Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing. We point out here the prominent role played by Galois fields viewed as cyclotomic extensions of the integers modulo a prime characteristic $p$. They can be used to generate efficient cyclic encoding, for transmitting secrete quantum keys, for quantum state recovery and for error correction in quantum computing. Finite projective planes and their generalization are the geometric counterpart to cyclotomic concepts, their coordinatization involves Galois fields, and they have been used repetitively for enciphering and coding. Finally the characters over Galois fields are fundamental for generating complete sets of mutually unbiased bases, a generic concept of quantum information processing and quantum entanglement. Gauss sums over Galois fields ensure minimum uncertainty under such protocols. Some Galois rings which are cyclotomic extensions of the integers modulo $4$ are also becoming fashionable for their role in time encoding and mutual unbiasedness.

1 : Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (FEMTO-ST)
CNRS : UMR6174 – Université de Franche-Comté – Université de Technologie de Belfort-Montbeliard – Ecole Nationale Supérieure de Mécanique et des Microtechniques
2 : Astronomical Institute (ASTRONOMICAL INSTITUTE)
Slovak Academy of Sciences

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Contributeur : Michel Planat <>
Soumis le : Mardi 7 Juin 2005, 15:07:42
Dernière modification le : Lundi 9 Janvier 2006, 14:11:49

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%% hal-00004513, version 2
%% http://hal.archives-ouvertes.fr/hal-00004513
@incollection{planat:hal-00004513,
    hal_id = {hal-00004513},
    url = {http://hal.archives-ouvertes.fr/hal-00004513},
    title = {{Abstract algebra, projective geometry and time encoding of quantum information}},
    author = {Planat, Michel, R. P. and Saniga, Metod},
    abstract = {{Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing. We point out here the prominent role played by Galois fields viewed as cyclotomic extensions of the integers modulo a prime characteristic $p$. They can be used to generate efficient cyclic encoding, for transmitting secrete quantum keys, for quantum state recovery and for error correction in quantum computing. Finite projective planes and their generalization are the geometric counterpart to cyclotomic concepts, their coordinatization involves Galois fields, and they have been used repetitively for enciphering and coding. Finally the characters over Galois fields are fundamental for generating complete sets of mutually unbiased bases, a generic concept of quantum information processing and quantum entanglement. Gauss sums over Galois fields ensure minimum uncertainty under such protocols. Some Galois rings which are cyclotomic extensions of the integers modulo $4$ are also becoming fashionable for their role in time encoding and mutual unbiasedness.}},
    keywords = {Mutually unbiased bases; Finite Geometries; Galois Fields; Fourier Transforms; QuantumInformation Theory},
    language = {Anglais},
    affiliation = {Franche-Comt{\'e} {\'E}lectronique M{\'e}canique, Thermique et Optique - Sciences et Technologies - FEMTO-ST , Astronomical Institute - ASTRONOMICAL INSTITUTE},
    booktitle = {{Endophysics, Time, Quantum and the Subjective}},
    publisher = {World Scientific},
    pages = {pp. 409-426},
    editor = {World Scientific },
    series = {eds R. Buccheri, A.C. Elitzur and M. Saniga },
    note = {To appear in R. Buccheri, A.C. Elitzur and M. Saniga (eds.), "Endophysics, Time, Quantum and the Subjective," World Scientific, Singapore. 16 pages. },
    year = {2005},
    pdf = {http://hal.archives-ouvertes.fr/hal-00004513/PDF/bielefeld.pdf},
}