Abstract algebra, projective geometry and time encoding of quantum information
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چکیده
algebra, projective geometry and time encoding of quantum information Michel Planat, Metod Saniga To cite this version: Michel Planat, Metod Saniga. Abstract algebra, projective geometry and time encoding of quantum information. World Scientific. Endophysics, Time, Quantum and the Subjective, World Scientific, pp. 409-426, 2005, eds R. Buccheri, A.C. Elitzur and M. Saniga. HAL Id: hal-00004513 https://hal.archives-ouvertes.fr/hal-00004513v2 Submitted on 7 Jun 2005 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. cc sd -0 00 04 51 3, v er si on 2 7 J un 2 00 5 R. Buccheri et al. (eds.); Endophysics, Time, Quantum and the Subjective; 409–426 c © 2005 World Scientific Publishing Co. All rights reserved. ABSTRACT ALGEBRA, PROJECTIVE GEOMETRY AND TIME ENCODING OF QUANTUM INFORMATIONALGEBRA, PROJECTIVE GEOMETRY AND TIME ENCODING OF QUANTUM INFORMATION MICHEL PLANAT FEMTO-ST, University of Franche-Comté, 32 Avenue de l’Observatoire 25044, Besançon, France ([email protected]) METOD SANIGA Astronomical Institute, Slovak Academy of Sciences SK-05960 Tatranská Lomnica, Slovak Republic ([email protected]) 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. 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.
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