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Imprimitivity Systems and Quantum Codes

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Stochastic Analysis and Mathematical Physics II

Part of the book series: Trends in Mathematics ((TM))

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Abstract

It is shown how classical error correcting codes can be quantized to yield quantum error correcting codes as defined by E. Knill and R. Laflamme [2] by using the notion of an imprimitivity system for a group action in the sense of Frobenius and Mackey [4], [8].

The author thanks the Indian National Science Academy for financial support in the form of INSA C.V. Raman Research Professorship

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References

  1. A.B. Calderbank, E.M. Rains, P.W. Shor and N.J.A. Sloane, “Quantum error correction via codes over GF (4)”, IEEE TRans. Information Theory, vol. 44, pp. 1369–1387, 1998.

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  2. E. Knill and R. Laflamme, “A theory of quantum error-correcting codes”, Phys. Rev. A, vol. 55, pp. 900–911, 1997; LANL e-print quantph/9604034.

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  3. E. Knill, “Group representations, error bases and quantum codes”, LANL e-print quant-ph/9608048.

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  4. G.W. Mackey, The Theory of Unitary Group Representations in Physics, Probability and Number Theory, Benjamin/Cummings, Reading, Mass., 1978.

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  5. G.W. Mackey, The Mathematical Foundations of Quantum Mechanics, W.A. Benjamin, Inc., New York, 1963.

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  6. F.J. Macwilliams and N.J.A. Sloane, The Theory of Error-correcting Codes, North-Holland, Amsterdam 1978.

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  7. K.R. Parthasarathy, “Projective unitary antiunitary representations of locally compact groups”, Commun. Math. Phys., vol. 15, pp. 305–328, 1969.

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  8. V.S. Varadarajan, Geometry of Quantum Theory, second edition, Springer-Verlag, Berlin, 1985.

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Author information

Authors and Affiliations

  1. Indian Statistical Institute Delhi Centre, S.J.S. Snsanwal Marg, New Delhi, 110 016, India

    Kalyanapuram Rangachari Parthasarathy

Authors
  1. Kalyanapuram Rangachari Parthasarathy
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Editor information

Editors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, C.P. 6904411, Santiago, Chile

    Rolando Rebolledo

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© 2003 Springer Basel AG

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Parthasarathy, K.R. (2003). Imprimitivity Systems and Quantum Codes. In: Rebolledo, R. (eds) Stochastic Analysis and Mathematical Physics II. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8018-3_9

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  • DOI: https://doi.org/10.1007/978-3-0348-8018-3_9

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9405-0

  • Online ISBN: 978-3-0348-8018-3

  • eBook Packages: Springer Book Archive

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