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A Criterion for Attaining the Welch Bounds with Applications for Mutually Unbiased Bases

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Mathematical Methods in Computer Science

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5393))

Abstract

The paper gives a short introduction to mutually unbiased bases and the Welch bounds and demonstrates that the latter is a good technical tool to explore the former. In particular, a criterion for a system of vectors to satisfy the Welch bounds with equality is given and applied for the case of MUBs. This yields a necessary and sufficient condition on a set of orthonormal bases to form a complete system of MUBs.

This condition takes an especially elegant form in the case of homogeneous systems of MUBs. We express some known constructions of MUBs in this form. Also it is shown how recently obtained results binding MUBs and some combinatorial structures (such as perfect nonlinear functions and relative difference sets) naturally follow from this criterion.

Some directions for proving non-existence results are sketched as well.

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Author information

Authors and Affiliations

  1. Department of Combinatorics and Optimization, and Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

    Aleksandrs Belovs

  2. Department of Computer Science, University of Latvia, Raiņa bulvāris 19, Rīga, Latvia

    Juris Smotrovs

Authors
  1. Aleksandrs Belovs
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  2. Juris Smotrovs
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Editor information

Editors and Affiliations

  1. Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, 76128, Karlsruhe, Germany

    Jacques Calmet  & Jörn Müller-Quade  & 

  2. Institut für Algorithmen und Kognitive Systeme, Institut für Algorithmen und Kognitive Universität Karlsruhe, 76128, Karlsruhe, Germany

    Willi Geiselmann

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Belovs, A., Smotrovs, J. (2008). A Criterion for Attaining the Welch Bounds with Applications for Mutually Unbiased Bases. In: Calmet, J., Geiselmann, W., Müller-Quade, J. (eds) Mathematical Methods in Computer Science. Lecture Notes in Computer Science, vol 5393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89994-5_6

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  • DOI: https://doi.org/10.1007/978-3-540-89994-5_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-89993-8

  • Online ISBN: 978-3-540-89994-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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