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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A gap for the maximum number of mutually unbiased bases
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by Mihály Weiner PDF
Proc. Amer. Math. Soc. 141 (2013), 1963-1969 Request permission

Abstract:

A collection of pairwise mutually unbiased bases (in short: MUB) in $d>1$ dimensions may consist of at most $d+1$ bases. Such “complete” collections are known to exist in $\mathbb {C}^d$ when $d$ is a power of a prime. However, in general, little is known about the maximum number $N(d)$ of bases that a collection of MUB in $\mathbb {C}^d$ can have.

In this work it is proved that a collection of $d$ MUB in $\mathbb {C}^d$ can always be completed. Hence $N(d)\neq d$, and when $d>1$ we have a dichotomy: either $N(d)=d+1$ (so that there exists a complete collection of MUB) or $N(d)\leq d-1$. In the course of the proof an interesting new characterization is given for a linear subspace of $M_d(\mathbb {C})$ to be a subalgebra.

References
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Additional Information
  • Mihály Weiner
  • Affiliation: Department of Analysis, Mathematical Institute, Budapest University of Economics and Technology (BME), Pf. 91, H-1521 Budapest, Hungary
  • Email: mweiner@renyi.hu
  • Received by editor(s): July 16, 2010
  • Received by editor(s) in revised form: October 4, 2011
  • Published electronically: January 23, 2013
  • Additional Notes: Supported in part by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory” and the Momentum Fund of the Hungarian Academy of Sciences.
  • Communicated by: Marius Junge
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 1963-1969
  • MSC (2010): Primary 15A30, 47L05, 81P70
  • DOI: https://doi.org/10.1090/S0002-9939-2013-11487-5
  • MathSciNet review: 3034423