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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A gap for the maximum number of mutually unbiased bases


Author: Mihály Weiner
Journal: Proc. Amer. Math. Soc. 141 (2013), 1963-1969
MSC (2010): Primary 15A30, 47L05, 81P70
Published electronically: January 23, 2013
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11487-5
PII: S 0002-9939(2013)11487-5
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.