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An application of positive definite functions to the problem of MUBs


Authors: Mihail N. Kolountzakis, Máté Matolcsi and Mihály Weiner
Journal: Proc. Amer. Math. Soc. 146 (2018), 1143-1150
MSC (2010): Primary 43A35; Secondary 15A30, 05B10
DOI: https://doi.org/10.1090/proc/13829
Published electronically: October 12, 2017
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Abstract: We present a new approach to the problem of mutually unbiased bases (MUBs), based on positive definite functions on the unitary group. The method provides a new proof of the fact that there are at most $ d+1$ MUBs in $ \mathbb{C}^d$, and it may also lead to a proof of non-existence of complete systems of MUBs in dimension 6 via a conjectured algebraic identity.


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Additional Information

Mihail N. Kolountzakis
Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, Voutes Campus, 700 13 Heraklion, Greece
Email: kolount@gmail.com

Máté Matolcsi
Affiliation: Department of Analysis, Budapest University of Technology and Economics (BME), H-1111, Egry J. u. 1, Budapest, Hungary — and — Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053, Realtanoda u 13-15, Budapest, Hungary
Email: matomate@renyi.hu

Mihály Weiner
Affiliation: Department of Analysis, Budapest University of Technology and Economics (BME), H-1111, Egry J. u. 1, Budapest, Hungary
Email: mweiner@renyi.hu

DOI: https://doi.org/10.1090/proc/13829
Keywords: Mutually unbiased bases, positive definite functions, unitary group
Received by editor(s): January 5, 2017
Received by editor(s) in revised form: April 16, 2017
Published electronically: October 12, 2017
Additional Notes: The first author was partially supported by grant No 4725 of the University of Crete
The second author was supported by the ERC-AdG 321104 and by NKFIH-OTKA Grant No. K104206
The third author was supported by the ERC-AdG 669240 QUEST “Quantum Algebraic Structures and Models” and by NKFIH-OTKA Grant No. K104206
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2017 American Mathematical Society

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