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Designs, Codes and Cryptography
Published in: Designs, Codes and Cryptography 1/2016

01-07-2016

Constructions of complex equiangular lines from mutually unbiased bases

Authors: Jonathan Jedwab, Amy Wiebe

Published in: Designs, Codes and Cryptography | Issue 1/2016

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Abstract

A set of vectors of equal norm in \(\mathbb {C}^d\) represents equiangular lines if the magnitudes of the Hermitian inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is \(d^2\), and it is conjectured that sets of this maximum size exist in \(\mathbb {C}^d\) for every \(d \ge 2\). We take a combinatorial approach to this conjecture, using mutually unbiased bases (MUBs) in the following three constructions of equiangular lines:
(1)
adapting a set of \(d\) MUBs in \(\mathbb {C}^d\) to obtain \(d^2\) equiangular lines in \(\mathbb {C}^d\),
 
(2)
using a set of \(d\) MUBs in \(\mathbb {C}^d\) to build \((2d)^2\) equiangular lines in \(\mathbb {C}^{2d}\),
 
(3)
combining two copies of a set of \(d\) MUBs in \(\mathbb {C}^d\) to build \((2d)^2\) equiangular lines in \(\mathbb {C}^{2d}\).
 
For each construction, we give the dimensions \(d\) for which we currently know that the construction produces a maximum-sized set of equiangular lines.
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Footnotes
1
A function \(f\) from \(\mathbb {N}\) to \(\mathbb {R}^+\) is \(\Theta (d^2)\) if there are positive constants \(c\) and \(C\), independent of \(d\), for which \(c d^2 \le f(d) \le C d^2\) for all sufficiently large \(d\).
 
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Metadata
Title
Constructions of complex equiangular lines from mutually unbiased bases
Authors
Jonathan Jedwab
Amy Wiebe
Publication date
01-07-2016
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 1/2016
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-015-0064-8

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