
Zap. Nauchn. Sem. POMI, 2015, Volume 437, Pages 35–61
(Mi znsl6172)




Ortogonal pairs and mutually unbiased bases
A. Bondal^{abcd}, I. Zhdanovskiy^{ec} ^{a} Steklov Institute of Mathematics, Moscow, Russia
^{b} Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, Kashiwa, Chiba 2778583, Japan
^{c} HSE Laboratory of Algebraic Geometry, Moscow, Russia
^{d} The Institute of Fundamental Science, Moscow, Russia
^{e} Moscow Institute of Physics and Technology
Abstract:
The goal of our article is a study of related mathematical and physical objects: orthogonal pairs in $\mathrm{sl}(n)$ and mutually unbiased bases in $\mathbb C^n$. An orthogonal pair in a simple Lie algebra is a pair of Cartan subalgebras that are orthogonal with respect to the Killing form. The description of orthogonal pairs in a given Lie algebra is an important step in the classification of orthogonal decompositions, i.e., decompositions of the Lie algebra into a direct sum of Cartan subalgebras pairwise orthogonal with respect to the Killing form. One of the important notions of quantum mechanics, quantum information theory, and quantum teleportation is the notion of mutually unbiased bases in the Hilbert space $\mathbb C^n$. Two orthonormal bases $\{e_i\}^n_{i=1}$, $\{f_j\}^n_{j=1}$ are mutually unbiased if and only if $\langle e_if_j\rangle^2=\frac1n$ for any $i,j=1,…,n$. The notions of mutually unbiased bases in $\mathbb C^n$ and orthogonal pairs in $\mathrm{sl}(n)$ are closely related. The problem of classification of orthogonal pairs in $\mathrm{sl}(n)$ and the closely related problem of classification of mutually unbiased bases in $\mathbb C^n$ are still open even for the case $n=6$. In this article, we give a sketch of our proof that there is a complex fourdimensional family of orthogonal pairs in $\mathrm{sl}(6)$. This proof requires a lot of algebraic geometry and representation theory. Further, we give an application of the result on the algebraic geometric family to the study of mutually unbiased bases. We show the existence of a real fourdimensional family of mutually unbiased bases in $\mathbb C^6$, thus solving a longstanding problem.
Key words and phrases:
orthogonal pairs, mutually unbiased bases (MUB), complex Hadamard matrices, generalized Hadamard matrices.
Funding Agency 
Grant Number 
Russian Foundation for Basic Research 
130100234 140100416 155150045 
Ministry of Education and Science of the Russian Federation 

This work was done during the authors' visit to the Kavli IPMU and was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The reported study was partially supported by the RFBR, research projects 130100234, 140100416, and 155150045. The article was prepared within the framework of a subsidy
granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program. 
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English version:
Journal of Mathematical Sciences (New York), 2016, 216:1, 23–40
Bibliographic databases:
ArXiv:
1510.05317
UDC:
512.812+512.552+512.77+512.76 Received: 19.10.2015
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Citation:
A. Bondal, I. Zhdanovskiy, “Ortogonal pairs and mutually unbiased bases”, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXVI. Representation theory, dynamical systems, combinatorial methods, Zap. Nauchn. Sem. POMI, 437, POMI, St. Petersburg, 2015, 35–61; J. Math. Sci. (N. Y.), 216:1 (2016), 23–40
Citation in format AMSBIB
\Bibitem{BonZhd15}
\by A.~Bondal, I.~Zhdanovskiy
\paper Ortogonal pairs and mutually unbiased bases
\inbook Representation theory, dynamical systems, combinatorial and algoritmic methods. Part~XXVI. Representation theory, dynamical systems, combinatorial methods
\serial Zap. Nauchn. Sem. POMI
\yr 2015
\vol 437
\pages 3561
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6172}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3499907}
\elib{https://elibrary.ru/item.asp?id=27153878}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2016
\vol 216
\issue 1
\pages 2340
\crossref{https://doi.org/10.1007/s109580162885z}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2s2.084969765228}
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