RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Zap. Nauchn. Sem. POMI:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

Ortogonal pairs and mutually unbiased bases

A. Bondalabcd, I. Zhdanovskiyec

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 277-8583, 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 $|\langle e_i|f_j\rangle|^2=\frac1n$ 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 four-dimensional 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 four-dimensional family of mutually unbiased bases in $\mathbb C^6$, thus solving a long-standing 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 13-01-00234
14-01-00416
15-51-50045
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 13-01-00234, 14-01-00416, and 15-51-50045. 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.


Full text: PDF file (292 kB)
References: PDF file   HTML file

English version:
Journal of Mathematical Sciences (New York), 2016, 216:1, 23–40

Bibliographic databases:

ArXiv: 1510.05317
Document Type: Article
UDC: 512.812+512.552+512.77+512.76
Received: 19.10.2015
Language: English

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 35--61
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6172}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3499907}
\elib{http://elibrary.ru/item.asp?id=27153878}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2016
\vol 216
\issue 1
\pages 23--40
\crossref{https://doi.org/10.1007/s10958-016-2885-z}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84969765228}


Linking options:
  • http://mi.mathnet.ru/eng/znsl6172
  • http://mi.mathnet.ru/eng/znsl/v437/p35

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Записки научных семинаров ПОМИ
    Number of views:
    This page:138
    Full text:42
    References:14

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019