Proceedings of the American Mathematical Society

A two-parameter family of complex Hadamard matrices of order

induced by hypocycloids

Author(s): Ferenc Szöllosi
Journal: Proc. Amer. Math. Soc.
MSC (2000): Primary 46L10; Secondary 05B20
Posted: October 20, 2009
Retrieve article in: PDF

Abstract: We construct a

-parameter family of complex Hadamard matrices of order

by a natural block construction. We combine this family with an earlier result of Zauner to derive a

-parameter family of triplets of mutually unbiased bases (MUBs) in $\mathbb{C}^6$ . This invalidates some numerical evidence given by Brierley and Weigert and sheds new light on the problem of determining the maximal number of MUBs in $\mathbb{C}^6$ .

1.: K. Beauchamp, R. Nicoara, Orthogonal maximal Abelian $\ast$ -subalgebras of the $6\times 6$ matrices, Linear Algebra Appl. 428, No. 8-9, 1833-1853 (2008). MR 2398121 (2009b:16068)
2.: I. Bengtsson, W. Bruzda, Å. Ericsson, J. Larsson, W. Tadej, K. Życzkowski, Mutually un- biased bases and Hadamards of order six, J. Math. Phys. 48, 052106 (2007).
3.: G. Björck, Functions of modulus on $\mathbb{Z}_n$ , whose Fourier transform have constant modulus, and ``cyclic -roots''. Recent Advances in Fourier Analysis and Its Applications. NATO Adv. Sci. Int. Ser. C, Math. Phys. Sci., 315, Kluwer, 131-140 (1990). MR 1081347 (91k:42021)
4.: S. Brierley, S. Weigert, Maximal sets of mutually unbiased quantum states in dimension six, Phys. Rev. A (3) 78, no. 4 (2008). MR 2491052
5.: S. Brierley, S. Weigert, Constructing mutually unbiased bases in dimension six, preprint, arXiv:0901.4051v1 [quant-ph] (2009).
6.: P. Butterley, W. Hall, Numerical evidence for the maximum number of mutually unbiased bases in dimension six, Physics Letters A 369, 5-8 (2007).
7.: N. J. Cerf, M. Bourennane, A. Karlsson, N. Gisin, Security of quantum key distribution using -level systems, Phys. Rev. Lett. 88, 127902 (2002).
8.: P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A 37, no. 20, 5355-5374 (2004). MR 2065675 (2005b:15045)
9.: U. Haagerup, Orthogonal maximal Abelian $\ast$ -subalgebras of $n\times n$ matrices and cyclic -roots, Operator Algebras and Quantum Field Theory (Rome, 1996), International Press, Cambridge, MA, 296-322 (1997). MR 1491124 (98k:46087)
10.: P. de la Harpe, V. F. R. Jones, Paires de sous-algèbres semi-simples et graphes fortement réguliers, C.R. Acad. Sci. Paris 311, série I, 147-150 (1990). MR 1065880 (92c:16004)
11.: P. Jaming, M. Matolcsi, P. Móra, F. Szöllősi, M. Weiner, A generalized Pauli problem and an infinite family of MUB-triplets in dimension , J. Physics A: Math. and Theo. 42, no. 24 (2009), 245305.
12.: A. Klappenecker, M. Rötteler, Constructions of Mutually Unbiased Bases. Finite fields and applications, Lecture Notes in Comput. Sci., 2948, Springer, Berlin, 137-144 (2004). MR 2092627 (2005f:81040)
13.: M. Matolcsi, F. Szöllősi, Towards the classification of $6\times 6$ complex Hadamard matrices, Open Sys. & Inf. Dyn. 15:2, 93-108 (2008). MR 2451755
14.: A. Munemasa, Y. Watatani, Orthogonal pairs of $\ast$ -subalgebras and association schemes, C.R. Acad. Sci. Paris 314, série I, 329-331 (1992). MR 1153709 (92m:46090)
15.: S. Popa, Orthogonal pairs of $\ast$ -subalgebras in finite von Neumann algebras, J. Operator Theory 9, 253-268 (1983). MR 703810 (84h:46077)
16.: M. Saniga, M. Planat, Sets of mutually unbiased bases as arcs in finite projectives planes, Chaos Solitons Fractals 26, 1267-1270 (2005). MR 2149314 (2006d:81050)
17.: A. J. Skinner, V. A. Newell, R. Sanchez, Unbiased bases (Hadamards) for -level systems: Four ways from Fourier, preprint, arXiv:0810.1761v1 [quant-ph] (2008).
18.: W. Tadej, K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13, 133-177 (2006). MR 2244963 (2007f:15020)
19.: Website for complex Hadamard matrices: http://chaos.if.uj.edu.pl/ $\sim$ karol/hadamard/
20.: M. Weiner, A gap for the maximum number of mutually unbiased bases, preprint, arXiv:0902.0635v1 [math-ph] (2009).
21.: R. F. Werner, All teleportation and dense coding schemes, J. Phys. A 34, 7081-7094 (2001). MR 1863141 (2002i:81063)
22.: P. Wocjan, Th. Beth, New construction of mutually unbiased bases in square dimensions, Quantum Information & Computation 5:2, 93-101 (2005). MR 2132048 (2006g:05039)
23.: W. K. Wootters, B. D. Fields, Optimal state-determination by mutually unbiased measurements, Ann. Physics 191, 363-381 (1989). MR 1003014 (90e:81019)
24.: G. Zauner, Orthogonale Lateinische Quadrate und Anordnungen, Verallgemeinerte Hadamard-matrizen und Unabhängigkelt in der Quanten-Wahrscheinlichkeitestheorie, Master Thesis, Universität Wien (1991).

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L10, 05B20

Ferenc Szöllosi
Affiliation: Institute of Mathematics and Its Applications, Central European University (CEU), H-1051, Nádor u. 9, Budapest, Hungary
Email: szoferi@gmail.com

DOI: 10.1090/S0002-9939-09-10102-8
PII: S 0002-9939(09)10102-8
Received by editor(s): April 3, 2009.
Posted: October 20, 2009
Additional Notes: This work was supported by Hungarian National Research Fund OTKA-K77748
Communicated by: Marius Junge
Copyright of article: Copyright 2009, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS