We present a new method for constructing affine families of complex Hadamard matrices in every even dimension. This method has an intersection with Diţă’s construction and generalizes Szöllősi's method. We extend some known families and present new ones existing in even dimensions. In particular, we find more than 13 millon inequivalent affine families in dimension 32. We also find analytical restrictions for any set of four mutually unbiased bases existing in dimension six and for any family of complex Hadamard matrices existing in every odd dimension.
REFERENCES
1.
R.
Werner
, “All teleportation and dense coding schemes
,” J. Phys. A
34
, 7081
–7094
(2001
).2.
L.
Vaidman
, Y.
Aharonov
, and D. Z.
Albert
, Phys. Rev. Lett.
58
, 1385
(1987
).3.
B.-G.
Englert
and Y.
Aharonov
, “The mean King's problem: Prime degrees of freedom
,” Phys. Lett. A
284
, 1
–5
(2001
).4.
A.
Klappenecker
and M.
Rötteler
, “New tales of the mean King
,” e-print arXiv:quant-ph/0502138 (2005
).5.
I.
Heng
and C. H.
Cooke
, “Error correcting codes associated with complex Hadamard matrices
,” Appl. Math. Lett.
11
, 77
–80
(1998
).6.
G.
Zauner
, Ph.D. dissertation, University of Wien
, 1999
.7.
T.
Tao
, “Fuglede's conjecture is false in 5 and higher dimensions
,” Math. Res. Lett.
11
, 251
–258
(2004
).8.
M.
Matolcsi
, “Fuglede's conjecture fails in dimension 4
,” Proc. Am. Math. Soc.
133
, 3021
–3026
(2005
).9.
M. N.
Kolountzakis
and M.
Matolcsi
, “Tiles with no spectra
,” Forum Mathematicum
18
(3
), 519
–528
(2006
).10.
M. N.
Kolountzakis
and M.
Matolcsi
, “Complex Hadamard matrices and the spectral set conjecture
,” in the Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations
(El Escorial
, 2004
).11.
S.
Popa
, “Orthogonal pairs of *-subalgebras in finite von Neumann algebras
,” J. Oper. Theory
9
, 253
–268
(1983
).12.
P.
de la Harpe
and V. R. F.
Jones
, “Paires de sous-algebres semi-simples et graphes fortement reguliers
,” Acad. Sci., Paris, C. R.
311
, 147
–150
(1990
).13.
A.
Munemasa
and Y.
Watatani
, “Orthogonal pairs of *-subalgebras and association schemes
,” Acad. Sci., Paris, C. R.
314
, 329
–331
(1992
).14.
U.
Haagerup
, Ortogonal Maximal Abelian *-subalgebras of n × n Matrices and Cyclic n-roots
, Operator Algebras and Quantum Field Theory
, edited by S.
Doplicher
et al. (MA International
, Cambridge
, 1997
), pp. 296
–322
.15.
G.
Björk
and R.
Fröberg
, “A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-roots
,” J. Symb. Comput.
12
, 329
–336
(1991
).16.
G.
Björck
and B.
Saffari
, “New classes of finite unimodular sequences with unimodular Fourier transform. Circulant Hadamard matrices with complex entries
,” Acad. Sci., Paris , C. R.
320
, 319
–324
(1995
).17.
C. D.
Godsil
and A.
Roy
, “Equiangular lines, mutually unbiased bases, and spin models
,” Eur. J. Combinatorics
30
(1
), 246
–262
(2009
);e-print arXiv:quant-ph/0511004.
18.
W.
Wootters
and B.
Fields
, “Optimal state-determination by mutually unbiased measurements
,” Ann. Phys.
191
, 363
–381
(1989
).19.
K.
Horadam
, Hadamard Matrices and Their Applications
(Princeton University
, 2007
).20.
W.
Tadej
and K.
Życzkowski
, “A concise guide to complex Hadamard matrices
,” Open Syst. Inf. Dyn.
13
, 133
–177
(2006
).21.
P.
Diţă
, “Four-parameter families of complex Hadamard matrices of order six
,” e-print arXiv:1207.2593v1 [math-ph] (2012
).22.
W.
Tadej
and K.
Życzkowski
, “Defect of a unitary matrix
,” Numer. Linear Algebra Appl.
429
, 447
–481
(2008
).23.
M.
Matolcsi
, J.
Réffy
, and F.
Szöllősi
, “Constructions of complex Hadamard matrices via tiling Abelian groups
,” Open Syst. Inf. Dyn.
14
, 247
(2007
).24.
P.
Jaming
et al., “A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6
,” J. Phys. A: Math. Theor.
42
, 245305
(2009
).25.
P.
Diţă
, “Complex Hadamard matrices from Sylvester inverse orthogonal matrices
,” Open Sys. Inf. Dyn.
16
, 387
–405
(2009
);see the following errata e-print arXiv:0901.0982v2.
26.
H.
Kharaghania
and B.
Tayfeh-Rezaie
, “Hadamard matrices of order 32
,” J. Combinatorial Designs
(in press) (2013
).27.
See the BTZ catalog of complex Hadamard matrices http://chaos.if.uj.edu.pl/~karol/hadamard for finding the current status of the problem.
28.
D.
McNulty
and S.
Weigert
, “Isolated Hadamard matrices from mutually unbiased product bases
,” J. Math. Phys.
53
(12
), 122202
(2012
);e-print arXiv:1208.1057v1 [math-ph].
29.
F.
Szöllősi
, “Parametrizing complex Hadamard matrices
,” Eur. J. Comb.
29
, 1219
–1234
(2008
).30.
P.
Diţă
, “Some results on the parametrization of complex Hadamard matrices
,” J. Phys. A
37
, 5355
(2004
).31.
J.
Sylvester
, “Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tesselated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers
,” London Edinburgh, Dublin Philos. Mag. J. Sci.
34
, 461
–475
(1867
).32.
N.
Barros
and I.
Bengtsson
, “Families of complex Hadamard matrices
,” Linear Algebra and its Applications
438
(7
), 2929
–2957
(2013
);e-print arXiv:1202.1181v1 [math-ph].
33.
P.
Diţă
, “Hadamard matrices from mutually unbiased bases
,” J. Math. Phys.
51
, 072202
(2010
).34.
P.
Diţă
, “Circulant conference matrices for new complex Hadamard matrices
,” e-print arXiv:1107.1338v1 [math-ph] (2011
).© 2013 American Institute of Physics.
2013
American Institute of Physics
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