221 research outputs found
Symplectic spreads, planar functions and mutually unbiased bases
In this paper we give explicit descriptions of complete sets of mutually
unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras
obtained from commutative and symplectic semifields, and
from some other non-semifield symplectic spreads. Relations between various
constructions are also studied. We show that the automorphism group of a
complete set of MUBs is isomorphic to the automorphism group of the
corresponding orthogonal decomposition of the Lie algebra .
In the case of symplectic spreads this automorphism group is determined by the
automorphism group of the spread. By using the new notion of pseudo-planar
functions over fields of characteristic two we give new explicit constructions
of complete sets of MUBs.Comment: 20 page
SU(2) nonstandard bases: the case of mutually unbiased bases
This paper deals with bases in a finite-dimensional Hilbert space. Such a
space can be realized as a subspace of the representation space of SU(2)
corresponding to an irreducible representation of SU(2). The representation
theory of SU(2) is reconsidered via the use of two truncated deformed
oscillators. This leads to replace the familiar scheme {j^2, j_z} by a scheme
{j^2, v(ra)}, where the two-parameter operator v(ra) is defined in the
enveloping algebra of the Lie algebra su(2). The eigenvectors of the commuting
set of operators {j^2, v(ra)} are adapted to a tower of chains SO(3) > C(2j+1),
2j integer, where C(2j+1) is the cyclic group of order 2j+1. In the case where
2j+1 is prime, the corresponding eigenvectors generate a complete set of
mutually unbiased bases. Some useful relations on generalized quadratic Gauss
sums are exposed in three appendices.Comment: 33 pages; version2: rescaling of generalized Hadamard matrices,
acknowledgment and references added, misprints corrected; version 3:
published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA/ (22 pages
MUBs inequivalence and affine planes
There are fairly large families of unitarily inequivalent complete sets of
N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The
number of such sets is not bounded above by any polynomial as a function of N.
While it is standard that there is a superficial similarity between complete
sets of MUBs and finite affine planes, there is an intimate relationship
between these large families and affine planes. This note briefly summarizes
"old" results that do not appear to be well-known concerning known families of
complete sets of MUBs and their associated planes.Comment: This is the version of this paper appearing in J. Mathematical
Physics 53, 032204 (2012) except for format changes due to the journal's
style policie
Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets
H. Cohn et. al. proposed an association scheme of 64 points in R^{14} which
is conjectured to be a universally optimal code. We show that this scheme has a
generalization in terms of Kerdock codes, as well as in terms of maximal real
mutually unbiased bases. These schemes also related to extremal line-sets in
Euclidean spaces and Barnes-Wall lattices. D. de Caen and E. R. van Dam
constructed two infinite series of formally dual 3-class association schemes.
We explain this formal duality by constructing two dual abelian schemes related
to quaternary linear Kerdock and Preparata codes.Comment: 16 page
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