51 research outputs found
Tight t-Designs and Squarefree Integers
The authors prove, using a variety of number-theoretical methods, that tight t-designs in the projective spaces FPn of ‘lines’ through the origin in Fn+1 (F = ℂ, or the quarternions H) satisfy t ⩽ 5.Such a design is a generalisation of a combinatorial t-design. It is known that t ⩽ 5 in the cases F=ℝ,O (the octonions) and that t ⩽ 11 for tight spherical t-designs; hence the author's result essentially completes the classification of tight t-designs in compact connected symmetric spaces of rank 1
Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
We introduce the problem of constructing weighted complex projective
2-designs from the union of a family of orthonormal bases. If the weight
remains constant across elements of the same basis, then such designs can be
interpreted as generalizations of complete sets of mutually unbiased bases,
being equivalent whenever the design is composed of d+1 bases in dimension d.
We show that, for the purpose of quantum state determination, these designs
specify an optimal collection of orthogonal measurements. Using highly
nonlinear functions on abelian groups, we construct explicit examples from d+2
orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10,
and 12, for example, where no complete sets of mutually unbiased bases have
thus far been found.Comment: 28 pages, to appear in J. Math. Phy
On the minimum diameter of plane integral point sets
Since ancient times mathematicians consider geometrical objects with integral
side lengths. We consider plane integral point sets , which are
sets of points in the plane with pairwise integral distances where not all
the points are collinear.
The largest occurring distance is called its diameter. Naturally the question
about the minimum possible diameter of a plane integral point set
consisting of points arises. We give some new exact values and describe
state-of-the-art algorithms to obtain them. It turns out that plane integral
point sets with minimum diameter consist very likely of subsets with many
collinear points. For this special kind of point sets we prove a lower bound
for achieving the known upper bound up to a
constant in the exponent.
A famous question of Erd\H{o}s asks for plane integral point sets with no 3
points on a line and no 4 points on a circle. Here, we talk of point sets in
general position and denote the corresponding minimum diameter by
. Recently could be determined via an
exhaustive search.Comment: 12 pages, 5 figure
Tight informationally complete quantum measurements
We introduce a class of informationally complete positive-operator-valued
measures which are, in analogy with a tight frame, "as close as possible" to
orthonormal bases for the space of quantum states. These measures are
distinguished by an exceptionally simple state-reconstruction formula which
allows "painless" quantum state tomography. Complete sets of mutually unbiased
bases and symmetric informationally complete positive-operator-valued measures
are both members of this class, the latter being the unique minimal rank-one
members. Recast as ensembles of pure quantum states, the rank-one members are
in fact equivalent to weighted 2-designs in complex projective space. These
measures are shown to be optimal for quantum cloning and linear quantum state
tomography.Comment: 20 pages. Final versio
Recommended from our members
Optimal and Near Optimal Configurations on Lattices and Manifolds
Optimal configurations of points arise in many contexts, for example classical ground states for interacting particle systems, Euclidean packings of convex bodies, as well as minimal discrete and continuous energy problems for general kernels. Relevant questions in this area include the understanding of asymptotic optimal configurations, of lattice and periodic configurations, the development of algorithmic constructions of near optimal configurations, and the application of methods in convex optimization such as linear and semidefinite programming
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Gitter und Anwendungen
The meeting focussed on lattices and their applications in mathematics and information technology. The research interests of the participants varied from engineering sciences, algebraic and analytic number theory, coding theory, algebraic geometry to name only a few
Fibonacci-Lucas SIC-POVMs
We present a conjectured family of SIC-POVMs which have an additional
symmetry group whose size is growing with the dimension. The symmetry group is
related to Fibonacci numbers, while the dimension is related to Lucas numbers.
The conjecture is supported by exact solutions for dimensions
d=4,8,19,48,124,323, as well as a numerical solution for dimension d=844.Comment: The fiducial vectors can be obtained from
http://sicpovm.markus-grassl.de as well as from the source files. v2:
precision for the numerical solution in dimension 844 increased to 150 digits
and new exact solution for dimension 323 adde
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