26,733 research outputs found
Complex spherical codes with three inner products
Let be a finite set in a complex sphere of dimension. Let be
the set of usual inner products of two distinct vectors in . A set is
called a complex spherical -code if the cardinality of is and
contains an imaginary number. We would like to classify the largest
possible -codes for given dimension . In this paper, we consider the
problem for the case . Roy and Suda (2014) gave a certain upper bound for
the cardinalities of -codes. A -code is said to be tight if
attains the bound. We show that there exists no tight -code except for
dimensions , . Moreover we make an algorithm to classify the largest
-codes by considering representations of oriented graphs. By this algorithm,
the largest -codes are classified for dimensions , , with a
current computer.Comment: 26 pages, no figur
Rigidity of spherical codes
A packing of spherical caps on the surface of a sphere (that is, a spherical
code) is called rigid or jammed if it is isolated within the space of packings.
In other words, aside from applying a global isometry, the packing cannot be
deformed. In this paper, we systematically study the rigidity of spherical
codes, particularly kissing configurations. One surprise is that the kissing
configuration of the Coxeter-Todd lattice is not jammed, despite being locally
jammed (each individual cap is held in place if its neighbors are fixed); in
this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic
lattice in three dimensions. By contrast, we find that many other packings have
jammed kissing configurations, including the Barnes-Wall lattice and all of the
best kissing configurations known in four through twelve dimensions. Jamming
seems to become much less common for large kissing configurations in higher
dimensions, and in particular it fails for the best kissing configurations
known in 25 through 31 dimensions. Motivated by this phenomenon, we find new
kissing configurations in these dimensions, which improve on the records set in
1982 by the laminated lattices.Comment: 39 pages, 8 figure
Experimental study of energy-minimizing point configurations on spheres
In this paper we report on massive computer experiments aimed at finding
spherical point configurations that minimize potential energy. We present
experimental evidence for two new universal optima (consisting of 40 points in
10 dimensions and 64 points in 14 dimensions), as well as evidence that there
are no others with at most 64 points. We also describe several other new
polytopes, and we present new geometrical descriptions of some of the known
universal optima.Comment: 41 pages, 12 figures, to appear in Experimental Mathematic
Three-point bounds for energy minimization
Three-point semidefinite programming bounds are one of the most powerful
known tools for bounding the size of spherical codes. In this paper, we use
them to prove lower bounds for the potential energy of particles interacting
via a pair potential function. We show that our bounds are sharp for seven
points in RP^2. Specifically, we prove that the seven lines connecting opposite
vertices of a cube and of its dual octahedron are universally optimal. (In
other words, among all configurations of seven lines through the origin, this
one minimizes energy for all potential functions that are completely monotonic
functions of squared chordal distance.) This configuration is the only known
universal optimum that is not distance regular, and the last remaining
universal optimum in RP^2. We also give a new derivation of semidefinite
programming bounds and present several surprising conjectures about them.Comment: 30 page
Symmetric Informationally Complete Quantum Measurements
We consider the existence in arbitrary finite dimensions d of a POVM
comprised of d^2 rank-one operators all of whose operator inner products are
equal. Such a set is called a ``symmetric, informationally complete'' POVM
(SIC-POVM) and is equivalent to a set of d^2 equiangular lines in C^d.
SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and
foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions
two, three, and four. We further conjecture that a particular kind of
group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical
results up to dimension 45 to bolster this claim.Comment: 8 page
Lecture notes: Semidefinite programs and harmonic analysis
Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th
International Workshop on High Performance Optimization Techniques (Algebraic
Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg
University, The Netherlands.Comment: 31 page
Complex spherical codes with two inner products
A finite set in a complex sphere is called a complex spherical -code
if the number of inner products between two distinct vectors in is equal to
. In this paper, we characterize the tight complex spherical -codes by
doubly regular tournaments, or skew Hadamard matrices. We also give certain
maximal 2-codes relating to skew-symmetric -optimal designs. To prove them,
we show the smallest embedding dimension of a tournament into a complex sphere
by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel
matrix.Comment: 10 pages, to appear in European Journal of Combinatoric
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