7,194 research outputs found
Resolvable designs with large blocks
Resolvable designs with two blocks per replicate are studied from an
optimality perspective. Because in practice the number of replicates is
typically less than the number of treatments, arguments can be based on the
dual of the information matrix and consequently given in terms of block
concurrences. Equalizing block concurrences for given block sizes is often, but
not always, the best strategy. Sufficient conditions are established for
various strong optimalities and a detailed study of E-optimality is offered,
including a characterization of the E-optimal class. Optimal designs are found
to correspond to balanced arrays and an affine-like generalization.Comment: Published at http://dx.doi.org/10.1214/009053606000001253 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Design Lines
The two basic equations satisfied by the parameters of a block design define
a three-dimensional affine variety in . A point
of that is not in some sense trivial lies on four lines lying in
. These lines provide a degree of organization for certain general
classes of designs, and the paper is devoted to exploring properties of the
lines. Several examples of families of designs that seem naturally to follow
the lines are presented.Comment: 16 page
Entanglement-assisted quantum low-density parity-check codes
This paper develops a general method for constructing entanglement-assisted
quantum low-density parity-check (LDPC) codes, which is based on combinatorial
design theory. Explicit constructions are given for entanglement-assisted
quantum error-correcting codes (EAQECCs) with many desirable properties. These
properties include the requirement of only one initial entanglement bit, high
error correction performance, high rates, and low decoding complexity. The
proposed method produces infinitely many new codes with a wide variety of
parameters and entanglement requirements. Our framework encompasses various
codes including the previously known entanglement-assisted quantum LDPC codes
having the best error correction performance and many new codes with better
block error rates in simulations over the depolarizing channel. We also
determine important parameters of several well-known classes of quantum and
classical LDPC codes for previously unsettled cases.Comment: 20 pages, 5 figures. Final version appearing in Physical Review
On the structure of the directions not determined by a large affine point set
Given a point set in an -dimensional affine space of size
, we obtain information on the structure of the set of
directions that are not determined by , and we describe an application in
the theory of partial ovoids of certain partial geometries
The curious nonexistence of Gaussian 2-designs
2-designs -- ensembles of quantum pure states whose 2nd moments equal those
of the uniform Haar ensemble -- are optimal solutions for several tasks in
quantum information science, especially state and process tomography. We show
that Gaussian states cannot form a 2-design for the continuous-variable
(quantum optical) Hilbert space L2(R). This is surprising because the affine
symplectic group HWSp (the natural symmetry group of Gaussian states) is
irreducible on the symmetric subspace of two copies. In finite dimensional
Hilbert spaces, irreducibility guarantees that HWSp-covariant ensembles (such
as mutually unbiased bases in prime dimensions) are always 2-designs. This
property is violated by continuous variables, for a subtle reason: the
(well-defined) HWSp-invariant ensemble of Gaussian states does not have an
average state because the averaging integral does not converge. In fact, no
Gaussian ensemble is even close (in a precise sense) to being a 2-design. This
surprising difference between discrete and continuous quantum mechanics has
important implications for optical state and process tomography.Comment: 9 pages, no pretty figures (sorry!
- …