583,194 research outputs found
Qubit-Qutrit Separability-Probability Ratios
Paralleling our recent computationally-intensive (quasi-Monte Carlo) work for
the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to
high numerical accuracy, the formulas of Sommers and Zyczkowski
(quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional
hyperarea of the (separable and nonseparable) N x N density matrices, based on
the Bures (minimal monotone) metric -- and also their analogous formulas
(quant-ph/0302197) for the (non-monotone) Hilbert-Schmidt metric. With the same
seven billion well-distributed (``low-discrepancy'') sample points, we estimate
the unknown volumes and hyperareas based on five additional (monotone) metrics
of interest, including the Kubo-Mori and Wigner-Yanase. Further, we estimate
all of these seven volume and seven hyperarea (unknown) quantities when
restricted to the separable density matrices. The ratios of separable volumes
(hyperareas) to separable plus nonseparable volumes (hyperareas) yield
estimates of the separability probabilities of generically rank-six (rank-five)
density matrices. The (rank-six) separability probabilities obtained based on
the 35-dimensional volumes appear to be -- independently of the metric (each of
the seven inducing Haar measure) employed -- twice as large as those (rank-five
ones) based on the 34-dimensional hyperareas. Accepting such a relationship, we
fit exact formulas to the estimates of the Bures and Hilbert-Schmidt separable
volumes and hyperareas.(An additional estimate -- 33.9982 -- of the ratio of
the rank-6 Hilbert-Schmidt separability probability to the rank-4 one is quite
clearly close to integral too.) The doubling relationship also appears to hold
for the N=4 case for the Hilbert-Schmidt metric, but not the others. We fit
exact formulas for the Hilbert-Schmidt separable volumes and hyperareas.Comment: 36 pages, 15 figures, 11 tables, final PRA version, new last
paragraph presenting qubit-qutrit probability ratios disaggregated by the two
distinct forms of partial transpositio
OPENMENDEL: A Cooperative Programming Project for Statistical Genetics
Statistical methods for genomewide association studies (GWAS) continue to
improve. However, the increasing volume and variety of genetic and genomic data
make computational speed and ease of data manipulation mandatory in future
software. In our view, a collaborative effort of statistical geneticists is
required to develop open source software targeted to genetic epidemiology. Our
attempt to meet this need is called the OPENMENDELproject
(https://openmendel.github.io). It aims to (1) enable interactive and
reproducible analyses with informative intermediate results, (2) scale to big
data analytics, (3) embrace parallel and distributed computing, (4) adapt to
rapid hardware evolution, (5) allow cloud computing, (6) allow integration of
varied genetic data types, and (7) foster easy communication between
clinicians, geneticists, statisticians, and computer scientists. This article
reviews and makes recommendations to the genetic epidemiology community in the
context of the OPENMENDEL project.Comment: 16 pages, 2 figures, 2 table
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