561 research outputs found
Metric Entropy of Homogeneous Spaces
For a (compact) subset of a metric space and , the {\em
covering number} is defined as the smallest number of
balls of radius whose union covers . Knowledge of the {\em
metric entropy}, i.e., the asymptotic behaviour of covering numbers for
(families of) metric spaces is important in many areas of mathematics
(geometry, functional analysis, probability, coding theory, to name a few). In
this paper we give asymptotically correct estimates for covering numbers for a
large class of homogeneous spaces of unitary (or orthogonal) groups with
respect to some natural metrics, most notably the one induced by the operator
norm. This generalizes earlier author's results concerning covering numbers of
Grassmann manifolds; the generalization is motivated by applications to
noncommutative probability and operator algebras. In the process we give a
characterization of geodesics in (or ) for a class of
non-Riemannian metric structures
How often is a random quantum state k-entangled?
The set of trace preserving, positive maps acting on density matrices of size
d forms a convex body. We investigate its nested subsets consisting of
k-positive maps, where k=2,...,d. Working with the measure induced by the
Hilbert-Schmidt distance we derive asymptotically tight bounds for the volumes
of these sets. Our results strongly suggest that the inner set of
(k+1)-positive maps forms a small fraction of the outer set of k-positive maps.
These results are related to analogous bounds for the relative volume of the
sets of k-entangled states describing a bipartite d X d system.Comment: 19 pages in latex, 1 figure include
Confidence regions for means of multivariate normal distributions and a non-symmetric correlation inequality for gaussian measure
Let be a Gaussian measure (say, on ) and let be such that K is convex, is a "layer" (i.e. for some , and ) and the
centers of mass (with respect to ) of and coincide. Then . This is motivated by the well-known
"positive correlation conjecture" for symmetric sets and a related inequality
of Sidak concerning confidence regions for means of multivariate normal
distributions. The proof uses an apparently hitherto unknown estimate for the
(standard) Gaussian cumulative distribution function: (valid for )
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