12,919 research outputs found
From Random Matrix Theory to Coding Theory : Volume of a Metric Ball in Unitary Group
Volume estimates of metric balls in manifolds find diverse applications in information and coding theory. In this paper, new results for the volume of a metric ball in unitary group are derived via tools from random matrix theory. The first result is an integral representation of the exact volume, which involves a Toeplitz determinant of Bessel functions. A simple but accurate limiting volume formula is then obtained by invoking Szego's strong limit theorem for large Toeplitz matrices. The derived asymptotic volume formula enables analytical evaluation of some coding-theoretic bounds of unitary codes. In particular, the Gilbert-Varshamov lower bound and the Hamming upper bound on the cardinality as well as the resulting bounds on code rate and minimum distance are derived. Moreover, bounds on the scaling law of code rate are found. Finally, a closed-form bound on the diversity sum relevant to unitary space-time codes is obtained, which was only computed numerically in the literature.Peer reviewe
Density of Spherically-Embedded Stiefel and Grassmann Codes
The density of a code is the fraction of the coding space covered by packing
balls centered around the codewords. This paper investigates the density of
codes in the complex Stiefel and Grassmann manifolds equipped with the chordal
distance. The choice of distance enables the treatment of the manifolds as
subspaces of Euclidean hyperspheres. In this geometry, the densest packings are
not necessarily equivalent to maximum-minimum-distance codes. Computing a
code's density follows from computing: i) the normalized volume of a metric
ball and ii) the kissing radius, the radius of the largest balls one can pack
around the codewords without overlapping. First, the normalized volume of a
metric ball is evaluated by asymptotic approximations. The volume of a small
ball can be well-approximated by the volume of a locally-equivalent tangential
ball. In order to properly normalize this approximation, the precise volumes of
the manifolds induced by their spherical embedding are computed. For larger
balls, a hyperspherical cap approximation is used, which is justified by a
volume comparison theorem showing that the normalized volume of a ball in the
Stiefel or Grassmann manifold is asymptotically equal to the normalized volume
of a ball in its embedding sphere as the dimension grows to infinity. Then,
bounds on the kissing radius are derived alongside corresponding bounds on the
density. Unlike spherical codes or codes in flat spaces, the kissing radius of
Grassmann or Stiefel codes cannot be exactly determined from its minimum
distance. It is nonetheless possible to derive bounds on density as functions
of the minimum distance. Stiefel and Grassmann codes have larger density than
their image spherical codes when dimensions tend to infinity. Finally, the
bounds on density lead to refinements of the standard Hamming bounds for
Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE
Transactions on Information Theor
Hilbert--Schmidt volume of the set of mixed quantum states
We compute the volume of the convex N^2-1 dimensional set M_N of density
matrices of size N with respect to the Hilbert-Schmidt measure. The hyper--area
of the boundary of this set is also found and its ratio to the volume provides
an information about the complex structure of M_N. Similar investigations are
also performed for the smaller set of all real density matrices. As an
intermediate step we analyze volumes of the unitary and orthogonal groups and
of the flag manifolds.Comment: 13 revtex pages, ver 3: minor improvement
Convex bodies of states and maps
We give a general solution to the question when the convex hulls of orbits of
quantum states on a finite-dimensional Hilbert space under unitary actions of a
compact group have a non-empty interior in the surrounding space of all density
states. The same approach can be applied to study convex combinations of
quantum channels. The importance of both problems stems from the fact that,
usually, only sets with non-vanishing volumes in the embedding spaces of all
states or channels are of practical importance. For the group of local
transformations on a bipartite system we characterize maximally entangled
states by properties of a convex hull of orbits through them. We also compare
two partial characteristics of convex bodies in terms of largest balls and
maximum volume ellipsoids contained in them and show that, in general, they do
not coincide. Separable states, mixed-unitary channels and k-entangled states
are also considered as examples of our techniques.Comment: 18 pages, 1 figur
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
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