13,327 research outputs found
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
Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
We investigate the set a) of positive, trace preserving maps acting on
density matrices of size N, and a sequence of its nested subsets: the sets of
maps which are b) decomposable, c) completely positive, d) extended by identity
impose positive partial transpose and e) are superpositive. Working with the
Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds
for the volumes of all five sets. A sample consequence is the fact that, as N
increases, a generic positive map becomes not decomposable and, a fortiori, not
completely positive.
Due to the Jamiolkowski isomorphism, the results obtained for quantum maps
are closely connected to similar relations between the volume of the set of
quantum states and the volumes of its subsets (such as states with positive
partial transpose or separable states) or supersets. Our approach depends on
systematic use of duality to derive quantitative estimates, and on various
tools of classical convexity, high-dimensional probability and geometry of
Banach spaces, some of which are not standard.Comment: 34 pages in Latex including 3 figures in eps, ver 2: minor revision
Online codes for analog signals
This paper revisits a classical scenario in communication theory: a waveform
sampled at regular intervals is to be encoded so as to minimize distortion in
its reconstruction, despite noise. This transformation must be online (causal),
to enable real-time signaling; and should use no more power than the original
signal. The noise model we consider is an "atomic norm" convex relaxation of
the standard (discrete alphabet) Hamming-weight-bounded model: namely,
adversarial -bounded. In the "block coding" (noncausal) setting, such
encoding is possible due to the existence of large almost-Euclidean sections in
spaces, a notion first studied in the work of Dvoretzky in 1961. Our
main result is that an analogous result is achievable even causally.
Equivalently, our work may be seen as a "lower triangular" version of
Dvoretzky theorems. In terms of communication, the guarantees are expressed in
terms of certain time-weighted norms: the time-weighted norm imposed
on the decoder forces increasingly accurate reconstruction of the distant past
signal, while the time-weighted norm on the noise ensures vanishing
interference from distant past noise. Encoding is linear (hence easy to
implement in analog hardware). Decoding is performed by an LP analogous to
those used in compressed sensing
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
Location of the spectrum of Kronecker random matrices
For a general class of large non-Hermitian random block matrices
we prove that there are no eigenvalues away from a deterministic set with very
high probability. This set is obtained from the Dyson equation of the
Hermitization of as the self-consistent approximation of the
pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation
from [arXiv:1604.08188v4] offers a unified treatment of many structured matrix
ensembles.Comment: 33 pages, 4 figures. Some assumptions in Section 3.1 and 3.2 relaxed.
Some typos corrected and references update
Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization
The l1/l2 ratio regularization function has shown good performance for
retrieving sparse signals in a number of recent works, in the context of blind
deconvolution. Indeed, it benefits from a scale invariance property much
desirable in the blind context. However, the l1/l2 function raises some
difficulties when solving the nonconvex and nonsmooth minimization problems
resulting from the use of such a penalty term in current restoration methods.
In this paper, we propose a new penalty based on a smooth approximation to the
l1/l2 function. In addition, we develop a proximal-based algorithm to solve
variational problems involving this function and we derive theoretical
convergence results. We demonstrate the effectiveness of our method through a
comparison with a recent alternating optimization strategy dealing with the
exact l1/l2 term, on an application to seismic data blind deconvolution.Comment: 5 page
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