5,386 research outputs found
Further Results on Geometric Properties of a Family of Relative Entropies
This paper extends some geometric properties of a one-parameter family of
relative entropies. These arise as redundancies when cumulants of compressed
lengths are considered instead of expected compressed lengths. These parametric
relative entropies are a generalization of the Kullback-Leibler divergence.
They satisfy the Pythagorean property and behave like squared distances. This
property, which was known for finite alphabet spaces, is now extended for
general measure spaces. Existence of projections onto convex and certain closed
sets is also established. Our results may have applications in the R\'enyi
entropy maximization rule of statistical physics.Comment: 7 pages, Prop. 5 modified, in Proceedings of the 2011 IEEE
International Symposium on Information Theor
Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions
The geometric mean is shown to be an appropriate statistic for the scale of a
heavy-tailed coupled Gaussian distribution or equivalently the Student's t
distribution. The coupled Gaussian is a member of a family of distributions
parameterized by the nonlinear statistical coupling which is the reciprocal of
the degree of freedom and is proportional to fluctuations in the inverse scale
of the Gaussian. Existing estimators of the scale of the coupled Gaussian have
relied on estimates of the full distribution, and they suffer from problems
related to outliers in heavy-tailed distributions. In this paper, the scale of
a coupled Gaussian is proven to be equal to the product of the generalized mean
and the square root of the coupling. From our numerical computations of the
scales of coupled Gaussians using the generalized mean of random samples, it is
indicated that only samples from a Cauchy distribution (with coupling parameter
one) form an unbiased estimate with diminishing variance for large samples.
Nevertheless, we also prove that the scale is a function of the geometric mean,
the coupling term and a harmonic number. Numerical experiments show that this
estimator is unbiased with diminishing variance for large samples for a broad
range of coupling values.Comment: 17 pages, 5 figure
A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the - Family
The so-called -z-\textit{R\'enyi Relative Entropies} provide a huge
two-parameter family of relative entropies which includes almost all well-known
examples of quantum relative entropies for suitable values of the parameters.
In this paper we consider a log-regularized version of this family and use it
as a family of potential functions to generate covariant symmetric
tensors on the space of invertible quantum states in finite dimensions. The
geometric formalism developed here allows us to obtain the explicit expressions
of such tensor fields in terms of a basis of globally defined differential
forms on a suitable unfolding space without the need to introduce a specific
set of coordinates. To make the reader acquainted with the intrinsic formalism
introduced, we first perform the computation for the qubit case, and then, we
extend the computation of the metric-like tensors to a generic -level
system. By suitably varying the parameters and , we are able to recover
well-known examples of quantum metric tensors that, in our treatment, appear
written in terms of globally defined geometrical objects that do not depend on
the coordinates system used. In particular, we obtain a coordinate-free
expression for the von Neumann-Umegaki metric, for the Bures metric and for the
Wigner-Yanase metric in the arbitrary -level case.Comment: 50 pages, 1 figur
Entropy: The Markov Ordering Approach
The focus of this article is on entropy and Markov processes. We study the
properties of functionals which are invariant with respect to monotonic
transformations and analyze two invariant "additivity" properties: (i)
existence of a monotonic transformation which makes the functional additive
with respect to the joining of independent systems and (ii) existence of a
monotonic transformation which makes the functional additive with respect to
the partitioning of the space of states. All Lyapunov functionals for Markov
chains which have properties (i) and (ii) are derived. We describe the most
general ordering of the distribution space, with respect to which all
continuous-time Markov processes are monotonic (the {\em Markov order}). The
solution differs significantly from the ordering given by the inequality of
entropy growth. For inference, this approach results in a convex compact set of
conditionally "most random" distributions.Comment: 50 pages, 4 figures, Postprint version. More detailed discussion of
the various entropy additivity properties and separation of variables for
independent subsystems in MaxEnt problem is added in Section 4.2.
Bibliography is extende
Universal quantum computation with little entanglement
We show that universal quantum computation can be achieved in the standard
pure-state circuit model while, at any time, the entanglement entropy of all
bipartitions is small---even tending to zero with growing system size. The
result is obtained by showing that a quantum computer operating within a small
region around the set of unentangled states still has universal computational
power, and by using continuity of entanglement entropy. In fact an analogous
conclusion applies to every entanglement measure which is continuous in a
certain natural sense, which amounts to a large class. Other examples include
the geometric measure, localizable entanglement, smooth epsilon-measures,
multipartite concurrence, squashed entanglement, and several others. We discuss
implications of these results for the believed role of entanglement as a key
necessary resource for quantum speed-ups
Holographic Entanglement Entropy
We review the developments in the past decade on holographic entanglement
entropy, a subject that has garnered much attention owing to its potential to
teach us about the emergence of spacetime in holography. We provide an
introduction to the concept of entanglement entropy in quantum field theories,
review the holographic proposals for computing the same, providing some
justification for where these proposals arise from in the first two parts. The
final part addresses recent developments linking entanglement and geometry. We
provide an overview of the various arguments and technical developments that
teach us how to use field theory entanglement to detect geometry. Our
discussion is by design eclectic; we have chosen to focus on developments that
appear to us most promising for further insights into the holographic map.
This is a draft of a few chapters of a book which will appear sometime in the
near future, to be published by Springer. The book in addition contains a
discussion of application of holographic ideas to computation of entanglement
entropy in strongly coupled field theories, and discussion of tensor networks
and holography, which we have chosen to exclude from the current manuscript.Comment: 154 pages. many figures. preliminary version of book chapters.
comments welcome. v2: typos fixed and references adde
Entropies from coarse-graining: convex polytopes vs. ellipsoids
We examine the Boltzmann/Gibbs/Shannon and the
non-additive Havrda-Charv\'{a}t / Dar\'{o}czy/Cressie-Read/Tsallis \
\ and the Kaniadakis -entropy \ \
from the viewpoint of coarse-graining, symplectic capacities and convexity. We
argue that the functional form of such entropies can be ascribed to a
discordance in phase-space coarse-graining between two generally different
approaches: the Euclidean/Riemannian metric one that reflects independence and
picks cubes as the fundamental cells and the symplectic/canonical one that
picks spheres/ellipsoids for this role. Our discussion is motivated by and
confined to the behaviour of Hamiltonian systems of many degrees of freedom. We
see that Dvoretzky's theorem provides asymptotic estimates for the minimal
dimension beyond which these two approaches are close to each other. We state
and speculate about the role that dualities may play in this viewpoint.Comment: 63 pages. No figures. Standard LaTe
- …