882 research outputs found
R\'enyi Entropy Power Inequalities via Normal Transport and Rotation
Following a recent proof of Shannon's entropy power inequality (EPI), a
comprehensive framework for deriving various EPIs for the R\'enyi entropy is
presented that uses transport arguments from normal densities and a change of
variable by rotation. Simple arguments are given to recover the previously
known R\'enyi EPIs and derive new ones, by unifying a multiplicative form with
constant c and a modification with exponent {\alpha} of previous works. In
particular, for log-concave densities, we obtain a simple transportation proof
of a sharp varentropy bound.Comment: 17 page. Entropy Journal, to appea
The information-theoretic meaning of Gagliardo--Nirenberg type inequalities
Gagliardo--Nirenberg inequalities are interpolation inequalities which were
proved independently by Gagliardo and Nirenberg in the late fifties. In recent
years, their connections with theoretic aspects of information theory and
nonlinear diffusion equations allowed to obtain some of them in optimal form,
by recovering both the sharp constants and the explicit form of the optimizers.
In this note, at the light of these recent researches, we review the main
connections between Shannon-type entropies, diffusion equations and a class of
these inequalities
On some interrelations of generalized -entropies and a generalized Fisher information, including a Cram\'er-Rao inequality
In this communication, we describe some interrelations between generalized
-entropies and a generalized version of Fisher information. In information
theory, the de Bruijn identity links the Fisher information and the derivative
of the entropy. We show that this identity can be extended to generalized
versions of entropy and Fisher information. More precisely, a generalized
Fisher information naturally pops up in the expression of the derivative of the
Tsallis entropy. This generalized Fisher information also appears as a special
case of a generalized Fisher information for estimation problems. Indeed, we
derive here a new Cram\'er-Rao inequality for the estimation of a parameter,
which involves a generalized form of Fisher information. This generalized
Fisher information reduces to the standard Fisher information as a particular
case. In the case of a translation parameter, the general Cram\'er-Rao
inequality leads to an inequality for distributions which is saturated by
generalized -Gaussian distributions. These generalized -Gaussians are
important in several areas of physics and mathematics. They are known to
maximize the -entropies subject to a moment constraint. The Cram\'er-Rao
inequality shows that the generalized -Gaussians also minimize the
generalized Fisher information among distributions with a fixed moment.
Similarly, the generalized -Gaussians also minimize the generalized Fisher
information among distributions with a given -entropy
Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems
The effect of quantum steering describes a possible action at a distance via
local measurements. Whereas many attempts on characterizing steerability have
been pursued, answering the question as to whether a given state is steerable
or not remains a difficult task. Here, we investigate the applicability of a
recently proposed method for building steering criteria from generalized
entropic uncertainty relations. This method works for any entropy which satisfy
the properties of (i) (pseudo-) additivity for independent distributions; (ii)
state independent entropic uncertainty relation (EUR); and (iii) joint
convexity of a corresponding relative entropy. Our study extends the former
analysis to Tsallis and R\'enyi entropies on bipartite and tripartite systems.
As examples, we investigate the steerability of the three-qubit GHZ and W
states.Comment: 27 pages, 8 figures. Published version. Title change
Unified entropic measures of quantum correlations induced by local measurements
We introduce quantum correlations measures based on the minimal change in
unified entropies induced by local rank-one projective measurements, divided by
a factor that depends on the generalized purity of the system in the case of
non-additive entropies. In this way, we overcome the issue of the artificial
increasing of the value of quantum correlations measures based on non-additive
entropies when an uncorrelated ancilla is appended to the system without
changing the computability of our entropic correlations measures with respect
to the previous ones. Moreover, we recover as limiting cases the quantum
correlations measures based on von Neumann and R\'enyi entropies (i.e.,
additive entropies), for which the adjustment factor becomes trivial. In
addition, we distinguish between total and semiquantum correlations and obtain
some relations between them. Finally, we obtain analytical expressions of the
entropic correlations measures for typical quantum bipartite systems.Comment: 10 pages, 1 figur
A simple probabilistic construction yielding generalized entropies and divergences, escort distributions and q-Gaussians
We give a simple probabilistic description of a transition between two states
which leads to a generalized escort distribution. When the parameter of the
distribution varies, it defines a parametric curve that we call an escort-path.
The R\'enyi divergence appears as a natural by-product of the setting. We study
the dynamics of the Fisher information on this path, and show in particular
that the thermodynamic divergence is proportional to Jeffreys' divergence.
Next, we consider the problem of inferring a distribution on the escort-path,
subject to generalized moments constraints. We show that our setting naturally
induces a rationale for the minimization of the R\'enyi information divergence.
Then, we derive the optimum distribution as a generalized q-Gaussian
distribution
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