83 research outputs found
Information theoretical properties of Tsallis entropies
A chain rule and a subadditivity for the entropy of type , which is
one of the nonadditive entropies, were derived by Z.Dar\'oczy. In this paper,
we study the further relations among Tsallis type entropies which are typical
nonadditive entropies. The chain rule is generalized by showing it for Tsallis
relative entropy and the nonadditive entropy. We show some inequalities related
to Tsallis entropies, especially the strong subadditivity for Tsallis type
entropies and the subadditivity for the nonadditive entropies. The
subadditivity and the strong subadditivity naturally lead to define Tsallis
mutual entropy and Tsallis conditional mutual entropy, respectively, and then
we show again chain rules for Tsallis mutual entropies. We give properties of
entropic distances in terms of Tsallis entropies. Finally we show
parametrically extended results based on information theory.Comment: The subsection on data processing inequality was deleted. Some typo's
were modifie
Entropic uncertainty relations and entanglement
We discuss the relationship between entropic uncertainty relations and
entanglement. We present two methods for deriving separability criteria in
terms of entropic uncertainty relations. Especially we show how any entropic
uncertainty relation on one part of the system results in a separability
condition on the composite system. We investigate the resulting criteria using
the Tsallis entropy for two and three qubits.Comment: 8 pages, 3 figures, v2: small change
Abstract composition rule for relativistic kinetic energy in the thermodynamical limit
We demonstrate by simple mathematical considerations that a power-law tailed
distribution in the kinetic energy of relativistic particles can be a limiting
distribution seen in relativistic heavy ion experiments. We prove that the
infinite repetition of an arbitrary composition rule on an infinitesimal amount
leads to a rule with a formal logarithm. As a consequence the stationary
distribution of energy in the thermodynamical limit follows the composed
function of the Boltzmann-Gibbs exponential with this formal logarithm. In
particular, interactions described as solely functions of the relative
four-momentum squared lead to kinetic energy distributions of the
Tsallis-Pareto (cut power-law) form in the high energy limit.Comment: Submitted to Europhysics Letters. LaTeX, 3 eps figure
Analysis of Velocity Derivatives in Turbulence based on Generalized Statistics
A theoretical formula for the probability density function (PDF) of velocity
derivatives in a fully developed turbulent flow is derived with the
multifractal aspect based on the generalized measures of entropy, i.e., the
extensive Renyi entropy or the non-extensive Tsallis entropy, and is used,
successfully, to analyze the PDF's observed in the direct numerical simulation
(DNS) conducted by Gotoh et al.. The minimum length scale r_d/eta in the
longitudinal (transverse) inertial range of the DNS is estimated to be
r_d^L/eta = 1.716 (r_d^T/eta = 2.180) in the unit of the Kolmogorov scale eta.Comment: 6 pages, 1 figur
Divergence Measure Between Chaotic Attractors
We propose a measure of divergence of probability distributions for
quantifying the dissimilarity of two chaotic attractors. This measure is
defined in terms of a generalized entropy. We illustrate our procedure by
considering the effect of additive noise in the well known H\'enon attractor.
Comparison of two H\'enon attractors for slighly different parameter values,
has shown that the divergence has complex scaling structure. Finally, we show
how our approach allows to detect non-stationary events in a time series.Comment: 9 pages, 6 figure
Entropic uncertainty relations for extremal unravelings of super-operators
A way to pose the entropic uncertainty principle for trace-preserving
super-operators is presented. It is based on the notion of extremal unraveling
of a super-operator. For given input state, different effects of each
unraveling result in some probability distribution at the output. As it is
shown, all Tsallis' entropies of positive order as well as some of Renyi's
entropies of this distribution are minimized by the same unraveling of a
super-operator. Entropic relations between a state ensemble and the generated
density matrix are revisited in terms of both the adopted measures. Using
Riesz's theorem, we obtain two uncertainty relations for any pair of
generalized resolutions of the identity in terms of the Renyi and Tsallis
entropies. The inequality with Renyi's entropies is an improvement of the
previous one, whereas the inequality with Tsallis' entropies is a new relation
of a general form. The latter formulation is explicitly shown for a pair of
complementary observables in a -level system and for the angle and the
angular momentum. The derived general relations are immediately applied to
extremal unravelings of two super-operators.Comment: 8 pages, one figure. More explanations are given for Eq. (2.19) and
Example III.5. One reference is adde
Properties of Classical and Quantum Jensen-Shannon Divergence
Jensen-Shannon divergence (JD) is a symmetrized and smoothed version of the
most important divergence measure of information theory, Kullback divergence.
As opposed to Kullback divergence it determines in a very direct way a metric;
indeed, it is the square of a metric. We consider a family of divergence
measures (JD_alpha for alpha>0), the Jensen divergences of order alpha, which
generalize JD as JD_1=JD. Using a result of Schoenberg, we prove that JD_alpha
is the square of a metric for alpha lies in the interval (0,2], and that the
resulting metric space of probability distributions can be isometrically
embedded in a real Hilbert space. Quantum Jensen-Shannon divergence (QJD) is a
symmetrized and smoothed version of quantum relative entropy and can be
extended to a family of quantum Jensen divergences of order alpha (QJD_alpha).
We strengthen results by Lamberti et al. by proving that for qubits and pure
states, QJD_alpha^1/2 is a metric space which can be isometrically embedded in
a real Hilbert space when alpha lies in the interval (0,2]. In analogy with
Burbea and Rao's generalization of JD, we also define general QJD by
associating a Jensen-type quantity to any weighted family of states.
Appropriate interpretations of quantities introduced are discussed and bounds
are derived in terms of the total variation and trace distance.Comment: 13 pages, LaTeX, expanded contents, added references and corrected
typo
Analysis of Velocity Fluctuation in Turbulence based on Generalized Statistics
The numerical experiments of turbulence conducted by Gotoh et al. are
analyzed precisely with the help of the formulae for the scaling exponents of
velocity structure function and for the probability density function (PDF) of
velocity fluctuations. These formulae are derived by the present authors with
the multifractal aspect based on the statistics that are constructed on the
generalized measures of entropy, i.e., the extensive R\'{e}nyi's or the
non-extensive Tsallis' entropy. It is revealed that there exist two scaling
regions separated by a crossover length, i.e., a definite length approximately
of the order of the Taylor microscale. It indicates that the multifractal
distribution of singularities in velocity gradient in turbulent flow is robust
enough to produce scaling behaviors even for the phenomena out side the
inertial range.Comment: 10 Pages, 5 figure
Improvement of the Heisenberg and Fisher-information-based uncertainty relations for D-dimensional central potentials
The Heisenberg and Fisher-information-based uncertainty relations are improved for stationary states of single-particle systems in a D-dimensional central potential. The improvement increases with the squared orbital hyperangular quantum number. The new uncertainty relations saturate for the isotropic harmonic oscillator wavefunction.We are very grateful for partial support to Junta de AndalucÃa (under the grants FQM-
0207 and FQM-481), Ministerio de Educaci´on y Ciencia (under the project FIS2005-00973),
and the European Research Network NeCCA (under the project INTAS-03-51-6637). RGF
acknowledges the support of Junta de AndalucÃa under the program of Retorno de Investigadores a Centros de Investigación Andaluces
Information Invariance and Quantum Probabilities
We consider probabilistic theories in which the most elementary system, a
two-dimensional system, contains one bit of information. The bit is assumed to
be contained in any complete set of mutually complementary measurements. The
requirement of invariance of the information under a continuous change of the
set of mutually complementary measurements uniquely singles out a measure of
information, which is quadratic in probabilities. The assumption which gives
the same scaling of the number of degrees of freedom with the dimension as in
quantum theory follows essentially from the assumption that all physical states
of a higher dimensional system are those and only those from which one can
post-select physical states of two-dimensional systems. The requirement that no
more than one bit of information (as quantified by the quadratic measure) is
contained in all possible post-selected two-dimensional systems is equivalent
to the positivity of density operator in quantum theory.Comment: 8 pages, 1 figure. This article is dedicated to Pekka Lahti on the
occasion of his 60th birthday. Found. Phys. (2009
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