24,046 research outputs found
Entanglement properties of bound and resonant few-body states
Studying the physics of quantum correlations has gained new interest after it
has become possible to measure entanglement entropies of few body systems in
experiments with ultracold atomic gases. Apart from investigating trapped atom
systems, research on correlation effects in other artificially fabricated
few-body systems, such as quantum dots or electromagnetically trapped ions, is
currently underway or in planning. Generally, the systems studied in these
experiments may be considered as composed of a small number of interacting
elements with controllable and highly tunable parameters, effectively described
by Schr\"odinger equation. In this way, parallel theoretical and experimental
studies of few-body models become possible, which may provide a deeper
understanding of correlation effects and give hints for designing and
controlling new experiments. Of particular interest is to explore the physics
in the strongly correlated regime and in the neighborhood of critical points.
Particle correlations in nanostructures may be characterized by their
entanglement spectrum, i.e. the eigenvalues of the reduced density matrix of
the system partitioned into two subsystems. We will discuss how to determine
the entropy of entanglement spectrum of few-body systems in bound and resonant
states within the same formalism. The linear entropy will be calculated for a
model of quasi-one dimensional Gaussian quantum dot in the lowest energy
states. We will study how the entanglement depends on the parameters of the
system, paying particular attention to the behavior on the border between the
regimes of bound and resonant states.Comment: 22 pages, 3 figure
Formal Groups and -Entropies
We shall prove that the celebrated R\'enyi entropy is the first example of a
new family of infinitely many multi-parametric entropies. We shall call them
the -entropies. Each of them, under suitable hypotheses, generalizes the
celebrated entropies of Boltzmann and R\'enyi.
A crucial aspect is that every -entropy is composable [1]. This property
means that the entropy of a system which is composed of two or more independent
systems depends, in all the associated probability space, on the choice of the
two systems only. Further properties are also required, to describe the
composition process in terms of a group law.
The composability axiom, introduced as a generalization of the fourth
Shannon-Khinchin axiom (postulating additivity), is a highly non-trivial
requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis
entropy are the only known composable cases. However, in the non-trace form
class, the -entropies arise as new entropic functions possessing the
mathematical properties necessary for information-theoretical applications, in
both classical and quantum contexts.
From a mathematical point of view, composability is intimately related to
formal group theory of algebraic topology. The underlying group-theoretical
structure determines crucially the statistical properties of the corresponding
entropies.Comment: 20 pages, no figure
Generalized (c,d)-entropy and aging random walks
Complex systems are often inherently non-ergodic and non-Markovian for which
Shannon entropy loses its applicability. In particular accelerating,
path-dependent, and aging random walks offer an intuitive picture for these
non-ergodic and non-Markovian systems. It was shown that the entropy of
non-ergodic systems can still be derived from three of the Shannon-Khinchin
axioms, and by violating the fourth -- the so-called composition axiom. The
corresponding entropy is of the form and depends on two system-specific scaling exponents, and . This
entropy contains many recently proposed entropy functionals as special cases,
including Shannon and Tsallis entropy. It was shown that this entropy is
relevant for a special class of non-Markovian random walks. In this work we
generalize these walks to a much wider class of stochastic systems that can be
characterized as `aging' systems. These are systems whose transition rates
between states are path- and time-dependent. We show that for particular aging
walks is again the correct extensive entropy. Before the central part
of the paper we review the concept of -entropy in a self-contained way.Comment: 8 pages, 5 eps figures. arXiv admin note: substantial text overlap
with arXiv:1104.207
Methods for calculating nonconcave entropies
Five different methods which can be used to analytically calculate entropies
that are nonconcave as functions of the energy in the thermodynamic limit are
discussed and compared. The five methods are based on the following ideas and
techniques: i) microcanonical contraction, ii) metastable branches of the free
energy, iii) generalized canonical ensembles with specific illustrations
involving the so-called Gaussian and Betrag ensembles, iv) restricted canonical
ensemble, and v) inverse Laplace transform. A simple long-range spin model
having a nonconcave entropy is used to illustrate each method.Comment: v1: 22 pages, IOP style, 7 color figures, contribution for the JSTAT
special issue on Long-range interacting systems. v2: Open problem and
references added, minor typos corrected, close to published versio
Beyond the Shannon-Khinchin Formulation: The Composability Axiom and the Universal Group Entropy
The notion of entropy is ubiquitous both in natural and social sciences. In
the last two decades, a considerable effort has been devoted to the study of
new entropic forms, which generalize the standard Boltzmann-Gibbs (BG) entropy
and are widely applicable in thermodynamics, quantum mechanics and information
theory. In [23], by extending previous ideas of Shannon [38], [39], Khinchin
proposed an axiomatic definition of the BG entropy, based on four requirements,
nowadays known as the Shannon-Khinchin (SK) axioms.
The purpose of this paper is twofold. First, we show that there exists an
intrinsic group-theoretical structure behind the notion of entropy. It comes
from the requirement of composability of an entropy with respect to the union
of two statistically independent subsystems, that we propose in an axiomatic
formulation. Second, we show that there exists a simple universal class of
admissible entropies. This class contains many well known examples of entropies
and infinitely many new ones, a priori multi-parametric. Due to its specific
relation with the universal formal group, the new family of entropies
introduced in this work will be called the universal-group entropy. A new
example of multi-parametric entropy is explicitly constructed.Comment: Extended version; 25 page
Entropic Uncertainty Relations in Quantum Physics
Uncertainty relations have become the trademark of quantum theory since they
were formulated by Bohr and Heisenberg. This review covers various
generalizations and extensions of the uncertainty relations in quantum theory
that involve the R\'enyi and the Shannon entropies. The advantages of these
entropic uncertainty relations are pointed out and their more direct connection
to the observed phenomena is emphasized. Several remaining open problems are
mentionedComment: 35 pages, review pape
Group entropies, correlation laws and zeta functions
The notion of group entropy is proposed. It enables to unify and generalize
many different definitions of entropy known in the literature, as those of
Boltzmann-Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals
are presented, related to nontrivial correlation laws characterizing
universality classes of systems out of equilibrium, when the dynamics is weakly
chaotic. The associated thermostatistics are discussed. The mathematical
structure underlying our construction is that of formal group theory, which
provides the general structure of the correlations among particles and dictates
the associated entropic functionals. As an example of application, the role of
group entropies in information theory is illustrated and generalizations of the
Kullback-Leibler divergence are proposed. A new connection between statistical
mechanics and zeta functions is established. In particular, Tsallis entropy is
related to the classical Riemann zeta function.Comment: to appear in Physical Review
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