46 research outputs found
Entropy in Dynamic Systems
In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed
Special Functions: Fractional Calculus and the Pathway for Entropy
Historically, the notion of entropy emerged in conceptually very distinct contexts. This book deals with the connection between entropy, probability, and fractional dynamics as they appeared, for example, in solar neutrino astrophysics since the 1970's (Mathai and Rathie 1975, Mathai and Pederzoli 1977, Mathai and Saxena 1978, Mathai, Saxena, and Haubold 2010). The original solar neutrino problem, experimentally and theoretically, was resolved through the discovery of neutrino oscillations and was recently enriched by neutrino entanglement entropy. To reconsider possible new physics of solar neutrinos, diffusion entropy analysis, utilizing Boltzmann entropy, and standard deviation analysis was undertaken with Super-Kamiokande solar neutrino data. This analysis revealed a non-Gaussian signal with harmonic content. The Hurst exponent is different from the scaling exponent of the probability density function and both Hurst exponent and scaling exponent of the Super-Kamiokande data deviate considerably from the value of Âœ, which indicates that the statistics of the underlying phenomenon is anomalous. Here experiment may provide guidance about the generalization of theory of Boltzmann statistical mechanics. Arguments in the so-called Boltzmann-Planck-Einstein discussion related to Planck's discovery of the black-body radiation law are recapitulated mathematically and statistically and emphasize from this discussion is pursued that a meaningful implementation of the complex âentropy-probability-dynamicsâ may offer two ways for explaining the results of diffusion entropy analysis and standard deviation analysis. One way is to consider an anomalous diffusion process that needs to use the fractional space-time diffusion equation (Gorenflo and Mainardi) and the other way is to consider a generalized Boltzmann entropy by assuming a power law probability density function. Here new mathematical framework, invented by sheer thought, may provide guidance for the generalization of Boltzmann statistical mechanics. In this book Boltzmann entropy, generalized by Tsallis and Mathai, is considered. The second one contains a varying parameter that is used to construct an entropic pathway covering generalized type-1 beta, type-2 beta, and gamma families of densities. Similarly, pathways for respective distributions and differential equations can be developed. Mathai's entropy is optimized under various conditions reproducing the well-known Boltzmann distribution, Raleigh distribution, and other distributions used in physics. Properties of the entropy measure for the generalized entropy are examined. In this process the role of special functions of mathematical physics, particularly the H-function, is highlighted
Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions
We present the main features of the mathematical theory generated by the Îș-deformed exponential function exp_Îș (x) with 0 †Îș < 1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The Îș-mathematics has its roots in special relativity and furnishes the theoretical foundations of the Îș-statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the Îș-algebra, we present the associated Îș-differential and Îș-integral calculus. Then, we obtain the corresponding Îș-exponential and Îș-logarithm functions and give the Îș-version of the main functions of the ordinary mathematics
Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels
This thesis combines two parallel research directions: an exploration into the
continuity properties of certain entropic quantities, and an investigation
into a simple class of physical systems whose time evolution
is given by the repeated application of a quantum channel.
In the first part of the thesis, we present a general technique for
establishing local and uniform continuity bounds for Schur concave functions;
that is, for real-valued functions which are decreasing in the majorization
pre-order. Continuity bounds provide a quantitative measure of robustness,
addressing the following question: If there is some uncertainty or error in
the input, how much uncertainty is there in the output? Our technique uses a
particular relationship between majorization and the trace distance between
quantum states (or total variation distance, in the case of probability
distributions). Namely, the majorization pre-order attains a maximum and a
minimum over Δ-balls in this distance. By tracing the path of the
majorization-minimizer as a function of the distance Δ, we obtain the
path of ``majorization flowââ. An analysis of the derivatives of Schur
concave functions along this path immediately yields tight continuity bounds
for such functions.
In this way, we find a new proof of the Audenaert-Fannes continuity bound for
the von Neumann entropy, and the necessary and sufficient conditions for its
saturation, in a universal framework which extends to the other functions,
including the RĂ©nyi and Tsallis entropies. In particular, we prove a novel
uniform continuity bound for the α-Rényi entropy with α > 1 with
much improved dependence on the dimension of the underlying system and the
parameter α compared to previously known bounds. We show that this
framework can also be used to provide continuity bounds for other Schur
concave functions, such as the number of connected components of a certain
random graph model as a function of the underlying probability distribution,
and the number of distinct realizations of a random variable in some fixed
number of independent trials as a function of the underlying probability mass
function. The former has been used in modeling the spread of epidemics, while
the latter has been studied in the context of estimating measures of
biodiversity from observations; in these contexts, our continuity bounds
provide quantitative estimates of robustness to noise or data collection
errors.
In the second part, we consider repeated interaction systems, in which a
system of interest interacts with a sequence of probes, i.e. environmental
systems, one at a time. The state of the system after each interaction is
related to the state of the system before the interaction by the so-called
reduced dynamics, which is described by the action of a quantum channel. When
each probe and the way it interacts with the system is identical, the reduced
dynamics at each step is identical. In this scenario, under the additional
assumption that the reduced dynamics satisfies a faithfulness property, we
characterize which repeated interaction systems break any initially-present
entanglement between the system and an untouched reference, after finitely
many steps. In this case, the reduced dynamics is said to be eventually
entanglement-breaking. This investigation helps improve our
understanding of which kinds of noisy time evolution destroy entanglement.
When the probes and their interactions with the system are slowly-varying
(i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality
between the entropy change of the system and the energy change of the probes,
in the limit in which the number of steps tends to infinity and both the
difference between consecutive probes and the difference between their
interactions vanishes. This analysis proceeds at a fine-grained level by means
of a two-time measurement protocol, in which the energy of the probes is
measured before and after each interaction. The quantities of interest are
then studied as random variables on the space of outcomes of the energy
measurements of the probes, providing a deeper insight into the interrelations
between energy and entropy in this setting.Cantab Capital Institute for the Mathematics of Informatio
Dynamical Systems
Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...
Highly symmetric POVMs and their informational power
We discuss the dependence of the Shannon entropy of normalized finite rank-1
POVMs on the choice of the input state, looking for the states that minimize
this quantity. To distinguish the class of measurements where the problem can
be solved analytically, we introduce the notion of highly symmetric POVMs and
classify them in dimension two (for qubits). In this case we prove that the
entropy is minimal, and hence the relative entropy (informational power) is
maximal, if and only if the input state is orthogonal to one of the states
constituting a POVM. The method used in the proof, employing the Michel theory
of critical points for group action, the Hermite interpolation and the
structure of invariant polynomials for unitary-antiunitary groups, can also be
applied in higher dimensions and for other entropy-like functions. The links
between entropy minimization and entropic uncertainty relations, the Wehrl
entropy and the quantum dynamical entropy are described.Comment: 40 pages, 3 figure
Applied Mathematics to Mechanisms and Machines
This book brings together all 16 articles published in the Special Issue "Applied Mathematics to Mechanisms and Machines" of the MDPI Mathematics journal, in the section âEngineering Mathematicsâ. The subject matter covered by these works is varied, but they all have mechanisms as the object of study and mathematics as the basis of the methodology used. In fact, the synthesis, design and optimization of mechanisms, robotics, automotives, maintenance 4.0, machine vibrations, control, biomechanics and medical devices are among the topics covered in this book. This volume may be of interest to all who work in the field of mechanism and machine science and we hope that it will contribute to the development of both mechanical engineering and applied mathematics
Quantum Nonlocality
This book presents the current views of leading physicists on the bizarre property of quantum theory: nonlocality. Einstein viewed this theory as âspooky action at a distanceâ which, together with randomness, resulted in him being unable to accept quantum theory. The contributions in the book describe, in detail, the bizarre aspects of nonlocality, such as EinsteinâPodolskyâRosen steering and quantum teleportationâa phenomenon which cannot be explained in the framework of classical physics, due its foundations in quantum entanglement. The contributions describe the role of nonlocality in the rapidly developing field of quantum information. Nonlocal quantum effects in various systems, from solid-state quantum devices to organic molecules in proteins, are discussed. The most surprising papers in this book challenge the concept of the nonlocality of Nature, and look for possible modifications, extensions, and new formulationsâfrom retrocausality to novel types of multiple-world theories. These attempts have not yet been fully successful, but they provide hope for modifying quantum theory according to Einsteinâs vision
A Framework for Uncertainty Relations
Uncertainty principle, which was first introduced by Werner Heisenberg
in 1927, forms a fundamental component of quantum mechanics.
A graceful aspect of quantum mechanics is that the uncertainty
relations between incompatible observables allow for succinct quan-
titative formulations of this revolutionary idea: it is impossible to
simultaneously measure two complementary variables of a particle in
precision. In particular, information theory offers two basic ways to
express the Heisenbergâs principle: variance-based uncertainty relations
and entropic uncertainty relations.
We first investigate the uncertainty relations based on the sum of
variances and derive a family of weighted uncertainty relations to
provide an optimal lower bound for all situations. Our work indicates
that it seems unreasonable to assume a priori that incompatible
observables have equal contribution to the variance-based sum form
uncertainty relations. We also study the role of mutually exclusive
physical states in the recent work and generalize the variance-based
uncertainty relations to mutually exclusive uncertainty relations.
Next, we develop a new kind of entanglement detection criteria within
the framework of marjorization theory and its matrix representation.
By virtue of majorization uncertainty bounds, we are able to construct
the entanglement criteria which have advantage over the scalar detect-
ing algorithms as they are often stronger and tighter.
Furthermore, we explore various expression of entropic uncertainty
relations, including sum of Shannon entropies, majorization uncer-
tainty relations and uncertainty relations in presence of quantum
memory. For entropic uncertainty relations without quantum side
information, we provide several tighter bounds for multi-measurements,
with some of them also valid for RĂ©nyi and Tsallis entropies besides
the Shannon entropy. We employ majorization theory and actions
of the symmetric group to obtain an admixture bound for entropic
uncertainty relations with multi-measurements. Comparisons among
existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical
side information. For modern uncertainty relations, those who allowed
for non-trivial amount of quantum side information, their bounds
have been strengthened by our recent result for both two and multi-
measurements.
Finally, we propose an approach which can extend all uncertainty
relations on Shannon entropies to allow for quantum side information
and discuss the applications of our entropic framework. Combined with
our uniform entanglement frames, it is possible to detect entanglement
via entropic uncertainty relations even if there is no quantum side in-
formation. With the rising of quantum information theory, uncertainty
relations have been established as important tools for a wide range of
applications, such as quantum cryptography, quantum key distribution,
entanglement detection, quantum metrology, quantum speed limit and
so on. It is thus necessary to focus on the study of uncertainty relations