Inequalities for Jensen–Sharma–Mittal and Jeffreys–Sharma–Mittal Type f–Divergences

In this paper, we introduce new divergences called Jensen–Sharma–Mittal and Jeffreys–Sharma–Mittal in relation to convex functions. Some theorems, which give the lower and upper bounds for two new introduced divergences, are provided. The obtained results imply some new inequalities corresponding to known divergences. Some examples, which show that these are the generalizations of Rényi, Tsallis, and Kullback–Leibler types of divergences, are provided in order to show a few applications of new divergences.     

Multiresolution information measures applied to speech recognition

Considerable advances in automatic speech recognition have been made in the last decades, thanks specially to the use of hidden Markov models. In the field of speech signal analysis, different techniques have been developed. However, deterioration in the performance of the speech recognizers has been observed when they are trained with clean signal and tested with noisy signals. This is still an open problem in this field. Continuous multiresolution entropy has been shown to be robust to additive noise in applications to different physiological signals. In previous works we have included Shannon and Tsallis entropies, and their corresponding divergences, in different speech analysis and recognition systems. In this paper we present an extension of the continuous multiresolution entropy to different divergences and we propose them as new dimensions for the pre-processing stage of a speech recognition system. This approach takes into account information about changes in the dynamics of speech signal at different scales. The methods proposed here are tested with speech signals corrupted with babble and white noise. Their performance is compared with classical mel cepstral parametrization. The results suggest that these continuous multiresolution entropy related measures provide valuable information to the speech recognition system and that they could be considered to be included as an extra component in the pre-processing stage.     

Entropy and information characteristics of qubit states

Shannon entropy, Rényi entropy, and Tsallis entropy are discussed for the tomographic probability distributions of qubit states. Relative entropy and its properties are considered for the tomographic probability distribution describing the states of multi-spin systems. New inequalities for Hermite polynomials are obtained.     

Escort entropies and divergences and related canonical distribution

We discuss two families of two-parameter entropies and divergences, derived from the standard Rényi and Tsallis entropies and divergences. These divergences and entropies are found as divergences or entropies of escort distributions. Exploiting the nonnegativity of the divergences, we derive the expression of the canonical distribution associated to the new entropies and a observable given as an escort-mean value. We show that this canonical distribution extends, and smoothly connects, the results obtained in nonextensive thermodynamics for the standard and generalized mean value constraints.     

On the construction of complex networks with optimal Tsallis entropy

In this work, we first formulate the Tsallis entropy in the context of complex networks. We then propose a network construction whose topology maximizes the Tsallis entropy. The growing network model has two main ingredients: copy process and random attachment mechanism (C-R model). We show that the resulting degree distribution exactly agrees with the required degree distribution that maximizes the Tsallis entropy. We also provide another example of network model using a combination of preferential and random attachment mechanisms (P-R model) and compare it with the distribution of the Tsallis entropy. In this case, we show that by adequately identifying the exponent factor q, the degree distribution can also be written in the q-exponential form. Taken together, our findings suggest that both mechanisms, copy process and preferential attachment, play a key role for the realization of networks with maximum Tsallis entropy. Finally, we discuss the interpretation of q parameter of the Tsallis entropy in the context of complex networks.     

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

Pinsker’s and Fannes’ type bounds on the Tsallis relative entropy are derived. The monotonicity property of the quantum f -divergence is used fot its estimation from below. For order $\alpha \in (0,1)$ , a family of lower bounds of Pinsker type is obtained. For $\alpha >1$ and the commutative case, upper continuity bounds on the relative entropy in terms of the minimal probability in its second argument are derived. Both the lower and upper bounds presented are reformulated for the case of Rényi’s entropies. The Fano inequality is extended to Tsallis’ entropies for all $\alpha >0$ . The deduced bounds on the Tsallis conditional entropy are used to obtain inequalities of Fannes’ type.     

From Rényi Entropy Power to Information Scan of Quantum States

In this paper, we generalize the notion of Shannon’s entropy power to the Rényi-entropy setting. With this, we propose generalizations of the de Bruijn identity, isoperimetric inequality, or Stam inequality. This framework not only allows for finding new estimation inequalities, but it also provides a convenient technical framework for the derivation of a one-parameter family of Rényi-entropy-power-based quantum-mechanical uncertainty relations. To illustrate the usefulness of the Rényi entropy power obtained, we show how the information probability distribution associated with a quantum state can be reconstructed in a process that is akin to quantum-state tomography. We illustrate the inner workings of this with the so-called “cat states”, which are of fundamental interest and practical use in schemes such as quantum metrology. Salient issues, including the extension of the notion of entropy power to Tsallis entropy and ensuing implications in estimation theory, are also briefly discussed.     

Refined Young Inequality and Its Application to Divergences

We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also studied some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we show some results on the Tsallis divergence, the Rényi divergence, the Jeffreys–Tsallis divergence and the Jensen–Shannon–Tsallis divergence.     

Uniform Estimates on the Tsallis Entropies

By means of a probabilistic coupling technique, we establish some tight upper bounds on the variations of the Tsallis entropies in terms of the uniform distance. We treat both classical and quantum cases. The results provide some quantitative characterizations of the uniform continuity and stability properties of the Tsallis entropies. As direct consequences, we obtain the corresponding results for the Shannon entropy and the von Neumann entropy, which are stronger than the conventional ones.      

Combinatorial basis and non-asymptotic form of the Tsallis entropy function

Using a q-analog of Boltzmann's combinatorial basis of entropy, the non-asymptotic non-degenerate and degenerate combinatorial forms of the Tsallis entropy function are derived. The new measures – supersets of the Tsallis entropy and the non-asymptotic variant of the Shannon entropy – are functions of the probability and degeneracy of each state, the Tsallis parameter q and the number of entities N. The analysis extends the Tsallis entropy concept to systems of small numbers of entities, with implications for the permissible range of q and the role of degeneracy.     

Inequalities for nonnegative numbers and information properties of qudit tomograms

We discuss some inequalities for N nonnegative numbers. We use these inequalities to obtain known inequalities for probability distributions and new entropic and information inequalities for quantum tomograms of qudit states. The inequalities characterize the degree of quantum correlations in addition to noncontextuality and quantum discord. We use the subadditivity and strong subadditivity conditions for qudit tomographic-probability distributions depending on the unitary-group parameters in order to derive new inequalities for Shannon, Rényi, and Tsallis entropies of spin states.     

Upper entropy axioms and lower entropy axioms

The paper suggests the concepts of an upper entropy and a lower entropy. We propose a new axiomatic definition, namely, upper entropy axioms, inspired by axioms of metric spaces, and also formulate lower entropy axioms. We also develop weak upper entropy axioms and weak lower entropy axioms. Their conditions are weaker than those of Shannon–Khinchin axioms and Tsallis axioms, while these conditions are stronger than those of the axiomatics based on the first three Shannon–Khinchin axioms. The subadditivity and strong subadditivity of entropy are obtained in the new axiomatics. Tsallis statistics is a special case of satisfying our axioms. Moreover, different forms of information measures, such as Shannon entropy, Daroczy entropy, Tsallis entropy and other entropies, can be unified under the same axiomatics.     

Effects of strain,magnetic field and temperature on entropy of a two dimensional GaAs quantum dot under spin–orbit interaction

In this paper, we have studied the effects of temperature, strain and magnetic field on non-extensive entropy of a two-dimensional (2D) quantum dot under spin–orbit interaction. To this end, we have obtained the energy levels and wave functions of the system in the presence of Bychkov–Rashba, Dresselhaus and strain effects by using diagonalization procedure. Then, we have used the Tsallis formalism and calculated the entropy of the system. It is found that the entropy is increased with enhancing the temperature with and without strain. The entropy increases with considering the negative strain. The strain has not strongly effect on the entropy at low temperatures.     

Hidden Correlations and Entanglement in Single-Qudit States<Superscript>†</Superscript>

We discuss the notion of hidden correlations in classical and quantum indivisible systems along with such characteristics of the correlations as the mutual information and conditional information corresponding to the entropic subadditivity condition and the entropic strong subadditivity condition. We present an analog of the Bayes formula for systems without subsystems, study entropic inequality for von Neumann entropy and Tsallis entropy of the single-qudit state, and discuss the inequalities for qubit and qutrit states as an example.     

Radial velocities of open stellar clusters: A new solid constraint favouring Tsallis maximum entropy theory

In the present study we apply a Tsallis maximum entropy distribution law to the study of the stellar residual radial velocity in a sample of 13 stellar open clusters. From a comparison between results obtained from the analysis based on Tsallis law and on the one based on the Maxwellian law we show that the generalized Tsallis distribution fits more closely the observed distribution of the stellar residual radial velocities for these stellar clusters. We have found clear evidences that the q-parameter in the Tsallis generalized distribution depends on stellar cluster ages for clusters older than . There is also some indication that q increases with cluster galactocentric distance. The results obtained in this work represent an additional solid constraint in the stellar astrophysics favoring the Tsallis maximum entropy theory.     

Tsallis $$\delta $$-entropy in an accelerating BIon

In this research, we consider thermodynamically the behaviour of an accelerating BIon and show that the entropy of this system has the form of Tsallis entropy. A BIon is a system that consists of a brane, an anti-brane and a wormhole which connects them together. By increasing the acceleration of branes, the area of BIon increases and its Tsallis entropy grows.     

Kernel Estimation of Cumulative Residual Tsallis Entropy and Its Dynamic Version under ρ-Mixing Dependent Data

Tsallis introduced a non-logarithmic generalization of Shannon entropy, namely Tsallis entropy, which is non-extensive. Sati and Gupta proposed cumulative residual information based on this non-extensive entropy measure, namely cumulative residual Tsallis entropy (CRTE), and its dynamic version, namely dynamic cumulative residual Tsallis entropy (DCRTE). In the present paper, we propose non-parametric kernel type estimators for CRTE and DCRTE where the considered observations exhibit an ρ-mixing dependence condition. Asymptotic properties of the estimators were established under suitable regularity conditions. A numerical evaluation of the proposed estimator is exhibited and a Monte Carlo simulation study was carried out.     

Some Parameterized Quantum Midpoint and Quantum Trapezoid Type Inequalities for Convex Functions with Applications

Quantum information theory, an interdisciplinary field that includes computer science, information theory, philosophy, cryptography, and entropy, has various applications for quantum calculus. Inequalities and entropy functions have a strong association with convex functions. In this study, we prove quantum midpoint type inequalities, quantum trapezoidal type inequalities, and the quantum Simpson’s type inequality for differentiable convex functions using a new parameterized q-integral equality. The newly formed inequalities are also proven to be generalizations of previously existing inequities. Finally, using the newly established inequalities, we present some applications for quadrature formulas.     

A note on bounded entropies

The aim of the paper is to study the link between non-additivity of some entropies and their boundedness. We propose an axiomatic construction of the entropy relying on the fact that entropy belongs to a group isomorphic to the usual additive group. This allows to show that the entropies that are additive with respect to the addition of the group for independent random variables are nonlinear transforms of the Rényi entropies, including the particular case of the Shannon entropy. As a particular example, we study as a group a bounded interval in which the addition is a generalization of the addition of velocities in special relativity. We show that Tsallis–Havrda–Charvat entropy is included in the family of entropies we define. Finally, a link is made between the approach developed in the paper and the theory of deformed logarithms.