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In this paper, we determine some new bounds for the generalized Tsallis relative operator entropy by providing some close and sharp bounds. In particular, we identify some bounds for the Tsallis relative operator entropy. Our main results confirm some results obtained in [16, 23]. Moreover, we reach some inequalities for the generalized relative operator entropy in some sense.

Information entropy and its extension, which are important generalizations of entropy, are currently applied to many research domains. In this paper, a novel generalized relative entropy is constructed to avoid some defects of traditional relative entropy. We present the structure of generalized relative entropy after the discussion of defects in relative entropy. Moreover, some properties of the provided generalized relative entropy are presented and proved. The provided generalized relative entropy is proved to have a finite range and is a finite distance metric. Finally, we predict nucleosome positioning of fly and yeast based on generalized relative entropy and relative entropy respectively. The experimental results show that the properties of generalized relative entropy are better than relative entropy.

Quantum relative entropy has been studied extensively, and many forms have been derived due to different parameters. Maximum relative entropy and minimum relative entropy are obtained by taking specific conditions for parameters. Our goal in this paper is to propose a new bipartite entanglement monotone based on minimum relative entropy of any bipartite quantum entanglement state. We also demonstrate that entanglement monotone satisfies some basic properties as an entanglement measure.

A highly versatile evaluation method is proposed for transient plasmas based on statistical physics. It would be beneficial in various industrial sectors, including semiconductors and automobiles. Our research focused on low-energy plasmas in laboratory settings, and they were assessed via our proposed method, which incorporates relative entropy and fractional Brownian motion, based on a revised collisional–radiative model. By introducing an indicator to evaluate how far a system is from its steady state, both the trend of entropy and the radiative process’ contribution to the lifetime of excited states were considered. The high statistical weight of some excited states may act as a bottleneck in the plasma’s energy relaxation throughout the system to a steady state. By deepening our understanding of how energy flows through plasmas, we anticipate potential contributions to resolving global environmental issues and fostering technological innovation in plasma-related industrial fields.

Belavkin–Staszewski relative entropy can naturally characterize the effects of the possible noncommutativity of quantum states. In this paper, two new conditional entropy terms and four new mutual information terms are first defined by replacing quantum relative entropy with Belavkin–Staszewski relative entropy. Next, their basic properties are investigated, especially in classical-quantum settings. In particular, we show the weak concavity of the Belavkin–Staszewski conditional entropy and obtain the chain rule for the Belavkin–Staszewski mutual information. Finally, the subadditivity of the Belavkin–Staszewski relative entropy is established, i.e., the Belavkin–Staszewski relative entropy of a joint system is less than the sum of that of its corresponding subsystems with the help of some multiplicative and additive factors. Meanwhile, we also provide a certain subadditivity of the geometric Rényi relative entropy.

Rényi-type generalizations of entropy, relative entropy and mutual information have found numerous applications throughout information theory and beyond. While there is consensus that the ways A. Rényi generalized entropy and relative entropy in 1961 are the “right” ones, several candidates have been put forth as possible mutual informations of order α . In this paper we lend further evidence to the notion that a Bayesian measure of statistical distinctness introduced by R. Sibson in 1969 (closely related to Gallager’s E 0 function) is the most natural generalization, lending itself to explicit computation and maximization, as well as closed-form formulas. This paper considers general (not necessarily discrete) alphabets and extends the major analytical results on the saddle-point and saddle-level of the conditional relative entropy to the conditional Rényi divergence. Several examples illustrate the main application of these results, namely, the maximization of α -mutual information with and without constraints.

In this paper, we present some inequalities for the generalized relative operator entropy according to the generalized Tsallis relative operator entropy. Our results are generalizations of some existing inequalities.

This study considers a new decomposition of an extended divergence on a foliation by deformed probability simplexes from the information geometry perspective. In particular, we treat the case where each deformed probability simplex corresponds to a set of q-escort distributions. For the foliation, different q-parameters and the corresponding α-parameters of dualistic structures are defined on each of the various leaves. We propose the divergence decomposition theorem that guides the proximity of q-escort distributions with different q-parameters and compare the new theorem to the previous theorem of the standard divergence on a Hessian manifold with a fixed α-parameter.

Given a probability measure, we consider the diffusion flows of probability measures associated with the partial differential equation (PDE) of Fokker–Planck. Our flows of the probability measures are defined as the solution of the Fokker–Planck equation for the same strictly convex potential, which means that the flows have the same equilibrium. Then, we shall investigate the time derivative for the relative entropy in the case where the object and the reference measures are moving according to the above diffusion flows, from which we can obtain a certain dissipation formula and also an integral representation of the relative entropy.

In this paper, we determine the bounds of the generalized relative operator entropy in general. In particular, we identify the bounds of the parametric extension of the Shannon entropy and of the generalized Tsallis relative operator entropy. Moreover, we improve the upper bound of the relative operator entropy in some sense.