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The Shannon entropy of a random variable $X$ has much behaviour analogous to a signed measure. Previous work has concretized this connection by defining a signed measure $\mu$ on an abstract information space $\tilde{X}$, which is taken to…
In its continuous version, the entropy functional measuring the information content of a given probability density may be plagued by a "measure" problem that results from improper weighting of phase space. This issue is addressed…
We obtain new entropy rigidity results for $u$-Gibbs measures by showing that whenever a $u$-Gibbs measure of a partially hyperbolic diffeomorphism admits an unstable Margulis family, the unstable Jacobian data of the system must to be…
The Shannon entropy is a fundamental measure for quantifying diversity and model complexity in fields such as information theory, ecology, and genetics. However, many existing studies assume that the number of species is known, an…
Entropy production is the hallmark of nonequilibrium physics, quantifying irreversibility, dissipation, and the efficiency of energy transduction processes. Despite many efforts, its measurement at the nanoscale remains challenging. We…
Thermodynamic uncertainty relations reveal a fundamental trade-off between the precision of a trajectory observable and entropy production, where the uncertainty of the observable is quantified by its variance. In information theory,…
When the von Neumann entropy (VNE) of a system increases due to measurements, certain information is lost, some of which may be recoverable. We define information retrievability (IR) and information loss (IL) as functions of the density…
We consider the information flow on a system's observable $X$ corresponding to a positive-operator valued measure under a quantum measurement process $Y$ described by a completely positive instrument from the viewpoint of the relative…
We consider stationary ergodic processes indexed by $\mathbb Z$ or $\mathbb Z^n$ whose finite dimensional marginals have laws which are absolutely continuous with respect to Lebesgue measure. We define an entropy theory for these continuous…
It is supposed that the exponential multiplier in the method of the non-equilibrium statistical operator (Zubarev`s approach) can be considered as a distribution density of the past lifetime of the system, and can be replaced by an…
We define a notion of entropy for an infinite family $\mathcal{C}$ of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for $\mathcal{C}$ under every ergodic transformation if and only if…
Information plays an important role in our understanding of the physical world. We hence propose an entropic measure of information for any physical theory that admits systems, states and measurements. In the quantum and classical world,…
We evaluate Shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal…
Recent theoretical results renewed the interest in charged particle multiplicity distributions. The Shannon entropy of such distributions is conjectured to be related to the entanglement or von Neumann entropy of partonic quantum system. In…
Shannon's fundamental bound for perfect secrecy says that the entropy of the secret message cannot be larger than the entropy of the secret key initially shared by the sender and the legitimate receiver. Massey gave an information theoretic…
We consider an abundant class of non-uniformly hyperbolic $C^2$-H\'enon like diffeomorphisms called strongly regular and which corresponds to Benedicks-Carleson parameters. We prove the existence of $m>0$ such that for any such…
Statements of Shannon's Noiseless Coding Theorem by various authors, including the original, are reviewed and clarified. Traditional statements of the theorem are often unclear as to when it applies. A new notation is introduced and the…
Entropies of mixing can be derived directly from the parent distributions of extreme value theory. They correspond to pseudo-additive entropies in the case of Pareto and power function distributions, while to the Shannon entropy in the case…
A two parameter generalization of Boltzmann-Gibbs-Shannon entropy based on natural logarithm is introduced. The generalization of the Shannon-Kinchinn axioms corresponding to the two parameter entropy is proposed and verified. We present…
The empirical measure of an interacting particle system is a purely atomic random probability measure. In the limit as the number of particles grows to infinity, we show for McKean-Vlasov systems with common noise that this measure becomes…