The measure-theoretic definition of Kullback-Leibler relative-entropy (KL-entropy) plays a basic role in the definitions of classical information measures. Entropy, mutual information and conditional forms of entropy can be expressed in terms of KL-entropy and hence properties of their measure-theoretic analogs will follow from those of measure-theoretic KL-entropy. These measure-theoretic defi...

We show that the class $\\mathscr{B}$, of discrete groups which satisfy conclusion Popa’s cocycle superrigidity theorem for Bernoulli actions, is invariant under measure equivalence. generalize this to setting probability preserving (p.m.p.) groupoids, and as a consequence we deduce any nonamenable lattice in product two noncompact, locally compact second countable must belong $\\mathscr{B}$. a...

Abstract Subshifts of deterministic substitutions are ubiquitous objects in dynamical systems and aperiodic order (the mathematical theory quasicrystals). Two their most striking features that they have low complexity (zero topological entropy) uniquely ergodic. Random a generalisation where the substituted image letter is determined by Markov process. In stark contrast to counterparts, subshif...

Quantum entropy is an important measure for describing the uncertainty of a quantum state, more in subsystems pure states implies stronger entanglement between subsystems. Our goal this work to quantify bipartite and entangled networks using dual entropy. We first propose type from Shannon define $$S^{t}$$ -entropy based on von Neumann its complementary dual. This analytic formula two-qubit sys...