Monotonicity Of Entropy Research Articles

Monotonicity with volume of entropy and of mean entropy for translationally invariant systems as consequences of strong subadditivity

We consider some questions concerning the monotonicity properties of entropy and mean entropy of states on translationally invariant systems (classical lattice, quantum lattice and quantum continuous). By taking the property of strong subadditivity, which for quantum systems was proven rather late in the historical development, as one of four primary axioms (the other three being simply positivity, subadditivity and translational invariance) we are able to obtain results, some new, some proved in a new way, which appear to complement in an interesting way results proved around thirty years ago on limiting mean entropy and related questions. In particular, we prove that as the sizes of boxes in Z^n or R^n increase in the sense of set inclusion, (1) their mean entropy decreases monotonically and (2) their entropy increases monotonically. Our proof of (2) uses the notion of m-point entropies which we introduce and which generalize the notion of index of correlation (see e.g. R. Horodecki, Phys. Lett. A 187 p145 1994). We mention a number of further results and questions concerning monotonicity of mean entropy for more general shapes than boxes and for more general translationally invariant (/homogeneous) lattices and spaces than Z^n or R^n.

A simple proof for monotonicity of entropy in the quadratic family

We give a simple proof for monotonicity of topological entropy in the quadratic family. We use a spectral property of a Ruelle operator.

Entropy of twist interval maps

We investigate the recently introduced notion of rotation numbers for periodic orbits of interval maps. We identify twist orbits, that is those orbits that are the simplest ones with given rotation number. We estimate from below the topological entropy of a map having an orbit with given rotation number. Our estimates are sharp: there are unimodal maps where the equality holds. We also discuss what happens for maps with larger modality. In the Appendix we present a new approach to the problem of monotonicity of entropy in one-parameter families of unimodal maps.

Entropy in classical and quantum physics

The enigma of ’’entropy’’ is reconsidered from the viewpoint of generalized information theory on a lattice generated from measurements that define the system. A small (incomplete) set of natural axioms for a global information measure is developed sufficiently to deduce as a special case a generalization of Segal’s entropy on a W*-algebra (classical and quantum). A simple relationship between monotonicity of entropy and a semigroup on [0,∞] (representing composibility of information) is presented. Various extensions of information-theoretic results are incidentally proven, including relations between regular composible informations (on an orthocomplemented complete lattice) and measures (on σ—ideals of the lattice).

Relative Entropy of States of von Neumann Algebras

Relative entropy of two states of a von Neumann algebra is defined in terms of the relative modular operator. The strict positivity, lower semicontinuity, convexity and monotonicity of relative entropy are proved. The Wigner-Yanase-Dyson-Lieb concavity is also proved for general von Neumann algebra.