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

Abstract

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.