  Persistent homology studies the evolution of k-dimensional holes along a
nested sequence of simplicial complexes (called a filtration). The set of bars
(i.e. intervals) representing birth and death times of k-dimensional holes
along such sequence is called the persistence barcode. k-Dimensional holes with
short lifetimes are informally considered to be "topological noise", and those
with long lifetimes are considered to be "topological features" associated to
the filtration. Persistent entropy is defined as the Shannon entropy of the
persistence barcode of a given filtration. In this paper we present new
important properties of persistent entropy of Cech and Vietoris-Rips
filtrations. Among the properties, we put a focus on the stability theorem that
allows to use persistent entropy for comparing persistence barcodes. Later, we
derive a simple method for separating topological noise from features in
Vietoris-Rips filtrations.
