Misc,
Tree Structures: A Variational Approach to Shannon--Wiener Information
(November 2003)
Abstract
Entanglement measures based on a logarithmic functional form naturally emerge
in any attempt to quantify the degree of entanglement in the state of a
multipartite quantum system. These measures can be regarded as generalizations
of the classical Shannon-Wiener information of a probability distribution into
the quantum regime. In the present work we introduce a previously unknown
approach to the Shannon-Wiener information which provides an intuitive
interpretation for its functional form as well as putting all entanglement
measures with a similar structure into a new context: By formalizing the
process of information gaining in a set-theoretical language we arrive at a
mathematical structure which we call ''tree structures'' over a given set. On
each tree structure, a tree function can be defined, reflecting the degree of
splitting and branching in the given tree. We show in detail that the
minimization of the tree function on, possibly constrained, sets of tree
structures renders the functional form of the Shannon-Wiener information. This
finding demonstrates that entropy-like information measures may themselves be
understood as the result of a minimization process on a more general underlying
mathematical structure, thus providing an entirely new interpretational
framework to entropy-like measures of information and entanglement. We suggest
three natural axioms for defining tree structures, which turn out to be related
to the axioms describing neighbourhood topologies on a topological space. The
same minimization that renders the functional form of the Shannon-Wiener
information from the tree function then assigns a preferred topology to the
underlying set, hinting at a deep relation between entropy-like measures and
neighbourhood topologies.
