Information and Its Metric

James P. Crutchfield
Physics Department
University of California
Berkeley, California 94720, USA

ABSTRACT: Information is taken as the primary physical entity from which probabilities can be derived. Information produced by a source is defined as the class of sources that are recoding-equivalent. Shannon entropy is one of a family of formal Renyi information measures on the space of unique sources. Each of these measures quantifies the volume of the source's recoding-equivalence class. A space of information sources is constructed from elements each of which is the class of recoding-equivalent, but otherwise unique sources. The norm in this space is the source entropy. A measure of distance between information sources is derived from an algebra of measurements. With this, the space of information sources is shown to be a metric space whose logic is described by a metric lattice. Applications of the information metric to quantum informational uncertainty and to information densities in multicomponent dynamical systems are outlined.

J. P. Crutchfield, "Information and Its Metric", in Nonlinear Structures in Physical Systems -- Pattern Formation, Chaos, and Waves, L. Lam and H. C. Morris, editors, Springer-Verlag, Berlin (1990) 119 - 130. [ps.gz]= 55kb [ps]= 199kb [pdf]= 136kb

NOTE: Appears in the proceedings of the Second Woodward Conference, San Jose State University, San Jose, California, 17 - 18 November 1989.