Universal geometric approach to uncertainty, entropy, and information

It is shown that a unique measure of volume is associated with any statistical ensemble, which directly quantifies the inherent spread or localization of the ensemble. It is applicable whether the ensemble is classical or quantum, continuous or discrete, and may be derived from a small number of theory-independent geometric postulates. Remarkably, this unique ensemble volume is proportional to the exponential of the ensemble entropy, and hence provides an interesting geometric characterization of the latter quantity. Applications include unified volume-based derivations of results in quantum and classical information theory, a precise geometric interpretation of thermodynamic entropy for equilibrium ensembles, a geometric derivation of semiclassical uncertainty relations, a means for defining classical and quantum localization for arbitrary evolution processes, and a proposed definition for the spot size of an optical beam. Advantages of ensemble volume over other measures of localization (root-mean-square deviation, Renyi entropies, and inverse participation ratio) are discussed.