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 localisation 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 a novel geometric characterisation of the latter quantity. Applications include unified volumebased derivations of the Holevo and Shannon bounds in quantum and classical information theory, a precise geometric interpretation of thermodynamic entropy for equilibrium ensembles, a geometric derivation of semi-classical uncertainty relations, a new means for defining classical and quantum localization for arbitrary evolution processes, a geometric interpretation of relative entropy, and a new 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. PACS Numbers: 03.65.Bz, 03.67.-a, 05.45+b, 42.60.Jf