Eur. Phys. J. B 17, 459-469 (2000)
https://doi.org/10.1007/s100510070125

Geometrical entropies. The extended entropy

Departamento de Optica, Escuela Universitaria de Optica, Universidad Complutense de Madrid, C/ Arcos de Jalón s/n, 28037 Madrid, Spain

Received: 4 November 1999
Published online: 15 October 2000

Abstract

By taking into account a geometrical interpretation of the measurement process [1,2], we define a set of measures of uncertainty. These measures will be called geometrical entropies. The amount of information is defined by considering the metric structure in the probability space. Shannon-von Neumann entropy is a particular element of this set. We show the incompatibility between this element and the concept of variance as a measure of the statistical fluctuations. When the probability space is endowed with the generalized statistical distance proposed in reference [3], we obtain the extended entropy. This element, which belongs to the set of geometrical entropies, is fully compatible with the concept of variance. Shannon-von Neumann entropy is recovered as an approximation of the extended entropy. The behavior of both entropies is compared in the case of a particle in a square-well potential.