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
Published online: 15 October 2000
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 , 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.
PACS: 03.65.Bz – Foundations, theory of measurement, miscellaneous theories / 89.70.+c – Information science
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2000