Jensen-Shannon divergence, Fisher information, and Wootters' hypothesis

Abstract

We discuss different statistical distances in probability space, with emphasis on the Jensen-Shannon divergence, vis-a-vis {\it metrics} in Hilbert space and their relationship with Fisher's information measure. This study provides further reconfirmation of Wootters' hypothesis concerning the possibility that statistical fluctuations in the outcomes of measurements be regarded (at least partly) as responsible for the Hilbert-space structure of quantum mechanics.