Wave functions' discernibility and the role of fluctuations

Abstract

The question of distinguishability of quantum states is at the heart of quantum information processing, an issue is here addressed with reference to different distances in probability space vis-a-vis metrics in Hilbert's one. We provide further reconfirmation of Wootters' hypothesis: the possibility that statistical fluctuations in the outcomes of measurements be regarded as responsible for the Hilbert-space structure of quantum mechanics, a view that becomes here considerably strengthened. We show that distances between neighboring states, whether of statistical or Hilbert's metric origin, have as a lower bound Fisher's measure, up to second-order approximation. As a consequence, the structure of the vicinitv of a given quantum state is to a large extent determined by the fluctuations of the pertinent observables. It is also shown that Tsallis' non-extensivity parameter q can be used as a tool for increasing discernibility between wave functions.