Geometric approach to extend Landau-Pollak uncertainty relations for positive operator-valued measures

G. M. Bosyk, S. Zozor, M. Portesi, T. M. Osán, P. W. Lamberti

Published 2014-06-13, updated 2014-10-01Version 2

We provide a twofold extension of Landau--Pollak uncertainty relations for mixed quantum states and for positive operator-valued measures, by recourse to geometric considerations. The generalization is based on metrics between pure states, having the form of a function of the square of the inner product between the states. The triangle inequality satisfied by such metrics plays a crucial role in our derivation. The usual Landau--Pollak inequality is thus a particular case (derived from Wootters metric) of the family of inequalities obtained, and, moreover, we show that it is the most restrictive relation within the family.