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

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.Comment: 9 pages, 2 figure

Geometric formulation of the uncertainty principle

A geometric approach to formulate the uncertainty principle between quantum observables acting on an N-dimensional Hilbert space is proposed. We consider the fidelity between a density operator associated with a quantum system and a projector associated with an observable, and interpret it as the probability of obtaining the outcome corresponding to that projector. We make use of fidelity-based metrics such as angle, Bures, and root infidelity to propose a measure of uncertainty. The triangle inequality allows us to derive a family of uncertainty relations. In the case of the angle metric, we recover the Landau-Pollak inequality for pure states and show, in a natural way, how to extend it to the case of mixed states in arbitrary dimension. In addition, we derive and compare alternative uncertainty relations when using other known fidelity-based metrics.publishedVersionFil: Bosyk, Gustavo Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Física La Plata; Argentina.Fil: Bosyk, Gustavo Martín. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Física; Argentina.Fil: Osán, Tristán Martín. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Osán, Tristán Martín. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina.Fil: Lamberti, Pedro Walter. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Lamberti, Pedro Walter. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina.Fil: Portesi, Mariela. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Física La Plata; Argentina.Fil: Portesi, Mariela. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Física; Argentina.Otras Ciencias Física