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

TL;DR

This paper extends Landau-Pollak uncertainty relations to mixed states and POVMs using geometric metrics between states, revealing the original inequality as the most restrictive case.

Contribution

It introduces a geometric framework to generalize Landau-Pollak uncertainty relations for mixed states and POVMs, broadening their applicability.

Findings

Generalized Landau-Pollak inequalities for mixed states and POVMs.

Identified the Wootters metric as the most restrictive case.

Established the triangle inequality as key to the derivation.

Abstract

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.