A geometric formulation of uncertainty principle

TL;DR

This paper introduces a geometric framework for the uncertainty principle in quantum mechanics using fidelity-based metrics, deriving new uncertainty relations and extending existing ones to mixed states.

Contribution

It proposes a novel geometric formulation of the uncertainty principle using fidelity metrics, extending Landau--Pollak inequality to mixed states in arbitrary dimensions.

Findings

Re-derivation of Landau--Pollak inequality for pure states

Extension of uncertainty relations to mixed states

Comparison of uncertainty bounds using different fidelity metrics

Abstract

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 ones, 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 re-obtain 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 novel uncertainty relations when using other known fidelity-based metrics.