A geometric formulation of uncertainty principle

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.

8 pages, 1 figure

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• Excepional Points from Hamiltonians of hybrid physical systems: Squeezing and anti-Squeezing
• Geometric uncertainty relation for quantum ensembles