Improving sum uncertainty relations with the quantum Fisher information

TL;DR

This paper enhances sum uncertainty relations by incorporating quantum Fisher information, leading to tighter bounds and improved understanding of quantum fluctuations, especially in angular momentum and phase estimation contexts.

Contribution

It introduces a method to tighten sum uncertainty relations using quantum Fisher information and applies it to angular momentum and phase estimation, revealing new bounds and optimal decompositions.

Findings

Stronger angular momentum uncertainty relations derived.

Optimal decompositions for variance achieved via analogy with rigid bodies.

Identification of classical and quantum limits in phase estimation.

Abstract

We show how preparation uncertainty relations that are formulated as sums of variances may be tightened by using the quantum Fisher information to quantify quantum fluctuations. We apply this to derive stronger angular momentum uncertainty relations, which in the case of spin-$1/2$ turn into equalities involving the purity. Using an analogy between pure-state decompositions in the Bloch sphere and the moment of inertia of rigid bodies, we identify optimal decompositions that achieve the convex- and concave-roof decomposition of the variance. Finally, we illustrate how these results may be used to identify the classical and quantum limits on phase estimation precision with an unknown rotation axis.