Implications and applications of the variance-based uncertainty equalities

TL;DR

This paper introduces new quantum uncertainty equalities that extend Heisenberg-type relations, providing deeper insights into incompatible observables, with applications in spin squeezing and interferometry.

Contribution

It formulates two universal quantum uncertainty equalities, offering a novel perspective beyond traditional inequalities and connecting them to intelligent states.

Findings

Derived state-independent bounds for variance sums

Applied inequalities to spin squeezing scenarios

Discussed implications for interferometric sensitivity

Abstract

In quantum mechanics, the variance-based Heisenberg-type uncertainty relations are a series of mathematical inequalities posing the fundamental limits on the achievable accuracy of the state preparations. In contrast, we construct and formulate two quantum uncertainty equalities, which hold for all pairs of incompatible observables and indicate the new uncertainty relations recently introduced by L. Maccone and A. K. Pati [Phys. Rev. Lett. 113, 260401 (2014)]. Furthermore, we present an explicit interpretation lying behind the derivations and relate these relations to the so-called intelligent states. As an illustration, we investigate the properties of these uncertainty inequalities in the qubit system and a state-independent bound is obtained for the sum of variances. Finally, we apply these inequalities to the spin squeezing scenario and its implication in interferometric sensitivity is also discussed.