Stronger classes of sum uncertainty and reverse uncertainy relations

TL;DR

This paper introduces a new uncertainty matrix to derive stronger uncertainty relations applicable to all quantum states and observables, with implications for quantum speed limits and fidelity bounds.

Contribution

It presents a novel uncertainty matrix linking purity and observable incompatibility, leading to tighter uncertainty relations for mixed and pure states, and extends these to multiple observables.

Findings

Derived stronger uncertainty relations for arbitrary states and observables.

Provided a reverse quantum speed limit for Markovian quantum evolution.

Established a lower bound for quantum state fidelity.

Abstract

Uncertainty relations are old, yet potentially rewarding to explore. By introducing a quantity called the uncertainty matrix, we provide a link between purity and observable incompatibility, and derive several stronger uncertainty relations in both forward and reverse directions for arbitrary quantum states, i.e., mixed as well as pure, and arbitrary incompatible quantum observables, none of which suffer from the problem of triviality. Besides the tightness, the interpretations of terms in these uncertainty relations may be of independent inter- est. We provide the possible generalization of stronger uncertainty relations to sum of variances of more than two observables. We also demonstrate applications of techniques used here to, firstly, obtain a simple reverse quantum speed limit for quantum states undergoing Markovian dynamical evolution, and secondly, to provide a lower bound for fidelity between two quantum states.