Optimal Uncertainty Relations for Extremely Coarse-grained Measurements

TL;DR

This paper derives improved and sharpened quantum uncertainty relations for position and momentum measurements under coarse-graining, ensuring non-trivial bounds even with extremely imprecise measurements.

Contribution

It introduces new bounds for uncertainty relations using Renyi entropy and variances, applicable to highly coarse-grained quantum measurements.

Findings

Improved lower bounds for coarse-grained uncertainty relations using Renyi entropy.

Sharpened Heisenberg-like uncertainty relation valid for any degree of coarse graining.

Non-trivial uncertainty bounds exist even with extremely coarse measurements.

Abstract

We derive two quantum uncertainty relations for position and momentum coarse-grained measurements. Building on previous results, we first improve the lower bound for uncertainty relations using the Renyi entropy, particularly in the case of coarse-grained measurements. We then sharpen a Heisenberg-like uncertainty relation derived previously in [Europhys. Lett. 97, 38003, (2012)] that uses variances and reduces to the usual one in the case of infinite precision measurements. Our sharpened uncertainty relation is meaningful for any amount of coarse graining. That is, there is always a non-trivial uncertainty relation for coarse-grained measurement of the non-commuting observables, even in the limit of extremely large coarse graining.