Entropic uncertainty relations for mutually unbiased periodic coarse-grained observables resemble their discrete counterparts

TL;DR

This paper extends entropic uncertainty relations to continuous systems with periodic coarse-grained measurements, showing they obey similar bounds as in discrete systems, using Rényi entropies.

Contribution

It proves that entropic uncertainty relations for mutually unbiased measurements hold in continuous domains with periodic coarse graining, generalizing discrete results.

Findings

Uncertainty relations hold for continuous periodic coarse-grained observables.

The sum of Rényi entropies is bounded by 1d for these observables.

Results unify discrete and continuous measurement frameworks.

Abstract

One of the most important and useful entropic uncertainty relations concerns a $d$ dimensional system and two mutually unbiased measurements. In such a setting, the sum of two information entropies is lower bounded by $\ln d$. It has recently been shown that projective measurements subject to operational mutual unbiasedness can also be constructed in a continuous domain, with the help of periodic coarse graining. Here we consider the whole family of R\'enyi entropies applied to these discretized observables and prove that such a scheme does also admit the uncertainty relation mentioned above.