# Uncertainty relation under information measurement and feedback control

## TL;DR

This paper establishes a fundamental uncertainty relation in classical systems under measurement and feedback, linking entropy production and information to the fluctuation limits of observables, applicable to various systems including Brownian ratchets.

## Contribution

It introduces a new uncertainty relation that incorporates information measures and applies to both continuous and discrete systems with feedback control.

## Key findings

- Entropy production and information constrain observable fluctuations.
- The relation applies to finite observation times.
- Demonstrated on a flashing Brownian ratchet.

## Abstract

Here, we investigate the uncertainty of dynamical observables in classical systems manipulated by repeated measurements and feedback control; the precision should be enhanced in the presence of an external controller but limited by the amount of information obtained from the measurements. We prove that the entropy production and the information quantity constrain from below the fluctuation of arbitrary observables that are antisymmetric under time reversal. The information term is the sum of the mutual entropy production and the Kullback--Leibler divergence, which characterises the irreversibility of the measurement outcomes. The result holds for finite observation times and for both continuous- and discrete-time systems. We apply the derived relation to study the precision of a flashing Brownian ratchet.