# Thermodynamic uncertainty relations including measurement and feedback

## TL;DR

This paper extends thermodynamic uncertainty relations to include measurement and feedback scenarios, showing how fluctuation relations imply bounds on the signal-to-noise ratio even when time-reversal symmetry is broken.

## Contribution

It demonstrates that fluctuation relations can be used to derive thermodynamic uncertainty relations in systems with measurement and feedback, broadening their applicability.

## Key findings

- Uncertainty relations apply to feedback-controlled systems.
- Backward experiments can compensate for large signal-to-noise ratios.
- Illustrations include Szilard engine and quantum dot work extraction.

## Abstract

Thermodynamic uncertainty relations quantify how the signal-to-noise ratio of a given observable is constrained by dissipation. Fluctuation relations generalize the second law of thermodynamics to stochastic processes. We show that any fluctuation relation directly implies a thermodynamic uncertainty relation, considerably increasing their range of applicability. In particular, we extend thermodynamic uncertainty relations to scenarios which include measurement and feedback. Since feedback generally breaks time-reversal invariance, the uncertainty relations involve quantities averaged over the forward and the backward experiment defined by the associated fluctuation relation. This implies that the signal-to-noise ratio of a given experiment can in principle become arbitrarily large as long as the corresponding backward experiment compensates, e.g. by being sufficiently noisy. We illustrate our results with the Szilard engine as well as work extraction by free energy reduction in a quantum dot.