# Generalized energy measurements and quantum work compatible with fluctuation theorems

## TL;DR

This paper explores whether generalized energy measurements can replace projective measurements in quantum fluctuation theorems, establishing conditions under which the Jarzynski and Crooks relations remain valid.

## Contribution

It identifies necessary and sufficient conditions for generalized measurements to preserve fluctuation theorems, extending previous results and correcting earlier errors.

## Key findings

- Jarzynski equality requires perfect initial measurement accuracy.
- Crooks relation tolerates only depolarizing channels as deviations.
- Projective measurements are necessary for Crooks relation in infinite-dimensional systems.

## Abstract

The probability densities of work that can be exerted on a quantum system initially staying in thermal equilibrium are constrained by the fluctuation relations of Jarzynski and Crooks, when the work is determined by two projective energy measurements. We investigate the question whether these fluctuation relations may still hold if one employs generalized energy measurements rather than projective ones. Restricting ourselves to a class of universal measurements which are independent of several details of the system on which the work is done, we find sets of necessary and sufficient conditions for the Jarzynski equality and the Crooks relation. The Jarzynski equality requires perfect accuracy for the initial measurement, while the final one can be erroneous. On the other hand, the Crooks relation can only tolerate a depolarizing channel as a deviation from the projective measurement for systems with a finite dimensional Hilbert space. For a separable infinite-dimensional space only projective measurements are compatible with the Crooks relation. The results we have obtained significantly extend those of [Venkatesh, Watanabe, and Talkner, New J. Phys. 16, 015032 (2014)] as well as avoid some errors present there.