Conditional work statistics of quantum measurements

TL;DR

This paper introduces a framework for defining and analyzing the conditional energy changes and work statistics in quantum measurements, generalizing existing notions and ensuring consistency with thermodynamic principles.

Contribution

It provides a general definition of conditional energies in quantum measurements, linking them to weak values and deriving the full conditional work statistics.

Findings

Conditional energy after measurement equals the expected Hamiltonian given the post-measurement state.

Conditional energy before measurement relates to the real part of the weak value of the Hamiltonian.

Non-recoverable work due to measurements is always non-negative, aligning with the second law.

Abstract

In this paper we introduce a definition for conditional energy changes due to general quantum measurements, as the change in the conditional energy evaluated before, and after, the measurement process. By imposing minimal physical requirements on these conditional energies, we show that the most general expression for the conditional energy after the measurement is simply the expected value of the Hamiltonian given the post-measurement state. Conversely, the conditional energy before the measurement process is shown to be given by the real component of the weak value of the Hamiltonian. Our definition generalises well-known notions of distributions of internal energy change, such as that given by stochastic thermodynamics. By determining the conditional energy change of both system and measurement apparatus, we obtain the full conditional work statistics of quantum measurements, and show that this vanishes for all measurement outcomes if the measurement process conserves the total energy. Additionally, by incorporating the measurement process within a cyclic heat engine, we quantify the non-recoverable work due to measurements. This is shown to always be non-negative, thus satisfying the second law, and will be independent of the apparatus specifics for two classes of projective measurements.