# A framework for sequential measurements and general Jarzynski equations

## TL;DR

This paper develops a comprehensive framework for sequential measurements in quantum and classical systems, deriving generalized Jarzynski equations and related theorems, expanding their applicability to various initial states and driven systems.

## Contribution

It introduces a unified J-equation for sequential measurements, extending the quantum Jarzynski equation to local equilibrium states and different ensembles, and connects to the Second Law.

## Key findings

- Derivation of a generalized J-equation for sequential measurements
- Extension of Jarzynski and Crooks relations to local equilibrium states
- Application to periodically driven quantum systems in contact with heat baths

## Abstract

We formulate a statistical model of two sequential measurements and prove a so-called J-equation that leads to various diversifications of the well-known Jarzynski equation including the Crooks dissipation theorem. Moreover, the J-equation entails formulations of the Second Law going back to Wolfgang Pauli. We illustrate this by an analytically solvable example of sequential discrete position-momentum measurements accompanied with the increase of Shannon entropy. The standard form of the J-equation extends the domain of applications of the quantum Jarzynski equation in two respects: It includes systems that are initially only in local equilibrium and it extends this equation to the cases where the local equilibrium is described by microcanononical, canonical or grand canonical ensembles. Moreover, the case of a periodically driven quantum system in thermal contact with a heat bath is shown to be covered by the theory presented here. Finally, we shortly consider the generalized Jarzynski equation in classical statistical mechanics.