# The Coherent Crooks Equality

## TL;DR

This paper extends the classical Crooks fluctuation theorem into the quantum regime by incorporating quantum control and energy superpositions, using an information theoretic approach that emphasizes energy conservation with time-independent Hamiltonians.

## Contribution

It introduces a quantum Crooks equality framework that explicitly models control and energy supply, including superpositions of energy eigenstates, within an information theoretic context.

## Key findings

- Derivation of a quantum Crooks-like equality for superposed energy states
- Framework accounts for energy conservation with time-independent Hamiltonians
- Generalizes classical fluctuation theorem to quantum control scenarios

## Abstract

This chapter reviews an information theoretic approach to deriving quantum fluctuation theorems. When a thermal system is driven from equilibrium, random quantities of work are required or produced: the Crooks equality is a classical fluctuation theorem that quantifies the probabilities of these work fluctuations. The framework summarised here generalises the Crooks equality to the quantum regime by modeling not only the driven system but also the control system and energy supply that enables the system to be driven. As is reasonably common within the information theoretic approach but high unusual for fluctuation theorems, this framework explicitly accounts for the energy conservation using only time independent Hamiltonians. We focus on explicating a key result derived by Johan {\AA}berg: a Crooks-like equality for when the energy supply is allowed to exist in a superposition of energy eigenstates states.