The tight Second Law inequality for coherent quantum systems and finite-size heat baths

Marcin Łobejko

doi:10.1038/s41467-021-21140-4

Abstract

In classical thermodynamics, the optimal work is given by the free energy difference, what according to the result of Skrzypczyk et al. can be generalized for individual quantum systems. The saturation of this bound, however, requires an infinite bath and ideal energy storage that is able to extract work from coherences. Here we present the tight Second Law inequality, defined in terms of the ergotropy (rather than free energy), that incorporates both of those important microscopic effects – the locked energy in coherences and the locked energy due to the finite-size bath. The former is solely quantified by the so-called control-marginal state, whereas the latter is given by the free energy difference between the global passive state and the equilibrium state. Furthermore, we discuss the thermodynamic limit where the finite-size bath correction vanishes, and the locked energy in coherences takes the form of the entropy difference. We supplement our results by numerical simulations for the heat bath given by the collection of qubits and the Gaussian model of the work reservoir.

Highlights

In classical thermodynamics, the optimal work is given by the free energy difference, what according to the result of Skrzypczyk et al can be generalized for individual quantum systems
This quantity on its own can be defined as the optimal work extracted from closed systems driven by the time-dependent and cyclic Hamiltonians, which proves an important connection between those two frameworks
We reveal that there is no full equivalence since models with implicit energy storage do not involve the concept of the locked energy in coherences, i.e., the off-diagonal part that contributes to ergotropy but cannot be extracted as a work

Summary

Introduction

The optimal work is given by the free energy difference, what according to the result of Skrzypczyk et al can be generalized for individual quantum systems. It provides a measure of how the corresponding thermal reservoir is able to extract free energy from the non-equilibrium quantum state, and we call it the locked energy in a finite-size bath.

Results

Conclusion