# The third law of thermodynamics in open quantum systems

## TL;DR

This paper proves the third law of thermodynamics for open quantum systems, showing that entropy approaches zero at zero temperature in equilibrium, with specific behavior for non-equilibrium steady states and localized states.

## Contribution

It establishes the third law for open quantum systems, including both equilibrium and non-equilibrium cases, and characterizes the entropy behavior related to localized states.

## Key findings

- Entropy approaches zero as temperature approaches zero in equilibrium systems.
- Localized states at the chemical potential lead to a residual entropy of g ln 2.
- In non-equilibrium steady states, local entropy vanishes as local temperature approaches zero.

## Abstract

We consider open quantum systems consisting of a finite system of independent fermions with arbitrary Hamiltonian coupled to one or more equilibrium fermion reservoirs (which need not be in equilibrium with each other). A strong form of the third law of thermodynamics, $S(T) \rightarrow 0$ as $T\rightarrow 0$, is proven for fully open quantum systems in thermal equilibrium with their environment, defined as systems where all states are broadened due to environmental coupling. For generic open quantum systems, it is shown that $S(T)\rightarrow g\ln 2$ as $T\rightarrow 0$, where $g$ is the number of localized states lying exactly at the chemical potential of the reservoir. For driven open quantum systems in a nonequilibrium steady state, it is shown that the local entropy $S({\bf x}; T) \rightarrow 0$ as $T({\bf x})\rightarrow 0$, except for cases of measure zero arising due to localized states, where $T({\bf x})$ is the temperature measured by a local thermometer.