The von Neumann Entropy for Mixed States

TL;DR

This paper extends the use of the Araki-Lieb inequality to calculate the von Neumann entropy of large systems when one subsystem starts in a mixed state, demonstrated through a two-level atom-field interaction.

Contribution

It introduces a method to compute the von Neumann entropy for mixed initial states using the Araki-Lieb inequality, which was previously limited to pure states.

Findings

Successfully applied the method to a two-level atom-field interaction.

Demonstrated the calculation of entropy for an infinite system with a mixed initial state.

Extended the applicability of the Araki-Lieb inequality in quantum entropy analysis.

Abstract

The Araki-Lieb inequality is commonly used to calculate the entropy of subsystems when they are initially in pure states as this forces the entropy of the two subsystems to be equal after the complete system evolves. Then, it is easy to calculate the entropy of a large subsystem by finding the entropy of the small one. To the best of our knowledge, there does not exist a way of calculating the entropy when one of the subsystems is initially in a mixed state. We show here that it is possible to use the Araki-Lieb inequality in this case and find the von Neumann entropy for the large (infinite) system. We show this in the two-level atom-field interaction.