  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.
