  In the expanding universe, two interacting fields are no longer in thermal
contact when the interaction rate becomes smaller than the Hubble expansion
rate. After decoupling, two subsystems are usually treated separately in
accordance with equilibrium thermodynamics and the thermodynamic entropy gives
a fiducial quantity conserved in each subsystem. In this paper, we discuss a
correction to this paradigm from quantum entanglement of two coupled fields.
The thermodynamic entropy is generalized to the entanglement entropy. We
formulate a perturbation theory to derive the entanglement entropy and present
Feynman rules in diagrammatic calculations. For specific models to illustrate
our formulation, an interacting scalar-scalar system, quantum electrodynamics,
and the Yukawa theory are considered. We calculate the entanglement entropy in
these models and find a quantum correction to the thermodynamic entropy. The
correction is revealed to be important in circumstances of instantaneous
decoupling.
