  We establish the fluctuation theorem in the presence of information exchange
between a nonequilibrium system and other degrees of freedom such as an
observer and a feedback controller, where the amount of information exchange is
added to the entropy production. The resulting generalized second law sets the
fundamental limit of energy dissipation and energy cost during the information
exchange. Our results apply not only to feedback-controlled processes but also
to a much broader class of information exchanges, and provides a unified
framework of nonequilibrium thermodynamics of measurement and feedback control.
