  Sagawa and Ueda established a fluctuation theorem of information exchange by
revealing the role of correlations in stochastic thermodynamics and unified the
non-equilibrium thermodynamics of measurement and feedback control [T. Sagawa
and M. Ueda, Phys. Rev. Lett. 109, 180602 (2012)]. They considered a process
where a non-equilibrium system exchanges information with other degrees of
freedom such as an observer or a feedback controller. They proved the
fluctuation theorem of information exchange under the assumption that the state
of the other degrees of freedom that exchange information with the system does
not change over time while the states of the system evolve in time. Here we
relax this constraint and prove that the same form of the fluctuation theorem
holds even if both subsystems co-evolve during information exchange processes.
This result may extend the applicability of the fluctuation theorem of
information exchange to a broader class of non-equilibrium processes, such as a
dynamic coupling in biological systems, where subsystems that exchange
information interact with each other.
