  With the increasing interest for the control of the system at the nano and
mesoscopic scales, studies have been focused on the limit of the energy
dissipation in an open system by refining the concept of the Maxwell's demon.
The well-known Sagawa-Ueda fluctuation theorem provides an explanation of the
demon: the absence of a part of demon's information leads to an improper
entropy production which violates the thermodynamic 2nd law. Realizing that the
demon contributes not only to the system but also to the environments, we
introduce the dissipative information to quantify the total contribution of the
demon, rather than using an improper entropy production. We prove a set of new
fluctuation theorems based on this, which can be used to uncover the truth
behind the demon: The controlled system does not violate the 2nd law at any
coarse-grained level for the demon's control. However, there exists an
inevitable demon-induced dissipative information which always increases the
entropy production. A consequence of these theorems is that, less work and more
heat can be extracted and generated respectively by a demon than the limits
predicted by the Ueda-Sagawa theorem. We also suggest a possible realization of
the experimental estimation of these work and heat bounds, which can be
measured and tested.
