  We put forth a unifying formalism for the description of the thermodynamics
of continuously monitored systems, where measurements are only performed on the
environment connected to a system. We show, in particular, that the conditional
and unconditional entropy production, which quantify the degree of
irreversibility of the open system's dynamics, are related to each other by the
Holevo quantity. This, in turn, can be further split into an information gain
rate and loss rate, which provide conditions for the existence of informational
steady-states (ISSs), i.e. stationary states of a conditional dynamics that are
maintained owing to the unbroken acquisition of information. We illustrate the
applicability of our framework through several examples.
