  The entropy production rate is a key quantity in non-equilibrium
thermodynamics of both classical and quantum processes. No universal theory of
entropy production is available to date, which hinders progress towards its
full grasping. By using a phase space-based approach, here we take the current
framework for the assessment of thermodynamic irreversibility all the way to
quantum regimes by characterizing entropy production -- and its rate --
resulting from the continuous monitoring of a Gaussian system. This allows us
to formulate a sharpened second law of thermodynamics that accounts for the
measurement back-action and information gain from a continuously monitored
system. We illustrate our framework in a series of physically relevant
examples.
