  The physics of driven-dissipative transitions is currently a topic of great
interest, particularly in quantum optical systems. These transitions occur in
systems kept out of equilibrium and are therefore characterized by a finite
entropy production rate. However, very little is known about how the entropy
production behaves around criticality and all of it is restricted to classical
systems. Using quantum phase-space methods, we put forth a framework that
allows for the complete characterization of the entropy production in
driven-dissipative transitions. Our framework is tailored specifically to
describe photon loss dissipation, which is effectively a zero temperature
process for which the standard theory of entropy production breaks down. As an
application, we study the open Dicke and Kerr models, which present continuous
and discontinuous transitions, respectively.We find that the entropy production
naturally splits into two contributions. One matches the behavior observed in
classical systems. The other diverges at the critical point.
