  Fluctuation theorems allow one to make generalised statements about the
behaviour of thermodynamic quantities in systems that are driven far from
thermal equilibrium. In this article we use Crooks' fluctuation theorem to
understand the entropy production of a continuously measured, zero-temperature
quantum system; namely an optical cavity measured via homodyne detection. At
zero temperature, if one uses the classical definition of inverse temperature
$\beta$, then the entropy production becomes divergent. Our analysis shows that
the entropy production can be well defined at zero temperature by considering
the entropy produced in the measurement record leading to an effective inverse
temperature $\beta_{\rm eff}$ which does not diverge. We link this result to
the Cram\'er-Rao inequality and show that the product of the Fisher information
of the work distribution with the entropy production is bounded below by half
of the square of the effective inverse temperature $\beta_{\rm eff}$. This
inequality indicates that there is a minimal amount of entropy production that
is paid to acquire information about the work done to a quantum system driven
far from equilibrium.
