  A general framework for analyzing the recently discovered phase transitions
in the steady state of dissipation-driven open quantum systems is still
missing. In order to fill this gap we extend the so-called fidelity approach to
quantum phase transitions to open systems whose steady state is a Gaussian
Fermionic state. We endow the manifold of correlations matrices of
steady-states with a metric tensor g measuring the distinguishability distance
between solutions corresponding to different set of control parameters. The
phase diagram can be then mapped out in terms of the scaling-behavior of g and
connections with the Liouvillean gap and the model correlation functions
unveiled. We argue that the fidelity approach, thanks to its
differential-geometric and information-theoretic nature, provides novel
insights on dissipative quantum critical phenomena as well as a general and
powerful strategy to explore them.
