  The phase diagram of a quantum XY spin chain with Gaussian-distributed random
anisotropies and transverse fields is investigated, with focus on the fidelity
susceptibility, a recently introduced quantum information theoretical measure.
Monitoring the finite-size scaling of the probability distribution of this
quantity as well as its average and typical values, we detect a
disorder-induced disappearance of criticality and the emergence of Griffiths
phases in this model. It is found that the fidelity susceptibility is not
self-averaging near the disorder-free quantum critical lines. At the Ising
critical point the fidelity susceptibility scales as a disorder-strength
independent stretched exponential of the system size, in contrast with the
quadratic scaling at the corresponding point in the disorder-free XY chain.
Along the line where the average anisotropy vanishes the fidelity
susceptibility appears to scale extensively, whereas in the disorder-free case
this point is quantum critical with quadratic finite-size scaling.
