  We study quantum spin systems with quenched Gaussian disorder. We prove that
the variance of all physical quantities in a certain class vanishes in the
infinite volume limit. We study also replica symmetry breaking phenomena, where
the variance of an overlap operator in the other class does not vanish in the
replica symmetric Gibbs state. On the other hand, it vanishes in a spontaneous
replica symmetry breaking Gibbs state defined by applying an infinitesimal
replica symmetry breaking field. We prove also that the finite variance of the
overlap operator in the replica symmetric Gibbs state implies the existence of
a spontaneous replica symmetry breaking.
