  We consider the asymmetric formulation of quantum hypothesis testing, where
two quantum hypotheses have different associated costs. In this problem, the
aim is to minimize the probability of false negatives and the optimal
performance is provided by the quantum Hoeffding bound. After a brief review of
these notions, we show how this bound can be simplified for pure states. We
then provide a general recipe for its computation in the case of multimode
Gaussian states, also showing its connection with other easier-to-compute lower
bounds. In particular, we provide analytical formulas and numerical results for
important classes of one- and two-mode Gaussian states.
