  We present a new variant of the quantum adversary method. All adversary
methods give lower bounds on the quantum query complexity of a function by
bounding the change of a progress function caused by one query. All previous
variants upper-bound the_difference_ of the progress function, whereas our new
variant upper-bounds the_ratio_ and that is why we coin it the multiplicative
adversary. The new method generalizes to all functions the new quantum
lower-bound method by Ambainis [Amb05, ASW06] based on the analysis of
eigenspaces of the density matrix. We prove a strong direct product theorem for
all functions that have a multiplicative adversary lower bound.
