  We find that ground states of the quantum Heisenberg antiferromagnet on the
geometrically frustrated pyrochlore checkerboard lattice are singlets and can
be expressed in terms of positive matrices. The magnetization at zero external
field vanishes for each frustrated tetrahedral unit separately and there is an
upper bound of 1/8 in natural units on the susceptibility both for the ground
state and at finite temperature. These results are the first exact ones in this
field and generalize to some other lattices; the approach is also of interest
for other spin systems.
