Ground state properties of a fully frustrated quantum spin system

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