A fundamental connection between superconductivity and quantum spin fluctua
tions in underdoped cuprates is revealed. A variational calculation shows t
hat Cooper pair hopping strongly reduces the local magnetization m(0). This
effect pertains to recent neutron scattering and muon spin rotation measur
ements in which m(0) varies weakly with hole doping in the poorly conductin
g regime, but drops precipitously above the onset of superconductivity.