  We exhibit string contributions to the $\nd{S}$ matrix relating in- and out-
state density matrices that do not factorize as a product of $S$ and
$S^\dagger$ matrices. They are associated with valley trajectories between
topological defects on the string world sheet, that appear as quantum
fluctuations in the space-time foam. Through their ultraviolet cut-off
dependences these valleys cause non-Hamiltonian time evolution and suppress
off-diagonal entries in the density matrix at large times.
