Valleys in string foam suppress quantum coherence

String contributions to the $ matrix are derived relating in- and out-state density matrices that do not factorize as a product of S and St 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. (authors). 33 refs., 1 fig