Stochastic dynamical reduction theories and superluminal communication

In theories which describe the reduction of the state vector as a physical process, the possibility exists, for certain experiments, of predictions which differ from those of quantum theory. These are ``interrupted reduction interference'' experiments, characterized by an interaction which triggers the reduction, followed rapidly (before the reduction is completed) by a measurement of interference between the superposed states that make up the state vector (possible examples: double Stern-Gerlach experiment, two-slit neutron interference). We consider the general class of stochastic reduction theories, and ask whether they allow superluminal communication by means of such experiments. We show that, if the state vector that precedes reduction is precisely reproducible, then superluminal communication can occur in certain circumstances, unless the off-diagonal elements of the density matrix decay exponentially, with a universal time constant. We also show, in that case, that no state vector ever reduces in a finite time, so such a theory is not satisfactory. However, superluminal communication can be avoided if reduction is triggered only in irreproducible state vectors, of such complexity that prior to reduction the ``effective'' density matrix, constructed from the ensemble of such state vectors and traced over the variables outside the experimenter's control, is diagonal. Then predictions are identical to those of quantum theory for ``interrupted reduction interference'' experiments and thus apparently for all experiments. The lesson of this paper is that the ``effective'' density matrix must always be used to make physical predictions in dynamical reduction theories. This supplies a resolution of the problem of reconciling state-vector reduction with relativity: even if the reduction dynamics is not relativistically invariant, its experimental predictions are. It also implies that the ``effective'' entropy increases during a measurement, but remains constant during reduction, which is the reverse of a common dictum.