Stochastic models for quantum state reduction give rise to statistical laws that are in many respects in agreement with those of standard quantum measurement theory. Here we construct a counterexample involving a Hamiltonian with degenerate eigenvalues such that the statistical predictions of stochastic reduction models differ from the predictions of quantum measurement theory. An idealised experiment is proposed whereby the validity of these predictions can be put to the test.
Stochastic models for quantum state reduction give rise to statistical laws that are in many respects in agreement with those of standard quantum measurement theory. Here we construct a counterexample involving a Hamiltonian with degenerate eigenvalues such that the statistical predictions of stochastic reduction models differ from the predictions of quantum measurement theory. An idealised experiment is proposed whereby the validity of these predictions can be put to the test.
Stochastic models for quantum state reduction give rise to statistical laws that are in most respects in agreement with those of quantum measurement theory. Here we examine the correspondence of the two theories in detail, making a systematic use of the methods of martingale theory. An analysis is carried out to determine the magnitude of the fluctuations experienced by the expectation of the observable during the course of the reduction process and an upper bound is established for the ensemble average of the greatest fluctuations incurred. We consider the general projection postulate of Lüders applicable in the case of a possibly degenerate eigenvalue spectrum, and derive this result rigorously from the underlying stochastic dynamics for state reduction in the case of both a pure and a mixed initial state. We also analyse the associated Lindblad equation for the evolution of the density matrix, and obtain an exact time-dependent solution for the state reduction that explicitly exhibits the transition from a general initial density matrix to the Lüders density matrix. Finally, we apply Girsanov's theorem to derive a set of simple formulae for the dynamics of the state in terms of a family of geometric Brownian motions, thereby constructing an explicit unravelling of the Lindblad equation.
We consider theories in which the Schr\"odinger equation is modified so that the reduction of the state vector becomes a dynamical process characterized by a mean reduction time ${\ensuremath{\tau}}_{R}$. It is generally believed that such a theory and quantum theory differently predict the outcome of any two experiments performed in rapid enough succession (separated by a time interval $\ensuremath{\lesssim}{\ensuremath{\tau}}_{R}$), such as Papaliolios's successive measurements of a photon's polarization. It is shown that the predictions of a plausible class of dynamical reduction theories (for which the ensemble average of ${|〈{\ensuremath{\varphi}}_{n}|\ensuremath{\psi}〉|}^{2}$ remains constant during the reduction process, where $|\ensuremath{\psi}〉$ is the state vector, and $|{\ensuremath{\varphi}}_{n}〉$ is any one of the basis states to which it may reduce) and quantum theory do not differ for any Papaliolios-type experiment. However, there are deviations from the predictions of quantum theory if the second experiment measures interference between the superposed states created by the first measurement. As an example, the result of a recent two-slit neutron interference experiment by Zeilinger et al. is applied to the theory of Pearle, placing an upper limit ${\ensuremath{\tau}}_{R}\ensuremath{\gtrsim}5$ sec on the neutron self-reduction time.
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.
An experimental test of the “special state” theory of quantum measurement is proposed. It should be feasible with present-day laboratory equipment and involves a slightly elaborated Stern–Gerlach setup. The “special state” theory is conservative with respect to quantum mechanics, but radical with respect to statistical mechanics, in particular regarding the arrow of time. In this article background material is given on both quantum measurement and statistical mechanics aspects. For example, it is shown that future boundary conditions would not contradict experience, indicating that the fundamental equal-a-priori-probability assumption at the foundations of statistical mechanics is far too strong (since future conditioning reduces the class of allowed states). The test is based on a feature of this theory that was found necessary in order to recover standard (Born) probabilities in quantum measurements. Specifically, certain systems should have “noise” whose amplitude follows the long-tailed Cauchy distribution. This distribution is marked by the occasional occurrence of extremely large signals as well as a non-self-averaging property. The proposed test is a variant of the Stern–Gerlach experiment in which protocols are devised, some of which will require the presence of this noise, some of which will not. The likely observational schemes would involve the distinction between detection and non-detection of that “noise”. The signal to be detected (or not) would be either single photons or electric fields (and related excitations) in the neighborhood of the ends of the magnets.
Theories including a collapse mechanism have been presented various years ago. They are based on a modification of standard quantum mechanics in which nonlinear and stochastic terms are added to the evolution equation. Their principal merits derive from the fact that they are mathematically precise schemes accounting, on the basis of a unique universal dynamical principle, both for the quantum behavior of microscopic systems as well as for the reduction associated to measurement processes and for the classical behavior of macroscopic objects. Since such theories qualify themselves not as new interpretations but as modifications of the standard theory they can be, in principle, tested against quantum mechanics. Recently, various investigations identifying possible crucial test have been discussed. In spite of the extreme difficulty to perform such tests it seems that recent technological developments allow at least to put precise limits on the parameters characterizing the modifications of the evolution equation. Here we will simply mention some of the recent investigations in this direction, while we will mainly concentrate our attention to the way in which collapse theories account for definite perceptual process. The differences between the case of reductions induced by perceptions and those related to measurement procedures by means of standard macroscopic devices will be discussed. On this basis, we suggest a precise experimental test of collapse theories involving conscious observers. We make plausible, by discussing in detail a toy model, that the modified dynamics can give rise to quite small but systematic errors in the visual perceptual process.