Weak Measurements, Quantum State Collapse and the Born Rule

Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigenstate will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. We revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. We describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigenstates. Investigation of the restrictions needed on the ensemble of quantum trajectories, so as to reproduce projective measurement in the appropriate limit, shows that the Born rule follows when the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation-dissipation relation. That implies that both the noise and the attraction have a common origin in the measurement interaction between the system and the apparatus. We analyse the quantum trajectory ensemble for the scenarios of quantum diffusion and binary quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter. This trajectory ensemble distribution can be tested in weak measurement experiments. We also comment on how the required noise may arise in the measuring apparatus.