Continuous measurements, quantum trajectories, and decoherent histories

Quantum open systems are described in the Markovian limit by master equations in Lindblad form. I argue that common ``quantum trajectory'' techniques representing continuous measurement schemes, which solve the master equation by unravelling its evolution into stochastic trajectories in Hilbert space, correspond closely to particular sets of decoherent (or consistent) histories. This is illustrated by a simple model of photon counting. An equivalence is shown for these models between standard quantum jumps and the orthogonal jumps of Di\'osi, which have already been shown to correspond to decoherent histories. This correspondence is compared to simple treatments of trajectories based on repeated or continuous measurements.