POVM generated quantum trajectories without stochastic differential equations

In this paper we examine the issue of quantum trajectories generated by QND-POVM's on {\it single} copies of unknown states. After an introduction to various aspects of quantum measurements, we discuss an earlier approach by one of us(NDH) based on Gaussian QND measurement operators that addressed the asymptotic behaviour of such trajectories showing the impossibility of determining the unknown state of a single copy from the statistics of such repeated measurements.The essence of our present work is the so called martingale and super-martingale properties of certain observables, and the consequent martingale convergence theorem. The main result is that asymptotically all trajectories approach either the non-degenerate eigenstates of the system observable, or,density matrices spanned by the degenerate eigenstates of the observable. The proofs given by us are very transparent..A unified treatment of both the degenerate and non-degenerate cases is given with the help of projectors of arbitrary dimensionalities.In the degenerate case we reproduce the Lüders prescription. The distribution of the trajectories is shown to be given exactly by the Born rule.Similar conclusions were reached, earlier to us, by Bauer et al on the one hand, and, by Amini et al on the other. A detailed comparison of the three approaches is given. A distinctive feature of all three approaches is that no use is made of stochastic differential equations and the conclusions follow directly from quantum mechanics. Alter and Yamomoto were the first to investigate repeated QND measurements on single copies in unknown states. We make detailed comparisons with their works too. We end with a brief discussion of i) the robustness of the results against free evolutions of both the system as well as the probe and ii) the anti-Zeno aspects of the results.