What do we learn from a quantum measurement?

The relation between the mathematical formalism of quantum mechanics and the underlying physical reality has been a source of intense debates since the early days of quantum theory. One point of contention involves the relation between measurements, represented by Hermitian operators, and what measurement result tell us about the real state of affairs, a problem which is often emphasized when we consider a sequence of measurements. The problem can be seen as a result of the unavoidable measurement back-action which can be reduced by increasing the measurement uncertainty.  In this talk, I will argue that weak measurements, with a negligible back-action and huge uncertainty, are a useful way to probe reality.  Starting from an operational approach to time symmetric quantum theory I will give an operational definition of weak measurements and show that the mean results of weak measurements (the so-called weak values) are a better probe of the world than strong measurements. I will then move on to a description of a recent experiment where we used weak measurements to construct the trajectories of quantum particles according to two `versions' of the de Broglie Bohm pilot wave theory.