  Weak values inferred from weak measurements have been proposed as a tool to
investigate trajectories of pre- and post-selected quantum systems. Are the
inferences drawn from the weak values about the past of a quantum particle
fully true? Can the two-state vector formalism predict everything that the
standard formalism of quantum mechanics can? To investigate these questions we
present a "which-path" gedanken experiment in which the information revealed by
a pre- and post-selected quantum system is surprisingly different from what one
would expect from the weak values computed using the two-state vector
formalism. In our gedanken experiment, a particle reveals its presence in
locations where the weak value of the projection operator onto those locations
was vanishingly small. Therefore our predictions turn out to be in
contradistinction to those made based on the nonvanishing weak values as the
presence indicators of the quantum particle. We propose a six port photon-based
interferometer setup as a possible physical realization of our gedanken
experiment.
