Impossible weak values

Weak values characterize quantum systems that are both pre- and post-selected. Such systems are described by two wavefunctions, the overlap of which defines the success probability of the experiment. In weak value theory, it is usually assumed that the weak value does not exist for zero-overlapping pre- and post-selected wavefunctions. In this paper, we show that weak values can be defined even when the success probability of the experiment tends to zero. We call such weak values impossible. Standard quantum expectation values can be constructed from weak values. Remarkably, impossible weak values vanish in the expectation value, but may appear in its dynamics. We show that impossible weak values are not unique for a given pre- and post-selected state pair. They depend on the path in state space along which the pair is approached.