  State selective protocols, like entanglement purification, lead to an
essentially non-linear quantum evolution, unusual in naturally occurring
quantum processes. Sensitivity to initial states in quantum systems, stemming
from such non-linear dynamics, is a promising perspective for applications.
Here we demonstrate that chaotic behaviour is a rather generic feature in state
selective protocols: exponential sensitivity can exist for all initial states
in an experimentally realisable optical scheme. Moreover, any complex rational
polynomial map, including the example of the Mandelbrot set, can be directly
realised. In state selective protocols, one needs an ensemble of initial
states, the size of which decreases with each iteration. We prove that
exponential sensitivity to initial states in any quantum system have to be
related to downsizing the initial ensemble also exponentially. Our results show
that magnifying initial differences of quantum states (a Schr\"odinger
microscope) is possible, however, there is a strict bound on the number of
copies needed.
