Nonlinear evolution is not a usual phenomenon in quantum physics. It is possible to define a time-evolution for an ensemble of equally prepared systems in a somewhat unusual way: take N systems, apply an entangling unitary transformation, measure all but one of the systems and, depending on the measurement results, keep or throw away the remaining system. This procedure applied to the whole ensemble results in a new ensemble, the state of which being a nonlinear transformation of the initial quantum state. Entanglement distillation protocols are a prime example of this procedure.
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Iterating the above type of protocols can lead to nontrivial time-evolution, which can become chaotic. We discuss various properties of the emerging dynamics for one- and two-qubit systems. For increasingly noisy initial states, a sudden change in the character of the evolution occurs, similar to a phase transition. In the case of qubit pairs, asymptotic entanglement can depend sensitively on the initial state. We report on the realization of a couple of steps for two of the protocols in optical experiments. Furthermore, we discuss possible applications, e.g. benchmarking quantum computers.