Speaker
Description
Closed quantum systems are described by a deterministic, linear evolution of their quantum states. Observation of a system breaks this rule: we gain information and the description becomes probabilistic. Provided we have multiple copies of a system in the same quantum state, we can design a protocol consisting of a joint unitary operation on all of them and a subsequent measurement on all but one of the outputs. Accepting only certain measurement outcomes, we arrive, in general, at a nonlinear quantum transformation on the remaining system. These nonlinear quantum transformations play an important role in quantum state distillation, aiding quantum communication schemes. We consider such iterated nonlinear protocols for qubits, involving a generalized CNOT, a measurement, and a single-qubit Hadamard gate processing n inputs in identical quantum states. In our work, we study the asymptotic properties of this special class of iterated nonlinear maps using numerical methods. We determine the fixed points, relevant fixed cycles and show the presence of a universal phase transition related to the disappearance of the fractalness of borders of different basins of attraction. The transition point is marked by a repelling fixed point, which has an increasing purity, tending to 1 as we increase the order n of the map. The generalized Julia set, defined as the set of border points between the basins of attraction for fixed purity initial states, has a constant fractal dimension as a function of the purity, above a critical value. Remarkably, we found that this fractal dimension is universal in this family, it is independent of the order n of the map.
