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Entanglement propagation and typicality of measurements in the quantum Kac ring

We study the time evolution of entanglement in a quantum version of the Kac ring. Our model consists of two spin chains and quantum gates instead of the classical markers. The gates take one qubit from each ring at a time as an input and entangle them. Subsequently, one ring is rotated. This protocol creates non-trivial entanglement between the two rings, which we measure by the entanglement entropy. The features of the entanglement evolution can best be understood by using knowledge about the behavior of an ensemble of classical Kac rings. For instance, the recurrence time of this quantum many-body system is twice the length of the chain and "thermalization" only occurs on time scales much smaller than the dimension of the Hilbert space. The model thus elucidates the relation between distribution of measurement results in quantum and classical systems: While in classical systems repeated measurements are performed over an ensemble of systems, the corresponding result is obtained by measuring the same quantum system prepared in an appropriate superposition repeatedly.