Entanglement entropy in a certain nonlinear discrete quantum walk model

Recently, an interesting model featuring a modified conditional shift operator (which causes nonlinearity by dependence on the local occupation probability of the quantum walker) in discrete-time quantum walks has been introduced. In this model, where a few curious dynamical behaviors like solitonlike propagation, self-trapping and chaos have been found, the problem of quantum entanglement of resulting walker-coin system has not been considered. Utilizing numerical simulations in the present work, we investigate the time dependence of entanglement entropy, a measure of quantum entanglement, for the walker-coin system in this model. It appears that for all values of the nonlinearity parameter characterizing this model, there exists a time above which the entanglement entropy is very close to unity. Usually, after relatively small number of time steps from the beginning, the entanglement entropy becomes significant. This is associated with the creation of two solitons moving away from zero in opposite directions. Moreover, as we show in the present work, the so-called soliton formation time is closely connected with the time for which very high quantum entanglement appears for the first time during the evolution of the walker-coin system.