We consider a class of quantum disordered XY chains where the eigenfunction correlator localization of the corresponding effective single particle Hamiltonian is satisfied. We consider the dynamics of the XY chain and we show that, starting from a broad class of product initial states, bipartite entanglement remains bounded for all times. Corollaries include area laws for the entanglement entropy of eigenstates and for the dynamics of any up-down configuration initial state. For the disordered isotropic XY chain we derive bounds for the particle number transport and prove that the expected number of particles that can penetrate out of a block of spins decays in distance. These results demonstrate the fact that the disordered XY chains are fully many-body localized.
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