Robust Entanglement in Anti-ferromagnetic Heisenberg Chains by Single-spin Optimal Control

We demonstrate how near-perfect entanglement (in fact arbitrarily close to maximal entanglement) can be generated between the end spins of an anti-ferromagnetic isotropic Heisenberg chain of length $N$, starting from the ground state in the $N/2$ excitation subspace, by applying a magnetic field along a given direction, acting on a single spin only. Temporally optimal magnetic fields to generate a singlet pair between the two end spins of the chain are calculated for chains up to length 20 using optimal control theory. The optimal fields are shown to remain effective in various non-ideal situations including thermal fluctuations, magnetic field leakage, random system couplings and decoherence. Furthermore, the quality of the entanglement generated can be substantially improved by taking these imperfections into account in the optimization. In particular, the optimal pulse of a given thermal initial state is also optimal for any other initial thermal state with lower temperature.