Efficient generation of the triplet Bell state between coupled spins using transitionless quantum driving and optimal control

We consider a pair of coupled spins with Ising interaction in the z direction and study the problem of generating efficiently the triplet Bell state. We initially analyze the transitionless quantum driving shortcut to adiabaticity method and point out its limitations when the available duration approaches zero. In this short time limit we explicitly calculate the fidelity of the method and find it to be much lower than unity, no matter how large the available control fields become. We find that there is a lower bound on the necessary time to complete this transfer, set by the finite value of the interaction between the spins. We then use numerical optimal control to find bang-bang pulse sequences, as well as, smooth controls, which can generate high levels of the target Bell state in the minimum possible time. The results of the present work are not restricted only to spin systems, but are expected also to find applications in other physical systems which can be modeled as interacting spins, for example, coupled quantum dots.