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We analytically investigate the role of entanglement in time-optimal stateevolution as an appli- cation of the quantum brachistochrone, a general methodfor obtaining the optimal time-dependent Hamiltonian for reaching a targetquantum state. As a model, we treat two qubits indirectly cou- pled through anintermediate qubit that is directly controllable, which represents a typicalsituation in quantum information processing. We find the time-optimal unitaryevolution law and quantify residual entanglement by the two-tangle between theindirectly coupled qubits, for all possible sets of initial pure quantum statesof a tripartite system. The integrals of the motion of the brachistochrone aredetermined by fixing the minimal time at which the residual entanglement ismaximized. Entan- glement plays a role for W and GHZ initial quantum states,and for the bi-separable initial state in which the indirectly coupled qubitshave a nonzero value of the 2-tangle.

Brachistochrone of Entanglement for Spin Chains