Analysis of the optimal solution of quantum metrology with quantum decoherence

Optimal control theory is applied to the quantum parameter estimation in the presence of decoherence. Concretely we look for the optimal control that maximizes quantum Fisher information for “twist and turn” problem, and the optimal solutions are confirmed by testing the optimality conditions.

Although the resulting optimal controls appear complicated, we identify two classes of problem-specific characterisitic states, the Heisenberg-limit (HL) state and the decoherence-free subspace, which are crucial in understanding the optimal quantum trajectories.

For the intemediate evolution time where the quantum decoherence is still small, the optimal control takes the system to the HL state. When the evolution time is long, the optimal control takes the system to the decoherence free subspace.