Abstract We investigate how to prepare a Bell entangled state deterministically by using two remote superconducting qubits coupled to a coplanar waveguide resonator via Lyapunov control. The results show that when spontaneous emission of the qubits is not considered, the qubits can be completely driven to the Bell entangled state by the control fields, while the spontaneous emission of the qubits is considered, the fidelity between the prepared state and the Bell entangled state can still reach a high value even in the case with a relatively large spontaneous emission rate in qubits. We also find that stronger control fields can accelerate the evolution of the state from the initial state to the Bell entangled state. In addition, there is no doubt that the larger spontaneous emission would lead to faster decoherence, which will accelerate the system to reach the target steady state. In other words, our scheme provides an efficient way for the preparation of entangled states in superconducting systems.
When a two-qubit system is initially maximally entangled, two independent decoherence channels, one per qubit, would greatly reduce the entanglement of the two-qubit system when it reaches its stationary state. We propose a method on how to minimize such a loss of entanglement in open quantum systems. We find that the quantum entanglement of general two-qubit systems with controllable parameters can be controlled by tuning both the single-qubit parameters and the two-qubit coupling strengths. Indeed, the maximum fidelity
Fmax between the stationary entangled state, ${\ensuremath{\rho}}_{\ensuremath{\infty}}$, and the maximally entangled state, ${\ensuremath{\rho}}_{m}$, can be about $2/3\ensuremath{\approx}\text{max}{\text{tr}({\ensuremath{\rho}}_{\ensuremath{\infty}}{\ensuremath{\rho}}_{m})}={F}_{\text{max}}$, corresponding to a maximum stationary concurrence,
Cmax, of about $1/3\ensuremath{\approx}C({\ensuremath{\rho}}_{\ensuremath{\infty}})={C}_{\text{max}}$. This is significant because the quantum entanglement of the two-qubit system can be produced and kept, even for a long time. We apply our proposal to several types of two-qubit superconducting circuits and show how the entanglement of these two-qubit circuits can be optimized by varying experimentally controllable parameters.