Two-qubit state recovery from amplitude damping based on weak measurement

In the quantum control process, arbitrary pure or mixed initial states need to be protected from amplitude damping through the noise channel using measurements and quantum control. However, how to achieve it on a two-qubit quantum system remains a challenge. In this paper, we propose a feed-forward control approach to protect arbitrary two-qubit pure or mixed initial states using the weak measurement. A feed-forward operation and measurements are used before the noise channel, and afterwards a reversed operation and measurements are applied to recover the state back to its initial state. In the case of two-qubit pure states, we use the unravelling trick to describe the state of the system in each step of the control procedure. For two-qubit mixed states, a completely-positive trace-preserving (CPTP) map is implemented. Finally, the fidelity and success probability are used to evaluate the effect of protection. The complete recovery conditions for the measurement strengths are derived, under which we achieve the optimal fidelity and the success probability of recovering the initial pure or mixed states.