Suppressing phase damping decoherence by periodical imperfect projective measurements

We explore how to overcome phase damping decoherence when imperfect projective measurements are available. It is demonstrated that a “relaxed” control goal may be achieved by combining simple open-loop coherent control with periodic imperfect projective measurements. Inspired by the idea of soft optimization, we propose to control the state of a qubit staying near a reference pure state with high probability for a sufficiently long time. This “relaxed” control goal is expressed in terms of three-parameter performance indexes, and it is in remarked contrast to the “hard” requirement that the state of the controlled qubit always stays at the reference pure state. We not only establish necessary conditions for relaxed robust problems, but also present sufficient conditions for the existence of a solution to the relaxed robust problems. With the help of main results, one can estimate how the maximal total time depends on the target state and the perturbation bounds. Furthermore, it is demonstrated that imperfect realization of projective measurements will worsen the three-parameter performance indexes.