Memory effects in a dynamical decoupling process

We establish a simple quantitative relationship between the environmental memory effects and the characteristics in a dynamical decoupling process. In contrast to previous works, our measures of non-Markovianity are tailored and extended to evaluate the strength of memory effects in dynamical decoupling. We find that if each kick commutes with the dynamical map of the uncontrolled system, then the change of the final dynamical map or the final state brought by the control (called the "effect of control") is upper (lower) bounded by the summation (difference) of the strengths of memory effects with and without control. We propose sufficient conditions for the commutation relation for parity kicks and illustrate our finding with a dissipative quantum Rabi model by numerical simulations where one or many cycles of parity kicks are implemented on the qubit. Besides, the results show that under certain conditions, the effect of control or the increase of performance by the control may be simply proportional to the strength of memory effects with or without control.