Entropic uncertainty relation under quantum channels with memory

Abstract: Recently, Xu et al. [Phys. Rev. A 86, 012113(2012)] explored the behavior of the entropic uncertainty relation under the influence of local unital and nonunital noisy channels for a class of Bell-diagonal states. We here reform their results and investigate the entropic uncertainty relation under the influence of unital and nonunital noisy channels with memory. Different types of noisy channels with memory, such as amplitude damping channel(nonunitary), phase-damping and depolarizing channels(unitary) have been taken into account. Some analytical or numerical results are presented. The effect of channels with memory on dynamics of the entropic uncertainties (or their lower bounds) has been discussed in detail. Compare with previous results, our results show that, the entropic uncertainties (or their lower bounds) subjecting to amplitude damping channel with memory will be reduced at first and then be lifted with the memory coefficient of channel $\mu$ increasing, however they will be only reduced under phase-damping and depolarizing channels with memory. Especially, in the limit of $\mu\rightarrow1$, the entropic uncertainties (or their lower bounds) could be well protected and immune to decoherence of channle. Moreover, the mechanism behind these phenomena are also explored by using the purity of state.