A Cost / Speed / Reliability Trade-off to Erasing

We present a KL-control treatment of the fundamental problem of erasing a bit. We introduce notions of "reliability" of information storage via a reliability timescale $\tau_r$, and "speed" of erasing via an erasing timescale $\tau_e$. Our problem formulation captures the tradeoff between speed, reliability, and the Kullback-Leibler (KL) cost required to erase a bit. We show that rapid erasing of a reliable bit costs at least $\log 2 - \log\left(1 - \operatorname{e}^{-\frac{\tau_e}{\tau_r}}\right) > \log 2$, which goes to $\frac{1}{2} \log\frac{2\tau_r}{\tau_e}$ when $\tau_r>>\tau_e$.