A Cost/Speed/Reliability Tradeoff 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 erasing a reliable bit fast costs at least \(\log 2 - \log \left( 1 - {\text {e}}^{-\frac{\tau _e}{\tau _r}}\right) > \log 2\), which goes to \(\frac{1}{2} \log \frac{2\tau _r}{\tau _e}\) when \(\tau _r>>\tau _e\).

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