Influence of measurement error on Maxwell's demon

In any general cycle of measurement, feedback, and erasure, the measurement will reduce the entropy of the system when information about the state is obtained, while erasure, according to Landauer's principle, is accompanied by a corresponding increase in entropy due to the compression of logical and physical phase space. The total process can in principle be fully reversible. A measurement error reduces the information obtained and the entropy decrease in the system. The erasure still gives the same increase in entropy, and the total process is irreversible. Another consequence of measurement error is that a bad feedback is applied, which further increases the entropy production if the proper protocol adapted to the expected error rate is not applied. We consider the effect of measurement error on a realistic single-electron box Szilard engine, and we find the optimal protocol for the cycle as a function of the desired power $P$ and error $\varepsilon$.

Figure 1

Schematic of reversible and irreversible measurement in a two-bit total system (system + memory). The four logical subspaces are 00/01/10/11.

Figure 2

How a system evolves from step $A$ to $D$ in Fig. 1 after a measurement error.

Figure 3

Level diagram after the first measurement and raising the potential of the empty island that does not cost any work.

Figure 4

Left: Example protocol during time $0<t<\tau$. Right: Level diagram at $t=\tau$.

Figure 5

Completing the cycle.

Figure 6

The main figure shows $\tau$ as a function of $P$ for different $\varepsilon$ (in steps of 0.02). The inset gives the scaled form of the same data, with $\tau$ as a function of $P/P^{max}$.

Figure 7

$V\left(t\right)$ for $\varepsilon =0.1$ and several values of $P$.

Figure 8

The fraction of the total entropy production $S_{\text{tot}}$ that is due to the measurement error $S_{\varepsilon }$ as a function of power for various values of $\varepsilon$.

Figure 9

$V_{\tau }$ as function of $P$ for different $\varepsilon$. The inset shows enlarged what happens for small errors and powers.

Figure 10

$p_{\tau }$ as a function of $P$ for different $\varepsilon$.

Figure 11

$\dot{S}/P$ as a function of $P$ with labels on the curves giving $\varepsilon$. For each curve, the value at $P=0$ and the slope of the tangent at that point will give the coefficients $c_1$ and $c_2$ of Eq. (A1). These are shown as functions of $\varepsilon$ in the inset, together with $c_1$ from Eq. (A2).