Finite-time erasing of information stored in fermionic bits

We address the issue of minimizing the heat generated when erasing the information stored in an array of quantum dots in finite time. We identify the fundamental limitations and trade-offs involved in this process and analyze how a feedback operation can help improve it.

Figure 1

(a) The information stored in the array of single-level quantum dots is erased by putting the quantum dots one after another in contact with the metallic lead and applying, during the contact time $t$, a time-dependent protocol on the lead chemical potential $\mu \left(t\right)$. (b) The decrease of the initial Shannon entropy $S_i=k_B\ln 2$ is accompanied by a heat release $-Q$ in the environment.

Figure 2

Minimal heat generated $-Q/T$ when erasing in a finite time $t$ initial information $S_i=T\ln 2$ ($p_i=1/2$) with a remaining error $S_f$ ($p_f$). The horizontal (red) curves on the surface correspond to $Q/T=0.05,0.3,\ln 2\text{,}\text{and}0.8$. The vertical plane (orange) corresponds to $p_f=p_c$ where the heat diverges. The (black) curve in the $t=20$ plane corresponds to Landauer's lower bound where $Q/T=\Delta S$.

Figure 3

Erasing efficiency $\eta$ corresponding to the erasure process in Fig. 2. The horizontal (red) curves on the surface correspond to $\eta =0.3,0.5\text{,}\text{and}0.7$.

Figure 4

Erasure power $\mathcal{P}$ as a function of heat generated $-Q/T$ and the duration of the erasure $t$. The (red) curves on the surface correspond to $Q/T=0.2,0.7\text{,}\text{and}1.2$.

Figure 5

Shown from top to bottom is the erasing efficiency at maximum erasing power (black), the corresponding contact time (green), final probability (red), and power (blue).

Figure 6

(a) As the lead enters in contact with the quantum dot to be erased, an imperfect measurement is performed. (b) The time-dependent protocol $\mu \left(t\right)$ applied during the contact time $t$ depends on the output of that measurement.

Figure 7

Ratio $Q_F/Q$ as a function of the final probability $p_f$ for different erasure times $t$. The curves bend down as $t$ increases and the dashed curve corresponds to the quasistatic limit (35). The vertical lines denote the critical probability $p_c$ reachable for each erasure time $t$. The measurement error is fixed at $\delta =0.02$.

Figure 8

Ratio $Q_F/Q$ as a function of the measurement error $\delta$ for different erasure times $t$. The final probability is fixed at $p_f=0.01$. The dotted curve represents the ratio at the critical time $t_{\min }=-\ln 2p_f$ and the dashed curve corresponds to the quasistatic limit (35).