Thermodynamics of memory erasure via a spin reservoir

Thermodynamics with multiple conserved quantities offers a promising direction for designing novel devices. For example, Vaccaro and Barnett's [J. A. Vaccaro and S. M. Barnett, Proc. R. Soc. A 467, 1770 (2011); S. M. Barnett and J. A. Vaccaro, Entropy 15, 4956 (2013)] proposed information erasure scheme, where the cost of erasure is solely in terms of a conserved quantity other than energy, allows for new kinds of heat engines. In recent work, we studied the discrete fluctuations and average bounds of the erasure cost in spin angular momentum. Here we clarify the costs in terms of the spin equivalent of work, called spinlabor, and the spin equivalent of heat, called spintherm. We show that the previously found bound on the erasure cost of ${\gamma }^{-1}ln2$ can be violated by the spinlabor cost, and only applies to the spintherm cost. We obtain three bounds for spinlabor for different erasure protocols and determine the one that provides the tightest bound. For completeness, we derive a generalized Jarzynski equality and probability of violation which shows that for particular protocols the probability of violation can be surprisingly large. We also derive an integral fluctuation theorem and use it to analyze the cost of information erasure using a spin reservoir.

Figure 1

An illustration of the erasure process for an arbitrary protocol. The upwards vertical direction represents increasing values of the $z$ component of angular momentum. The state of the spin reservoir is represented on the far left by a spin level diagram. The remaining spin level diagrams to its right represent the spin state of the memory-ancilla system at various stages of the erasure process. The value of $m$ is the number of cnot steps that have taken place. The illustration shows the specific case of $C=2$, where there are two probabilities at $m=C$, one before equilibration and one after equilibration. Other values are $p_{\uparrow }=0.5$, $p_{\downarrow }=1-p_{\uparrow }$, $Q_{\uparrow }\left(m\right)=\frac{e^{-(m+1)\gamma \hslash }}{1+e^{-(m+1)\gamma \hslash }}$ for $m\ge 0$, and $Q_{\downarrow }\left(m\right)=1-Q_{\uparrow }\left(m\right)$ for $m\ge 0$.

Figure 2

Spinlabor distribution for different protocols with $p=0.5$. The black line indicates the average value ${\langle {\mathcal{L}}_{\mathrm{s}}\rangle }_C$ of the spinlabor performed on the memory-ancilla system, and the pink line indicates the bound on the erasure cost, ${\gamma }^{-1}ln2$, derived in Refs. [8, 9] and quoted in (2). As discussed in the main text, a careful analysis shows that the erasure cost in Refs. [8, 9] is defined in terms of the spintherm absorbed by the reservoir, and it is demonstrated in (a), (c), and (d) that the bound does not apply to the average spinlabor. This highlights the need for care when considering the physical form of the erasure cost associated with a spin reservoir.

Figure 3

The values of $R$ in (20) as a function of $C$ and $\alpha$ for the maximal-stored-information case $p_{\downarrow }=p_{\uparrow }=0.5$. The value of the average spinlabor cost ${\langle {\mathcal{L}}_{\mathrm{s}}\rangle }_C$ is calculated using (11), and to enhance the graphical representation, the values of $R$ have been interpolated between the discrete values of $C$ (vertical gray lines). The black dots represent the four values chosen for Fig. 2.

Figure 4

A contour plot, similar to Fig. 3 and for the same maximal-stored-information case, of the right-hand side of (21) as a bound on the average spinlabor cost ${\langle {\mathcal{L}}_{\mathrm{s}}\rangle }_C$.

Figure 5

Spinlabor ${\mathcal{L}}_{\mathrm{s}}$ probability distribution for different protocols for $p_{\uparrow }=0.5$. The black line indicates the average value and the pink indicates the bound of ${\gamma }^{-1}(ln2-lnA)$. Notice that the average spinlabor cost in (c) is near the middle of the plot; this implies that there is no second peak in the distribution beyond the range plotted.

Figure 6

Probability of violation for different protocols with $p_{\uparrow }=p_{\downarrow }=0.5$. The variables are a one-to-one correspondence of the ordering in Fig. 5.

Figure 7

Spinlabor distribution for $C=10$. The black line indicates the average value and the pink ${\gamma }^{-1}ln{A}^{\prime }$.

Figure 8

Probability of violation for different asymmetric memory and $C=10$. The variables are a one-to-one correspondence to the ordering in Fig. 7.

Figure 9

Plot of $\mathrm{\Delta }B$ which compares Eq. (92) to Eq. (93) with $p_{\uparrow }=0.5$, $\alpha =0.01$–0.49, and $C=0$–10 discretely.

Figure 10

Spinlabor distribution with an overlapping Gaussian curve at $C=0$.