Fig. 2

The action of the cycle over the machine. The cycle S _m,n is represented in a pictorial way over the eigenstates of the d-dimensional machine (where d = m + n). Notice that the machine has a trivial Hamiltonian, and we order the eigenstates to only simplify the visualisation of the cycle. The upward arrow connecting two eigenstates of the machine represents a swap between these two states and the pair (\(\left| 0 \right\rangle _{\mathrm{P}}\),\(\left| 1 \right\rangle _{\mathrm{P}}\)) of the passive state. The downward arrow connecting two eigenstates of the machine represents a swap between this pair and the pair (\(\left| 1 \right\rangle _{\mathrm{P}}\),\(\left| 2 \right\rangle _{\mathrm{P}}\)) of the passive state. We initially perform m − 1 swaps between (\(\left| 0 \right\rangle _{\mathrm{P}}\),\(\left| 1 \right\rangle _{\mathrm{P}}\)) and \(\left\{ {\left( {\left| j \right\rangle_{\rm M}, \left| j + 1 \right\rangle_{\rm M}} \right)} \right\}_{j = 0}^{m - 2}\), and one swap between (\(\left| 0 \right\rangle _{\mathrm{P}}\),\(\left| 1 \right\rangle _{\mathrm{P}}\)) and \(\left( {\left| m - 1 \right\rangle_{\rm M}, \left| m + n - 1 \right\rangle_{\rm M}} \right)\). Then, we perform n − 1 swaps between (\(\left| 1 \right\rangle _{\mathrm{P}}\),\(\left| 2 \right\rangle _{\mathrm{P}}\)) and \(\left\{ {\left( {\left| j \right\rangle_{\rm M}, \left| j + 1 \right\rangle_{\rm M}} \right)} \right\}_{j = m}^{m + n - 2}\), and one swap between (\(\left| 1 \right\rangle _{\mathrm{P}}\),\(\left| 2 \right\rangle _{\mathrm{P}}\)) and \(\left( \left| 0 \right\rangle_{\rm M} , \left| m \right\rangle_{\rm M}\right)\). If we consider the arrow representation of swaps, we can see that the cycle is close, and this allows us to recover the local state of the machine M while extracting work