Energetics of correlations in interacting systems

A fundamental connection between thermodynamics and information theory arises from the fact that correlations exhibit an inherent work value. For noninteracting systems this translates to a work cost for establishing correlations. Here we investigate the relationship between work and correlations in the presence of interactions that cannot be controlled or removed. For such naturally coupled systems, which are correlated even in thermal equilibrium, we determine general strategies that can reduce the work cost of correlations, and illustrate these for a selection of exemplary physical systems.

Figure 1

Advantage in correlation cost. During step $\mathrm{II}$ of the protocol to generate correlations between two qubits, an advantage over the noninteracting case arises when $\mathrm{\Delta }{\tilde{F}}_{S_i}$ from Eq. (11) becomes negative. $\mathrm{\Delta }{\tilde{F}}_{S_i}$ is plotted here against the temperature $T$ in units of $\omega$ (recall that we use units where $\hslash =k_{\mathrm{B}}=1$) for ${\alpha }_{\mathrm{II}}=0.5$, and the different curves correspond to values of $\epsilon$ (also in units of $\omega$) from $\epsilon =0$ (top) to $\epsilon =-1$ (bottom) in steps of 0.1. The advantage increases with increasing coupling strength $\epsilon$, but does not monotonically decrease with the temperature. Instead, the advantage becomes maximal at a finite temperature. Although curves are only shown for a fixed value ${\alpha }_{\mathrm{II}}=0.5$, we have checked that other values yield analogous behavior and the advantage increases monotonically with ${\alpha }_{\mathrm{II}}$.

Figure 2

Fermionic newly generated correlation cost. The difference in correlations that can be newly generated for an available energy $W$ (in units of $W_{\min }$) in the presence $\left(\mathrm{\Delta }{\mathcal{I}}_{\epsilon \ne 0}\right)$ and absence $\left(\mathrm{\Delta }{\mathcal{I}}_{\epsilon =0}\right)$ of interactions is shown for temperatures $T=0.1,...,1$ (in units of $\hslash \omega /k_B$) in steps of 0.1 (blue to red, top to bottom) for the ratios ${\epsilon }_{\mathrm{even}}/{\epsilon }_{\mathrm{odd}}=2,1$, and 0.5 in (a), (b), and (c), respectively.

Figure 3

Fermionic correlation cost. The maximal correlation of the final state that is achievable for a fixed input energy $W$ [in units of $W_{\min }$ from Eq. (36)] is shown for temperatures $T=0.1,...,1$ (in units of $\hslash \omega /k_B$) in steps of 0.1 (blue to red, top to bottom) for the ratios ${\epsilon }_{\mathrm{even}}/{\epsilon }_{\mathrm{odd}}=2,1$, and 0.5 in (a), (b), and (c), respectively. In all cases, the achievable final correlation is larger in the presence of interactions (solid lines) than in their absence (dashed lines). In (a) this can be seen from the inset plot, where the horizontal axis is not scaled with $W_{\min }$. For (b) and (c) one may deduce this directly from the plots, since the solid lines are strictly above their corresponding dashed lines, and $W_{\min }$ is maximal when ${\epsilon }_{\mathrm{even}}={\epsilon }_{\mathrm{odd}}=0$.

Figure 4

Fermionic thermal state correlation. The correlation of the initial thermal state $\tau \left(\beta \right)$, as measured by the mutual information ${\mathcal{I}}_{S_1{S}_2}$, depend on the (relative) and absolute sizes of the couplings ${\epsilon }_{\mathrm{even}}$ and ${\epsilon }_{\mathrm{odd}}$. The correlation is plotted for temperatures $T=0\left(0.01\right),...,1$ (in units of $\hslash \omega /k_B$) in steps of 0.05 (blue to red, top to bottom) for the ratios ${\epsilon }_{\mathrm{even}}/{\epsilon }_{\mathrm{odd}}=2,1$, and 0.5 in (a), (b), and (c), respectively.