Thermodynamics of creating correlations: Limitations and optimal protocols

We establish a rigorous connection between fundamental resource theories at the quantum scale. Correlations and entanglement constitute indispensable resources for numerous quantum information tasks. However, their establishment comes at the cost of energy, the resource of thermodynamics, and is limited by the initial entropy. Here, the optimal conversion of energy into correlations is investigated. Assuming the presence of a thermal bath, we establish general bounds for arbitrary systems and construct a protocol saturating them. The amount of correlations, quantified by the mutual information, can increase at most linearly with the available energy, and we determine where the linear regime breaks down. We further consider the generation of genuine quantum correlations, focusing on the fundamental constituents of our universe: fermions and bosons. For fermionic modes, we find the optimal entangling protocol. For bosonic modes, we show that while Gaussian operations can be outperformed in creating entanglement, their performance is optimal for high energies.

Figure 1

Illustration of the general setup. Two quantum systems, $S_1$ and $S_2$, at thermal equilibrium with a bath at temperature $T$ are acted upon either by a unitary $U_S$ on the bipartite system or by a more general unitary $U_{SB}$ that also involves the bath. The application of these unitaries, which correlate the system, requires a supply of external energy. In this general setting, we determine the optimal amount of correlations and entanglement that can be generated in the system for any given amount of energy.

Figure 2

Illustration of the protocol. In the first step, the system is cooled down by a controlled interaction with the bath, and the heat $Q$ is transferred to the bath. The associated work cost is $W_{\mathrm{I}}$. In the second step, the system is isolated from the bath before it is correlated though a unitary operation, which effectively heats up the subsystems. The energy cost of the second step is $W_{\mathrm{II}}$.

Figure 3

Fermionic entanglement cost. The solid curves in (a) show the amount of entanglement (of formation) that can maximally be generated in the even subspace of two fermionic modes that are initially in a thermal state, for a given energy cost $W$. The curves are plotted for initial temperatures varying from $T=0$ to $T=1$ in steps of $0.1$ (top to bottom) in units $\hslash \omega /k_B$. The horizontal axis shows the relative energy cost, i.e., the fraction of $W$ and the minimal energy cost $W_{\max }=[2Tln(e^{\beta }+1)-\omega ]$ to generate a maximally entangled pure state. (b) The corresponding effective final temperature $T_{\mathrm{II}}\ge T$ of the marginals after the protocol.

Figure 4

Optimal bosonic entanglement. The curves in (a) show the optimal amount of entanglement (of formation) that can be generated by Gaussian operations on two bosonic modes, ${}_{S_1}$ and ${}_{S_2}$, of frequency $\omega$, which are initially in a thermal state of temperature $T$. The horizontal axis shows the supplied energy $W$ in units of $\omega$. (b) The local temperature $T_{\mathrm{II}}$ of the modes after the protocol for values of $W$ for which entanglement can be generated. The curves in both (a) and (b) are plotted for initial temperatures varying from $T=0$ to $T=1$ in steps of $0.1$ (top to bottom) in units $\hslash \omega /k_B$.

Figure 5

Optimal fermionic entanglement. The solid curves in (a) show the amount of entanglement (of formation) that can maximally be generated between two fermionic modes, $S_1$ and $S_2$, that are initially in a thermal state of temperature $T$, for a given energy cost $W$. The horizontal axis shows the relative energy cost, i.e., the fraction of $W$ and the energy cost $W_{\max }$. (b) The average particle numbers $N_{S_1}^{\mathrm{I}}$ (dashed lines) and $N_{S_2}^{\mathrm{I}}$ (solid lines) after the first step of the protocol, where we have assumed $N_{S_2}^{\mathrm{I}}\ge N_{S_1}^{\mathrm{I}}$ without loss of generality. The curves in both (a) and (b) are plotted for initial temperatures varying from $T=0$ to $T=1$ in steps of $0.1$ and for the limit $T\to \infty$ (bottom to top) in units $\hslash \omega /k_B$. The dashed curves in (a) show the corresponding curves of Fig. 3 for temperatures varying from $T=0$ to $T=1$ in steps of $0.1$ (top to bottom) as a comparison.

Figure 6

Comparison of Gaussian and non-Gaussian operations. The plot shows the amount of entanglement (of formation) that can maximally be generated between two bosonic modes in step $\mathrm{II}$ of the protocol. Both modes are assumed to have been cooled to temperature $T_{\mathrm{I}}$ in the first step. Using the energy $W_{\mathrm{II}}$, the solid curves show the optimal entanglement generated by Gaussian operations, while the dashed curves show the amount of entanglement generated by the non-Gaussian protocol. In both cases, the curves are plotted for temperatures varying from $T_{\mathrm{I}}=0$ to $T_{\mathrm{I}}=1$ in steps of $0.1$ (top to bottom) in units $\hslash \omega /k_B$.