Quantum Thermodynamics of Correlated-Catalytic State Conversion at Small Scale

The class of possible thermodynamic conversions can be extended by introducing an auxiliary system called catalyst, which assists in state conversion while its own state remains unchanged. We reveal a complete characterization of catalytic state conversion in quantum and single-shot thermodynamics by allowing an infinitesimal correlation between the system and the catalyst. Specifically, we prove that a single thermodynamic potential, which provides the necessary and sufficient condition for the correlated-catalytic state conversion, is given by the standard nonequilibrium free energy defined with the Kullback-Leibler divergence. This resolves the conjecture raised by Wilming, Gallego, and Eisert [Entropy 19, 241 (2017)] and by Lostaglio and Müller [Phys. Rev. Lett. 123, 020403 (2019)] in the positive. Moreover, we show that, with the aid of the work storage, any quantum state can be converted into another by paying the work cost equal to the nonequilibrium free energy difference. Our result would serve as a step towards establishing resource theories of catalytic state conversion in the fully quantum regime.

Figure 1

Schematic of our setup. We convert the system $S$ from $\rho$ to ${\rho }^{\prime }$ with the aid of the catalyst $C$ and the work storage $W$. The catalyst $C$ returns to its original state while it can correlate with the system. The work storage $W$ changes its state with energy difference $w\le F(\rho )-F({\rho }^{\prime })$ with probability arbitrarily close to unity.

Figure 2

(a) Schematic of the criterion $S_{\infty }({\rho }^{\prime }||{\rho }_{\text{Gibbs}})<S_{\infty }(\rho ||{\rho }_{\text{Gibbs}})$ represented by the dashed line in the $x\text{−}z$ plane of the Bloch sphere. We draw the states inconvertible from $\rho$ within error $ϵ$ in gray. (b) Schematic of the criterion $S_1({\rho }^{\prime }||{\rho }_{\text{Gibbs}})<S_1(\rho ||{\rho }_{\text{Gibbs}})$ in the $x\text{−}z$ plane of the Bloch sphere, where we draw the convertible and inconvertible states with a correlated catalyst from $\rho$ in red and gray, respectively. In particular, there exists a Gibbs-preserving map with the correlated catalyst converting $\rho \to {\rho }^{\prime }$.

Figure 3

Schematic of Step 3 of the proof. (a) The initial state of the composite $SC$. The vertical direction represents different systems $S_1,\dots ,S_8$, and the horizontal direction means their classical mixture. (b) Schematic of how the CPTP map $\mathrm{\Lambda }$ gives the desired catalytic state conversion.