Correlating Thermal Machines and the Second Law at the Nanoscale

Thermodynamics at the nanoscale is known to differ significantly from its familiar macroscopic counterpart: The possibility of state transitions is not determined by free energy alone but by an infinite family of free-energy-like quantities; strong fluctuations (possibly of quantum origin) allow one to extract less work reliably than what is expected from computing the free-energy difference. However, these known results rely crucially on the assumption that the thermal machine is not only exactly preserved in every cycle but also kept uncorrelated from the quantum systems on which it acts. Here, we lift this restriction: We allow the machine to become correlated with the microscopic systems on which it acts while still exactly preserving its own state. Surprisingly, we show that this possibility restores the second law in its original form: Free energy alone determines the possible state transitions, and the corresponding amount of work can be invested or extracted from single systems exactly and without any fluctuations. At the same time, the work reservoir remains uncorrelated from all other systems and parts of the machine. Thus, microscopic machines can increase their efficiency via clever “correlation engineering” in a perfectly cyclic manner, which is achieved by a catalytic system that can sometimes be as small as a single qubit (though some setups require very large catalysts). Our results also solve some open mathematical problems on majorization which may lead to further applications in entanglement theory.

Popular Summary

The second law of thermodynamics, which states that the entropy of a closed system cannot spontaneously decrease, is a bedrock of physics. But without modification, it might not apply in the quantum world that is relevant to so much of modern technology. Recent research has shown that the second law has to be supplemented by a family of many second laws. However, here we present a surprising twist: The standard second law does uniquely characterize what is possible in the microscopic world if machines act on each particle only once. While textbook thermodynamics claims that entropy and free energy only apply to large numbers of particles on average, these results show that they also determine what is possible if machines act on single particles, quantifying the exact amount of work to be spent or extracted.

For large numbers of weakly interacting particles, a process is allowed if entropy does not decrease. But for single or strongly correlated particles, this criterion alone is not sufficient; quantum fluctuations will tend to dominate, hence the need for additional “second laws.” Yet, we show that all these additional constraints vanish if the machine is allowed to build up correlations with the particles on which it acts, a strategy that is not available or even harmful in the many-particle case.

This result restores the sufficiency of the original second law, points out clever strategies that machines may use in the microscopic world, and gives a novel microscopic meaning to entropy and free energy, which may have additional applications in entanglement theory.

Figure 1

Thermal operation of the form that we are considering in this paper; compare Fig. 1 in Ref. [10]. We have a system $A$ that we would like to act on, by transforming its state ${\rho }_A$ into another state ${\rho }_A^{\prime }$. We have access to a heat bath with an arbitrary Hamiltonian $H_B$, which is in its thermal state ${\gamma }_B$ at some fixed temperature $T$. The thermal machine contains a quantum system in state ${\sigma }_M$, and it controls a unitary transformation $U_{AMB}$ (symbolized by the pentagon), acting on the system $A$, heat bath $B$, and its internal system $M$. Crucially, this transformation is fully energy preserving, i.e., $[U_{AMB},H_A+H_M+H_B]=0$. By tracing over the heat bath, we obtain the map ${\sigma }_{AM}={\mathrm{Tr}}_B[U_{AMB}({\rho }_A\otimes {\sigma }_M\otimes {\gamma }_B)U_{AMB}^{†}]$, which is, in total, a thermal operation, ${\rho }_A\otimes {\sigma }_M\mapsto {\sigma }_{AM}$. We demand that the machine’s internal state ${\sigma }_M$ is exactly preserved (hence, ${\sigma }_M$ is often called a “catalyst”: It enables the transformation but is not consumed in the process), and we would like the resulting state of $A$, ${\mathrm{Tr}}_M{\sigma }_{AM}$, to be identical (or very close) to the desired target state ${\rho }_A^{\prime }$. The difference to Ref. [10] is that we allow correlations to build up between $A$ and $M$. If work is spent or extracted, we model this by an additional two-level system (“work bit”) $W$ which, initially as well as finally, is enforced to be exactly in an energy eigenstate (ground state $|g⟩$ or excited state $|e⟩$). This restriction ensures that $W$ remains uncorrelated with all other systems that are involved in this process; hence, the resulting work $\mathrm{\Delta }$ can be reliably transferred to or from an external battery.

Figure 2

Example of a work cost scenario without allowing correlations to build up. A qubit $A$, initially in equilibrium, is supposed to be heated up to an infinite temperature by spending some energy $\mathrm{\Delta }$ and by using a (potentially large) catalytic system $M$ that remains uncorrelated with $AW$ (and unchanged by the process). A transition of this form is possible only at a work cost of at least $\mathrm{\Delta }⪆0.4k_{B}T$.

Figure 3

Example work cost if correlations between $M$ and $A$ are allowed to build up. Since $M$ is locally exactly preserved, it can be reused on further states (just not on those ones on which it has already acted before). This transition is possible at a work cost of only $\mathrm{\Delta }⪆0.26k_{B}T$ (about $1/3$ less than in Fig. 2), even though the catalyst $M$ consists only of a single qubit.

Figure 4

The extension $p_{X{Y}_1}^{\prime }$ of $p_X^{\prime }$ that is used in the main text to establish sufficiency of the entropy conditions in the main theorem. According to Eq. (7), the goal is to build an extension such that the blue curve (that is, the $\alpha$-Rényi entropy balance) attains only positive values. The plot is for $m=3$, $\delta ={10}^{-3}$, $p=p_X=(\frac{91}{100},\frac{1}{20},\frac{1}{25})$, $Y_1={\mathbb{R}}^{n^2+n+1}$ with $n={10}^{15}$, and $p^{\prime }=p_X^{\prime }=(\frac{17}{20},\frac{7}{50},\frac{1}{100})$. Since $H_{\alpha }(p)>H_{\alpha }(p^{\prime })$ for $0<\alpha \le \frac{1}{3}$, there does not exist $c_M$ such that Eq. (5) holds true; i.e., no standard catalytic thermal operation can transform $p$ into $p^{\prime }$. Nevertheless, the transition can be achieved by a correlating catalytic thermal operation. The shaded colors show how different entries of $p_{X{Y}_1}^{\prime }$ are responsible for (the positivity of) different parts of the curve, as explained in the main text. In the limit $n\to \infty$, only positivity at $\alpha =1$, i.e., positive balance of Shannon entropy, remains as a necessary condition.

Figure 5

The thermal Lorenz curves signify the possibility of state transition (d1) by a thermal operation.