Landauer Versus Nernst: What is the True Cost of Cooling a Quantum System?

Thermodynamics connects our knowledge of the world to our capability to manipulate and thus to control it. This crucial role of control is exemplified by the third law of thermodynamics, Nernst’s unattainability principle, which states that infinite resources are required to cool a system to absolute zero temperature. But what are these resources and how should they be utilized? And how does this relate to Landauer’s principle that famously connects information and thermodynamics? We answer these questions by providing a framework for identifying the resources that enable the creation of pure quantum states. We show that perfect cooling is possible with Landauer energy cost given infinite time or control complexity. However, such optimal protocols require complex unitaries generated by an external work source. Restricting to unitaries that can be run solely via a heat engine, we derive a novel Carnot-Landauer limit, along with protocols for its saturation. This generalizes Landauer’s principle to a fully thermodynamic setting, leading to a unification with the third law and emphasizes the importance of control in quantum thermodynamics.

Popular Summary

Thermodynamics provides the link between information processing and physics. The success of information theory resides in its ability to transcend physics and abstractly deal with information, independently of physical information carriers. In practice, however, one must always consider the physical cost to processing information. The most famous connection to this end is given by Landauer's principle, which states that any irreversible manipulation of information—such as the erasure of a bit—is inevitably accompanied by heat production.

More specifically, to erase a bit, one would need to dissipate at least $k_{B}T\log (2)$ units of heat. In principle, one could saturate this bound by performing the erasure in an infinitely slow manner. This fundamental limit holds true beyond the classical setting and also applies to erasing quantum bits (qubits). Here, an intriguing apparent contradiction with another thermodynamic law, namely Nernst's "unattainability principle" (or 3rd law), emerges: the 3rd law implies an infinite work cost to cool any finite-temperature system to absolute zero temperature. In the case of quantum systems, the law is a consequence of global unitary evolution between the target system and a thermal environment and therefore also applies to the erasure of qubits.

While this apparent paradox is resolved by restating the unattainability principle in terms of diverging resources—time for Landauer erasure; energy for conventional finite-time cooling—here we point out another oft-overlooked resource: complexity. Technically, perfect erasure would be possible with both finite time and energy resources given an unbounded level of control complexity. We explore this asymptotic trinity of informational and thermodynamic resources for quantum systems of various dimensions.

Figure 1

Framework. We consider the task of cooling a quantum system in two extremal control scenarios, with each step of both paradigms comprising two primitives. The top panel depicts the coherent-control scenario: in the control step (left), an agent can use a work source $\mathcal{W}$ to implement any global unitary on the system $\mathcal{S}$ and machine $\mathcal{M}$, which both begin thermal at inverse temperature $\beta$; in cooling the target, energy and entropy is transferred to the machine. The machine then rethermalizes with its environment (right), thereby dissipating the energy it gained in the control step. The bottom panel depicts the incoherent-control scenario: the machine is bipartitioned into a cold part at inverse temperature $\beta$ and a hot part at inverse temperature ${\beta }_H<\beta$. In the control step, the agent switches on an interaction between the three systems, represented by a global energy-conserving unitary $U_{\mathrm{EC}}$. In the rethermalization step, the interaction is turned off and both subsystems of the machine rethermalize to their respective initial temperatures; the hot part draws energy from the heat bath while the cold part dissipates heat to its environment. In both paradigms, we quantify the control complexity as the effective dimension accessed by the unitary operation in a given control step (i.e., the dimension of the system-machine Hilbert space upon which the unitary acts nontrivially).

Figure 2

Complexity. We consider structural (left) and control complexity (right). Structural complexity concerns properties of the machine Hamiltonian. For perfect cooling it is necessary that the largest energy gap diverges [see Eq. (5)]. Moreover, an infinite-dimensional machine with particular energy-level structure is required for saturation of the Landauer bound. Control complexity refers to properties of the unitary that represents a protocol. The yellow box in the foreground represents a unitary $U$ involving the entire machine, whereas the smaller yellow columns in the background represent a potential decomposition (e.g., of the diverging-time protocol) into unitaries $U_i$ involving certain subspaces of the overall machine. Not only must the target system interact with all levels of an infinite-dimensional machine for Landauer-cost cooling, it must do so in a fine-tuned way.

Figure 3

Imperfect cooling. We compare the cooling performance of a degenerate qubit target system using either $N$ machine qubits of linearly increasing energy accessed sequentially or a single unitary on a $2^N$-dimensional machine, the latter being a finite adaptation of a protocol presented in Ref. [29]. We set $\beta =1$, choose units such that $\hslash =k_B=1$, and fix $1-ϵ$ to be the desired final ground-state population of the target. We plot the inverse of the excess work cost above the Landauer limit, $W-\beta \tilde{\mathrm{\Delta }}{S}_{\mathcal{S}}$ (in units of the smallest machine energy gap, ${\omega }_{\mathcal{M}}^{\min }$), confirming that the surplus work cost in both cases scales with $N^{-1}$. Interestingly, we see that the protocol in which the target is sequentially swapped with machine qubits outperforms that which uses a high-dimensional unitary (at equal overall control complexity) in terms of energy cost required to reach a desired temperature.

Figure 4

Imperfect cooling with coherent and incoherent control. We compare the performance of coherent and incoherent control protocols for cooling a degenerate qubit target by swapping it with machine qubits with linearly increasing energy. The final ground-state population is fixed to be $0.99$. The inverse of the surplus work cost $W-\beta \tilde{\mathrm{\Delta }}{S}_{\mathcal{S}}$ (with $\beta =1$) is plotted (in units of the smallest machine energy gap, ${\omega }_{\mathcal{M}}^{\min }$) against the total number of unitary operations, with the temperature of the hot bath in the incoherent control protocols set to ${\beta }_H=\mathrm{\infty }$ in order to make meaningful comparison to the coherent control case. We see that the coherent control protocol (blue) outperforms the two incoherent ones (purple, red) at any given time. As discussed in the text, there are two choices for how to implement an incoherent control protocol of this type with fixed control complexity: The red line corresponds to a protocol in which a machine (subspace) with the same energy gap is reused 5 times before moving on to the next; on the other hand, the purple line depicts the case where there are no repetitions within each stage defined by a distinct energy gap in the machine. By inspection, the single-use incoherent protocol (purple) requires approximately $3$ times more unitaries to achieve the same efficiency as the coherent one (blue), whereas the five-repetition incoherent protocol (red) requires approximately 5.3 times as many unitaries as the coherent one.