Catalytic transformations with finite-size environments: applications to cooling and thermometry
Fritz Haber Research Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel
Published: | 2021-09-21, volume 5, page 547 |
Eprint: | arXiv:2010.09070v3 |
Doi: | https://doi.org/10.22331/q-2021-09-21-547 |
Citation: | Quantum 5, 547 (2021). |
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Abstract
The laws of thermodynamics are usually formulated under the assumption of infinitely large environments. While this idealization facilitates theoretical treatments, real physical systems are always finite and their interaction range is limited. These constraints have consequences for important tasks such as cooling, not directly captured by the second law of thermodynamics. Here, we study catalytic transformations that cannot be achieved when a system exclusively interacts with a finite environment. Our core result consists of constructive conditions for these transformations, which include the corresponding global unitary operation and the explicit states of all the systems involved. From this result we present various findings regarding the use of catalysts for cooling. First, we show that catalytic cooling is always possible if the dimension of the catalyst is sufficiently large. In particular, the cooling of a qubit using a hot qubit can be maximized with a catalyst as small as a three-level system. We also identify catalytic enhancements for tasks whose implementation is possible without a catalyst. For example, we find that in a multiqubit setup catalytic cooling based on a three-body interaction outperforms standard (non-catalytic) cooling using higher order interactions. Another advantage is illustrated in a thermometry scenario, where a qubit is employed to probe the temperature of the environment. In this case, we show that a catalyst allows to surpass the optimal temperature estimation attained only with the probe.
![](https://quantum-journal.org/wp-content/uploads/2021/09/Quantum_image-1.png)
Featured image: (a) Cooling and no-cooling conditions to cool a two-level system (system labeled as c) using a three-level environment (system labeled as h). For cooling to be possible a global eigenstate where the system c is excited must be more populated than a global eigenstate where it is in the ground state, as shown in the largest box. The no-cooling condition can be bypassed through a catalyst, which enables cooling and is kept unchanged. (b) In our framework correlations between the catalyst and the rest of the system are allowed. (c) The catalytic transformations comprise an initial operation that performs cooling and perturbs the catalyst, followed by a restoring operation that recovers the catalyst. (d) An analogous procedure can be applied in a thermometry setting. (Left) A two-level probe (labeled as P) optimally probes the temperature of the environment. (Right) The interaction with a catalyst increases the temperature information transferred to the probe.
Popular summary
In this work we introduce a framework for the construction of explicit catalytic transformations. Focusing on cooling, we derive sufficient conditions for catalytic cooling when this is otherwise impossible. A key result in this respect is that a catalyst of sufficiently large dimension allows cooling regardless of the environment dimension, as long as the later does not start in a fully mixed state. In addition, we demonstrate cooling enhancements where a catalyst can increase cooling and at the same time reduce the complexity of the interactions with the environment.
Beyond the thermodynamic scenario, our results unveil catalytic transformations that no coupling between the system and the environment can achieve. We illustrate this finding in a thermometry setting. Here, we show that a catalyst can reduce the error in estimating the temperature of a thermal environment. On top of these applications, we hope that the technical aspects of our work also help to shed light on the fascinating phenomenon of catalysis.
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