Catalytic transformations with finite-size environments: applications to cooling and thermometry

Ivan Henao and Raam Uzdin

Fritz Haber Research Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel

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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.

Catalysts are a key component of the modern chemical industry, with an impact in increased production rates, energy saving, and reduction of waste material. Concomitant with these advantages is the fact that catalysts remain ideally unaltered, thereby providing an extremely efficient method to assist diverse chemical reactions. This appealing feature has inspired the study of catalysts for applications in Quantum Information and Quantum Thermodynamics. Despite important progress, a crucial step towards a deeper understanding of the mechanisms that underpin catalysis is the derivation of explicit catalytic transformations, where information about the state of the catalyst and the corresponding evolution is available. In contrast, the current paradigm emphasizes the existence of a given transformation over the details of its implementation.

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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