Correlated thermal machines in the micro-world R. Gallego, A. Riera, and J. Eisert 1 1 Dahlem Center for Complex Quantum Systems, Freie Universit¨ at Berlin, 14195 Berlin, Germany How much work can be extracted from a heat bath using a thermal machine operating at the quantum level? In this work, we establish a framework of work extraction in the presence of quantum correlations as they naturally emerge in quantum thermodynamics beyond the weak-coupling regime. Quantum correlations and entanglement, as they are ubiquitously encountered when putting systems into contact with a bath, emerge as a limitation to work extraction, rather than as an advantage. We bring together concepts of quantum thermody- namics with those of non-equilibrium dynamics, in a rigorous but physically motivated approach. We discuss various limits that relate to known results and put our work into context of approaches to finite-time quantum thermodynamics. The theory of thermodynamics originates from the study of thermal machines in the early industrial age, when it was of utmost importance to find out what rates of work extrac- tion could ultimately be achieved. Early on, it became clear that the study of thermal machines would be intimately related with topics of fundamental physics such as statistical mechan- ics and with notions of classical information theory [1]. Here, the interplay and relations between widely-studied notions of work, entropy and of statistical ensembles are in the focus of attention. Concomitant with the technological development, the theory also became more intricate and addressed more elaborate situations. Famous thought experiments such as the ones associated with Maxwell’s demon, Landauer’s principle, and Slizard’s engine have not only puzzled researchers for a long time, but today also serve as a source of inspiration for quantitative studies of achievable rates when employing ther- mal machines [2–5]. And indeed, with nano-machines oper- ating at or close to the quantum level coming into reach, there has recently been an explosion of interest on the question what role quantum effects may possibly play. It is the potential and limits of work extraction with physically plausible operations which respect quantum correlations that are established in this work. As far as the methodological development is concerned, this renewed interest is in part – but not only – triggered by the impetus of the availability of new methods from quantum in- formation theory. Our understanding of thermal machines has benefited from two novel concepts. In particular, the formal- ism of asymptotic resource theories applied to thermal states [8–10] and the use of smooth-entropies [6, 11] have provided new insights into protocols of work extraction. The former allows one to obtain ultimate bounds on work extraction of quantum thermal machines, without having to resort to any specific model. The latter provides grounds to the quantita- tive study of work extraction in a single-shot regime, rather than expectation values. These approaches rely only on the existence of systems captured by the usual Gibbs ensemble playing the role of a thermal bath and standard rules of ma- nipulation of closed quantum systems or Hamiltonians. In a related by different development, new methods have also allowed significant progress on another old problem, namely the very emergence of statistical ensembles by means of non-equilibrium many-body dynamics itself, specifically the canonical or Gibbs ensemble. The idea is that quenched Figure 1. Scheme of the setting of the work extraction problem. closed, non-integrable many-body systems, however de- scribed by a unitarily evolving pure state, are generically ex- pected to equilibrate [12–16] and behave – for the overwhelm- ing majority of times – as if they were described by a ther- mal state when considering expectation values of local observ- ables [15, 17, 18]. The eigenstate thermalization hypothesis [17, 19, 20] gives further substance to this expectation. This means that reduced states of any sub-system S with an inter- acting Hamiltonian H SB are then described by ρ S = tr B (ω(H SB )) (1) where ω(H) := e -βH /Z with Z := tr(e -βH ), that is, as a reduction of the Gibbs state of a larger system composed of S and its complement B. While a complete characterization of the precise fine-print ensuring that a system exhibits such be- havior is still missing, and while integrability alone is strictly speaking not sufficient for thermalization to occur [21], there exists significant theoretical and experimental evidence that typical many body systems indeed show such a behavior [15]. Strikingly, one can detect a certain tension between these two novel approaches: One aims at explaining thermal ma- chines by relying entirely on states described by a Gibbs en- semble; whereas evidence accumulates that the Gibbs ensem- ble should not be used to describe the sub-systems themselves. This will only be approximately true in extremely weakly- interacting systems. In any regime involving strong interac- tion regime, the Gibbs ensemble emerges as in Eq. (1). Also, it is an important enterprise to understand what rates can in principle be achieved, resorting to intricate schemes remi- niscent of protocols of quantum information processing and quantum computing. At the same time, it seems fair to say, it is equally important to flesh out how to grasp work extrac- tion with schemes that very much resemble what one would expect when actually putting physical systems together and