Thermodynamics of Quantum Causal Models: An Inclusive, Hamiltonian Approach

Philipp Strasberg

Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

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Operational quantum stochastic thermodynamics is a recently proposed theory to study the thermodynamics of open systems based on the rigorous notion of a quantum stochastic process or quantum causal model. In there, a stochastic trajectory is defined solely in terms of experimentally accessible measurement results, which serve as the basis to define the corresponding thermodynamic quantities. In contrast to this observer-dependent point of view, a `black box', which evolves unitarily and can simulate a quantum causal model, is constructed here. The quantum thermodynamics of this big isolated system can then be studied using widely accepted arguments from statistical mechanics. It is shown that the resulting definitions of internal energy, heat, work, and entropy have a natural extension to the trajectory level. The canonical choice of them coincides with the proclaimed definitions of operational quantum stochastic thermodynamics, thereby providing strong support in favour of that novel framework. However, a few remaining ambiguities in the definition of stochastic work and heat are also discovered and in light of these findings some other proposals are reconsidered. Finally, it is demonstrated that the first and second law hold for an even wider range of scenarios than previously thought, covering a large class of quantum causal models based solely on a single assumption about the initial system-bath state.

Small systems fluctuate and behave stochastically. Reconciling this picture with the traditional framework of macroscopic thermodynamics is challenging. During the past decades a framework called stochastic thermodynamics emerged that equips even a single stochastic trajectory observed in an experiment with the traditional notions of internal energy, heat, work, and entropy. This framework has proven very successful, at least if the fluctuating system behaves classically.

The extension of classical stochastic thermodynamics to the quantum regime is far from clear, in particular as there is no universal consensus about what a "quantum stochastic trajectory" actually means. An often employed strategy imagines to split the dynamics of the quantum system into a fictitious ensemble of trajectories. Consequently, a problem appears if one attempts to measure those trajectories: due to the invasiveness of quantum measurements the dynamics of the system changes and is no longer described by the same ensemble.

In this paper we propose a similar strategy, which, however, includes the detector in the description and generates an ensemble of trajectories based on the measurement results obtained in a fictitious experiment. The theory thus takes into account the measurement backaction and can be readily tested experimentally. Interestingly, the definitions we find for internal energy, heat, work, and entropy agree with the ones recently proposed in a framework called "operational quantum stochastic thermodynamics". Nevertheless, we also find a few remaining ambiguities, which cannot be fixed with the present ensemble approach. This suggests that quantum stochastic thermodynamics is more than a simple extension of classical stochastic thermodynamics: it is a much richer theory.

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