The tight Second Law inequality for coherent quantum systems and finite-size heat baths

Abstract

In classical thermodynamics, the optimal work is given by the free energy difference, what according to the result of Skrzypczyk et al. can be generalized for individual quantum systems. The saturation of this bound, however, requires an infinite bath and ideal energy storage that is able to extract work from coherences. Here we present the tight Second Law inequality, defined in terms of the ergotropy (rather than free energy), that incorporates both of those important microscopic effects – the locked energy in coherences and the locked energy due to the finite-size bath. The former is solely quantified by the so-called control-marginal state, whereas the latter is given by the free energy difference between the global passive state and the equilibrium state. Furthermore, we discuss the thermodynamic limit where the finite-size bath correction vanishes, and the locked energy in coherences takes the form of the entropy difference. We supplement our results by numerical simulations for the heat bath given by the collection of qubits and the Gaussian model of the work reservoir.

Quantum and Classical Ergotropy from Relative Entropies

“…For the sake of simplicity, we further assume that the eigenenergies are ordered in ascending magnitude, . As an alternative expression, the quantum ergotropy can be expressed as the difference of relative entropies [ 14 , 48 ] (see Appendix B for the proof) …”

Second law of thermodynamics for batteries with vacuum state

“…In order to take full advantage of these techniques it is crucial to understand how thermodynamic laws translate into the non-equilibrium domain, where fluctuations of thermodynamic quantities begin to play a significant role and averaged quantities are no longer enough to characterize their thermodynamic behaviour. This motivates extending thermodynamic framework to systems driven out of equilibrium, a setting which has been extensively studied in the recent literature [7][8][9][10][11][12][13][14][15].…”

Work and Fluctuations: Coherent vs. Incoherent Ergotropy Extraction

“…We stress that this is unavoidable if one wants to incorporate interference effects since the processing of coherences of the system strongly depends on coherences of the work reservoir. In particular, treating the energy-storage device explicitly leads to the so-called work-locking [18,19] (i.e., an inability of work extraction from coherences), which non-autonomous frameworks cannot even capture.…”

“…[24] where are discussed corrections if translational invariance is violated). Moreover, it has also been shown that the optimal work extracted by the quantum weight is limited by the ergotropy [19,25], which, with the notion of passivity, is another building block of quantum thermodynamics (see, e.g., [26,27]). Hence, taking the weight as a proper model for fluctuation theorems allows us to analyze the quantum coherences, from a thermodynamic perspective, within a fully quantum setup.…”

“…Hence, taking the weight as a proper model for fluctuation theorems allows us to analyze the quantum coherences, from a thermodynamic perspective, within a fully quantum setup. So far, the work-locking for a quantum weight has been completely described as a dephasing process of a coherent contribution to the ergotropy, described by an effective, the so-called control-marginal state [19]. Since this particular example of the work-locking refers to a loss of the ergotropy due to a dumping of coherences, we call it the ergotropy decoherence.…”