Abstract
This paper considers work extraction from a quantum system to a work storage system (or weight) following Horodecki and Oppenheim (2013 Nat. Commun. 4 2059). An alternative approach is here developed that relies on the comparison of subspace dimensions without a need to introduce thermo-majorization used previously. Optimal single shot work for processes where a weight transfers from (a) a single energy level to another single energy level is then re-derived. In addition we discuss the final state of the system after work extraction and show that the system typically ends in its thermal state, while there are cases where the system is only close to it. The work of formation in the single level transfer setting is also re-derived. The approach presented now allows the extension of the single shot work concept to work extraction (b) involving multiple final levels of the weight. A key conclusion here is that the single shot work for case (a) is appropriate only when a resonance of a particular energy is required. When wishing to identify ‘work extraction’ with finding the weight in a specific available energy or any higher energy a broadening of the single shot work concept is required. As a final contribution we consider transformations of the system that (c) result in general weight state transfers. Introducing a transfer-quantity allows us to formulate minimum requirements for transformations to be at all possible in a thermodynamic framework. We show that choosing the free energy difference of the weight as the transfer-quantity one recovers various single shot results including single level transitions (a), multiple final level transitions (b), and recent results on restricted sets of multi-level to multi-level weight transfers.
Highlights
The neat characterization of general classical non-equilibrium processes in terms of fluctuation relations [2,3,4, 8] has rapidly advanced the general understanding of thermodynamic processes and properties at the mesoscopic scale
E.g. [1, 15], derive upper bounds on the amount of work that can be drawn from a quantum system that starts in a non-equilibrium state in a ‘single shot’
The optimal work stated in equation (1) corresponds to the following task: for the weight initially entirely in its ground state the full system is transferred to a final quantum state such that the probability for the weight to be found in an energy eigenstate with energy EWfin is 1 − ε where 1 > ε ⩾ 0, see figure 1(a)
Summary
The neat characterization of general classical non-equilibrium processes in terms of fluctuation relations [2,3,4, 8] has rapidly advanced the general understanding of thermodynamic processes and properties at the mesoscopic scale. The single shot work done by the system is here associated [1] with the transition of the weight from a single energy eigenstate (of energy 0) to another single energy eigenstate (of energy w) The proof of the bounds provided in [1] relies on established mathematical concepts from quantum information theory and a new majorization concept called ‘thermo-majorization’ Researchers, in particular those outside of quantum information theory, may find the proof mathematically heavy and are unable to follow the detailed logic.
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