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.
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.
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.
Abstract We revisit the problem of work extraction from a system in contact with a heat bath to a work storage system, and the reverse problem of state formation from a thermal system state in single-shot quantum thermodynamics. A physically intuitive and mathematically simple approach using only elementary majorization theory and matrix analysis is developed, and a graphical interpretation of the maximum extractable work, minimum work cost of formation, and corresponding single-shot free energies is presented. This approach provides a bridge between two previous methods based respectively on the concept of thermomajorization and a comparison of subspace dimensions. In addition, a conceptual inconsistency with regard to general work extraction involving transitions between multiple energy levels of the work storage system is clarified and resolved. It is shown that an additional contribution to the maximum extractable work in those general cases should be interpreted not as work extracted from the system, but as heat transferred from the heat bath. Indeed, the additional contribution is an artifact of a work storage system (essentially a suspended ‘weight’ that can be raised or lowered) that does not truly distinguish work from heat. The result calls into question the common concept that a work storage system in quantum thermodynamics is simply the quantum version of a suspended weight in classical thermodynamics.
We consider the task of extracting work from quantum systems in the resource theory perspective of thermodynamics, where free states are arbitrary thermal states, and allowed operations are energy conserving unitary transformations. Taking as our work storage system a 'weight' we prove the second law and then present simple protocols which extract average work equal to the free energy change of the system - the same amount as in classical thermodynamics. Crucially, for systems in 'classical' states (mixtures of energy eigenstates) our protocol works on a single copy of the system. This is in sharp contrast to previous results, which showed that in case of almost-deterministic work extraction, collective actions on multiple copies are necessary to extract the free energy. This establishes the fact that free energy is a meaningful notion even for individual systems in classical states. However, for non-classical states, where coherences between energy levels exist, we prove that collective actions are necessary, so long as no external sources of coherence are used.
The minimum amount of thermodynamic work required in order to implement a quantum computation or a quantum state transformation can be quantified using frameworks based on the resource theory of thermodynamics, deeply rooted in the works of Landauer and Bennett. For instance, the work we need to invest in order to implement n independent and identically distributed (i.i.d.) copies of a quantum channel is quantified by the thermodynamic capacity of the channel when we require the implementation's accuracy to be guaranteed in diamond norm over the n-system input. Recent work showed that work extraction can be implemented universally, meaning the same implementation works for a large class of input states, while achieving a variable work cost that is optimal for each individual i.i.d. input state. Here, we revisit some techniques leading to derivation of the thermodynamic capacity, and leverage them to construct a thermodynamic implementation of n i.i.d. copies of any time-covariant quantum channel, up to some process decoherence that is necessary because the implementation reveals the amount of consumed work. The protocol uses so-called thermal operations and achieves the optimal per-input work cost for any i.i.d. input state; it relies on the conditional erasure protocol in our earlier work, adjusted to yield variable work. We discuss the effect of the work-cost decoherence. While it can significantly corrupt the correlations between the output state and any reference system, we show that for any time-covariant i.i.d. input state, the state on the output system faithfully reproduces that of the desired process to be implemented. As an immediate consequence of our results, we recover recent results for optimal work extraction from i.i.d. states up to the error scaling and implementation specifics, and propose an optimal preparation protocol for time-covariant i.i.d. states.
We analyze work extraction protocols using the long-lived quantum coherence of a three-level quantum system, which is coupled to a thermal bath through dipole-monopole interactions. We identify situations where persistent quantum coherence arises, i.e., for systems with degenerate excited states with aligned transition dipoles or nearly degenerate systems with small energy splittings. By designing two innovative thermodynamic protocols involving energy-preserving unitary operations, we show that quantum coherence can be transformed into population asymmetry, serving as a quantum resource for work extraction. As the system approaches final thermal equilibrium, the initial quantum coherence effectively acts as fuel, being progressively consumed. Specifically, we propose an optimized protocol capable of extracting the maximal extractable work (MEW), measured by the free energy difference (FED), from quantum coherence in a single-shot thermodynamic cycle. Our results highlight the thermodynamic advantages of long-lived coherence in a three-level quantum system and could influence future designs of coherence-driven quantum thermal machines.
We present a critical examination of the difficulties with the quantum versions of a lifted weight that are widely used as work storage systems in quantum thermodynamics. To overcome those difficulties, we turn to the strong connections between information and thermodynamics illuminated by Szilard’s engine and Landauer’s principle, and consider the concept of informational work storage. This concept is in sharp contrast with the usual one of mechanical work storage underlying the idealization of a quantum weight. An informational work storage system based on maximally mixed qubits that does not act as an entropy sink and is capable of truly distinguishing work from heat is studied. Applying it to the problem of single-shot work extraction in various extraction schemes, we show that for a given system state the maximum extractable work is independent of extraction scheme, in accordance with the second law of thermodynamics.