We characterize the impact that the application of two maps in a quantum-controlled order has on the process of work extraction via unitary cycles and its optimization. The control is based on the quantum switch model that applies maps in an order not necessarily compatible with the underlying causal structure and, in principle, can be implemented experimentally. First, we show that the activation of quantum maps through the quantum switch model always entails a non-negative gain in ergotropy compared to their consecutive application. We also establish a condition that the maps should fulfill in order to achieve a nonzero ergotropic gain. We then perform a thorough analysis of maps applied to a two-level system and provide general conditions for achieving a positive gain on the incoherent part of ergotropy. Our results are illustrated with several examples and applied to qubit and $d$-dimensional quantum systems. In particular, we demonstrate that a nonzero work can be extracted from a system thermalized by two coherently controlled reservoirs.
We characterize the impact that the application of two maps in a quantum-controlled order has on the process of work extraction via unitary cycles and its optimization. The control is based on the quantum switch model that applies maps in an order not necessarily compatible with the underlying causal structure and, in principle, can be implemented experimentally. First, we show that the activation of quantum maps through the quantum switch model always entails a non-negative gain in ergotropy compared to their consecutive application. We also establish a condition that the maps should fulfill in order to achieve a nonzero ergotropic gain. We then perform a thorough analysis of maps applied to a two-level system and provide general conditions for achieving a positive gain on the incoherent part of ergotropy. Our results are illustrated with several examples and applied to qubit and d-dimensional quantum systems. In particular, we demonstrate that a nonzero work can be extracted from a system thermalized by two coherently controlled reservoirs.
We characterize the impact that the application of two consecutive quantum channels or their quantum superposition (thus, without a definite causal order) has on ergotropy, i.e. the maximum work that can be extracted from a system through a cyclic unitary transformation. First of all, we show that commutative channels always lead to a non-negative gain; non-commutative channels, on the other hand, can entail both an increase and a decrease in ergotropy. We then perform a thorough analysis for qubit channels and provide general conditions for achieving a positive gain on the incoherent part of ergotropy. Finally, we extend our results to d-dimensional quantum systems undergoing a pair of completely depolarizing channels.
The quantum switch implements indefinite causal order via a coherent control of the orderings of multiple quantum operations, leading to various advantages and applications, one of which being in work extraction where enhancements to daemonic ergotropy is possible. Motivated by recent developments in the connections between non-Markovianity and the quantum switch, we construct a non-Markovian process that reduces to the two-party quantum switch in the fully non-Markovian limit. By controlling the amount of non-Markovianity in the process, we identified two operational regimes with differing behaviors. One has its daemonic ergotropy dependent on the presence and amount of non-Markovianity, achieving the maximum in the quantum switch case of full non-Markovianity. The other regime, however, has no advantages from non-Markovianity. We compare this non-Markovian process with the case of a superposition of independent channels, where two channels are placed in a coherent superposition without indefinite causal order, uncovering the advantages of non-Markovianity. Finally, the conditions required for the production of positive daemonic ergotropy are also derived for the case of fully non-Markovian and fully Markovian limits, where we compare against the conditions required for the superposition of independent channels.
Constraints on work extraction are fundamental to our operational understanding of the thermodynamics of both classical and quantum systems. In the quantum setting, finite-time control operations typically generate coherence in the instantaneous energy eigenbasis of the dynamical system. Thermodynamic cycles can, in principle, be designed to extract work from this nonequilibrium resource. Here, we isolate and study the quantum coherent component to the work yield in such protocols. Specifically, we identify a coherent contribution to the ergotropy (the maximum amount of unitarily extractable work via cyclical variation of Hamiltonian parameters). We show this by dividing the optimal transformation into an incoherent operation and a coherence extraction cycle. We obtain bounds for both the coherent and incoherent parts of the extractable work and discuss their saturation in specific settings. Our results are illustrated with several examples, including finite-dimensional systems and bosonic Gaussian states that describe recent experiments on quantum heat engines with a quantized load.
Abstract Thermalization processes degrade the states of any working medium, turning any initial state into a passive state from which no work can be extracted. Recently, it has been shown that this degradation can be avoided if two identical thermalization processes take place in coherently controlled order, in a scenario known as the quantum SWITCH. In some situations, control over the order even enables work extraction when the medium was initially in a passive state. This activation phenomenon, however, is subject to a limitation: to extract non-zero work, the initial temperature of the medium should be less than half of the temperature of the reservoirs. Here we analyze this limitation, showing that it still holds true even when the medium interacts with N ⩾ 2 reservoirs in a coherently-controlled order. Then, we show that the limitation can be lifted when the medium and the control systems are initially correlated. In particular, when the medium and control are entangled, work extraction becomes possible for every initial value of the local temperature of the medium.
We consider a quasi-probability distribution of work for an isolated quantum system coupled to the energy-storage device given by the ideal weight. Specifically, we analyze a trade-off between changes in average energy and changes in weight's variance, where work is extracted from the coherent and incoherent ergotropy of the system. Primarily, we reveal that the extraction of positive coherent ergotropy can be accompanied by the reduction of work fluctuations (quantified by a variance loss) by utilizing the non-classical states of a work reservoir. On the other hand, we derive a fluctuation-decoherence relation for a quantum weight, defining a lower bound of its energy dispersion via a dumping function of the coherent contribution to the system's ergotropy. Specifically, it reveals that unlocking ergotropy from coherences results in high fluctuations, which diverge when the total coherent energy is unlocked. The proposed autonomous protocol of work extraction shows a significant difference between extracting coherent and incoherent ergotropy: The former can decrease the variance, but its absolute value diverges if more and more energy is extracted, whereas for the latter, the gain is always non-negative, but a total (incoherent) ergotropy can be extracted with finite work fluctuations. Furthermore, we present the framework in terms of the introduced quasi-probability distribution, which has a physical interpretation of its cumulants, is free from the invasive nature of measurements, and reduces to the two-point measurement scheme (TPM) for incoherent states. Finally, we analytically solve the work-variance trade-off for a qubit, explicitly revealing all the above quantum and classical regimes.
Understanding the generation and activation of ergotropy, the maximum work extractable by cyclic unitary operations, is central to quantum battery research. Positive-temperature thermal states are passive, whereas nonequilibrium environments can sustain nonpassive steady states with nonzero ergotropy. However, the full secular approximation commonly used in Lindblad descriptions can suppress steady-state coherence and obscure its contribution to ergotropy. Using a Bloch-Redfield master equation beyond the full secular approximation, we investigate the ergotropy of the nonequilibrium steady state (NESS) of two exchange-coupled qubits connected to independent thermal reservoirs. The NESS ergotropy of two qubits with the same frequency is entirely coherent and grows quadratically with temperature bias near equilibrium. In contrast, detuned qubits can exhibit incoherent ergotropy. We derive an exact analytical criterion for its activation and show that activation requires the hotter reservoir to be coupled to the qubit with the higher bare frequency. Comparing NESSs obtained with and without full secularization reveals quantitative and qualitative differences in ergotropy that reflect changes in both its coherent and incoherent contributions. The results clarify the mechanisms of NESS ergotropy activation and demonstrate the importance of retaining steady-state coherence beyond the full secular approximation when characterizing stationary work resources.