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.
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.
The amount of extractable work from a physical system is fundamentally connected to the information available about its state, as illustrated by Maxwell's demon and the Gibbs paradox. In standard thermodynamic protocols involving system--bath interactions, the maximum work is given by the free-energy difference between the initial state and the corresponding Gibbs state at the bath temperature. This motivates a natural question: does information also limit work extraction in closed quantum systems that do not involve a heat bath and where work is obtained through unitary operations generated by a time-dependent Hamiltonian? While ergotropy quantifies the maximum work extractable via unitary operations, it assumes complete knowledge of the quantum state, typically requiring full state tomography. In realistic scenarios, however, only partial information is accessible. In this case, the relevant figure of merit is observational ergotropy, which depends explicitly on the measurement used to probe the system. We show that observational ergotropy decreases under classical post-processing of measurement outcomes, implying that fine-grained measurements allow greater work extraction than coarse-grained ones. Moreover, maximizing observational ergotropy over all possible measurements recovers standard ergotropy, which decomposes into incoherent (classical) and coherent (quantum) contributions. Our results demonstrate that coherence in the measurement projectors constitutes the key resource, enabling work extraction beyond the incoherent limit and establishing coherence as the origin of quantum advantage in observational ergotropy extraction.
Ergotropy is defined as the maximum amount of work that can be extracted through a unitary cyclic evolution. It plays a crucial role in assessing the work capacity of a quantum system. Recently, the significance of quantum coherence in work extraction has been theoretically identified, revealing that quantum states with more coherence possess more ergotropy compared to their dephased counterparts. However, an experimental study of the coherent ergotropy remains absent. Here, we report an experimental investigation of the coherent ergotropy in a single spin system. Based on the method of measuring ergotropy with an ancilla qubit, both the coherent and incoherent components of the ergotropy for the nonequilibrium state were successfully extracted. The increase in ergotropy induced by the increase in the coherence of the system was observed by varying the coherence of the state. Our work reveals the interplay between quantum thermodynamics and quantum information theory, future investigations could further explore the role other quantum attributes play in thermodynamic protocols.
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.
Quantum coherence, encoded in the off-diagonal elements of a system's density matrix, is a key resource in quantum thermodynamics, fundamentally limiting the maximum extractable work known as ergotropy. While previous experiments have isolated coherence-related contributions to work extraction, it remains unclear how coherence can be harnessed in a controllable and energy-efficient manner. Here, we experimentally investigate the role of initial-state coherence in work extraction from a superconducting transmon qubit. By preparing a variety of pure states and implementing three complementary extraction protocols, we reveal how coherence governs the partitioning of ergotropy. We find that the choice of initial state depends on the dominant decoherence channel-energy relaxation or dephasing. By further accounting for thermodynamic costs, we identify optimal initial states that maximize the efficiency. Our results demonstrate that the initial-state design provides a scalable approach to coherence control and advances the development of efficient quantum thermodynamic devices.
We investigate the decomposition of ergotropy into incoherent and coherent contributions for quantum systems subject to typical Markovian noise channels. The incoherent part originates from population inversion in the energy eigenbasis after dephasing, while the coherent part captures the role of quantum coherence in work extraction. For single-qubit systems, we derive explicit conditions for freezing and enhancement of coherent ergotropy and obtain an analytical upper bound, showing that it cannot exceed one half of the state's quantum coherence. We then study two classes of separable two-qubit states under local noise. For Bell-diagonal states, which are locally completely passive and possess no local coherence, we prove that the total extractable work equals the average of geometric quantum and classical correlations. In this case, coherent ergotropy cannot be enhanced, although freezing occurs under specific noise conditions. By contrast, for separable states with local coherence, coherent ergotropy can increase under all considered noise channels, including phase-flip and depolarizing noise. Extending the analysis to multipartite systems, we show that both the magnitude and range of noise-induced enhancement grow with the number of qubits, indicating collective reinforcement. Finally, we demonstrate through an explicit example that entanglement does not prevent this enhancement: coherent ergotropy may increase under noise even for entangled states. Our results reveal that noise can assist energy storage, challenging the conventional view of noise as purely detrimental and suggesting compatibility between noise-assisted enhancement and fast entanglement-based charging mechanisms in quantum batteries.
A key concept in quantum thermodynamics is extractable work, which specifies the maximum amount of work that can be extracted from a quantum system. Different quantities are used to measure extractable work, the most prevalent of which are ergotropy and the difference between the non-equilibrium and equilibrium quantum free energies. Using the latter, we investigate the evolution of extractable work when an open quantum system undergoes a general quantum process described by a completely-positive and trace-preserving dynamical map. We derive a fundamental equation of thermodynamics for such processes as a relation between the distinct sorts of energy change in such a way that the first and the second law of thermodynamics are combined. We then identify the contributions from the reversible and irreversible processes in this equation and demonstrate that they are respectively responsible for the evolution of heat and extractable work of the open quantum system. Furthermore, we show how this correspondence between irreversibility and extractable work has the potential to provide a clear explanation of how the quantum properties of a system affect its extractable work evolution. Specifically, we establish this by directly connecting the change in extractable work with the change in standard quantifiers of quantum non-Markovianity and quantum coherence during a general quantum process. We illustrate these results with two examples.