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arXiv:2508.12797 · Europhysics Letters (EPL)

Unified theory of classical and quantum ergotropy

Michele Campisi

Published Apr 27, 2026• 2 citations
arXivHTMLPDF

Abstract

Abstract Quantifying the ergotropy (also known as available energy), namely the maximal amount of energy that can be extracted from a thermally isolated system, is a central problem in quantum thermodynamics. Notably, the same problem has been long studied for classical systems as well, e.g. , in plasma physics and astrophysics, where the basic principles for its solution are known for the case of collisionless fluids. Here we provide the general analytical expression of ergotropy of classical systems valid regardless of their size and the type of interparticle interactions, and show that it emerges as the classical limit of the quantum expression of ergotropy, for quantum systems that are classically ergodic. We thus establish a unified theory of classical and quantum ergotropy, whose applicability ranges from atomic to galactic scale. Such unified theory is indispensable for studying the genuine quantum signatures of ergotropy: We show that the celebrated decomposition of quantum ergotropy into coherent ant inchoherent parts survives in the classical regime, indicating that coherences do not necessarily reveal quantumness. The unified theory also allows to port tools and methods across the classical-quantum boundary to unlock the solution of standing problems. We apply this to swiftly solve the open problem of ergotropy extraction in the classical regime.

DOI
10.1209/0295-5075/ae652c

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Abstract Quantifying the ergotropy (also known as available energy), namely the maximal amount of energy that can be extracted from a thermally isolated system, is a central problem in quantum thermodynamics. Notably, the same problem has been long studied for classical systems as well, e.g. , in plasma physics and astrophysics, where the basic principles for its solution are known for the case of collisionless fluids. Here we provide the general analytical expression of ergotropy of classical systems valid regardless of their size and the type of interparticle interactions, and show that it emerges as the classical limit of the quantum expression of ergotropy, for quantum systems that are classically ergodic. We thus establish a unified theory of classical and quantum ergotropy, whose applicability ranges from atomic to galactic scale. Such unified theory is indispensable for studying the genuine quantum signatures of ergotropy: We show that the celebrated decomposition of quantum ergotropy into coherent ant inchoherent parts survives in the classical regime, indicating that coherences do not necessarily reveal quantumness. The unified theory also allows to port tools and methods across the classical-quantum boundary to unlock the solution of standing problems. We apply this to swiftly solve the open problem of ergotropy extraction in the classical regime.
The quantum ergotropy quantifies the maximal amount of work that can be extracted from a quantum state without changing its entropy. We prove that the ergotropy can be expressed as the difference of quantum and classical relative entropies of the quantum state with respect to the thermal state. This insight is exploited to define the classical ergotropy, which quantifies how much work can be extracted from distributions that are inhomogeneous on the energy surfaces. A unified approach to treat both, quantum as well as classical scenarios, is provided by geometric quantum mechanics, for which we define the geometric relative entropy. The analysis is concluded with an application of the conceptual insight to conditional thermal states, and the correspondingly tightened maximum work theorem.
The quantum ergotropy quantifies the maximal amount of work that can be extracted from a quantum state without changing its entropy. Given that the ergotropy can be expressed as the difference of quantum and classical relative entropies of the quantum state with respect to the thermal state, we define the classical ergotropy, which quantifies how much work can be extracted from distributions that are inhomogeneous on the energy surfaces. A unified approach to treat both quantum as well as classical scenarios is provided by geometric quantum mechanics, for which we define the geometric relative entropy. The analysis is concluded with an application of the conceptual insight to conditional thermal states, and the correspondingly tightened maximum work theorem.
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.
Abstract It is an established fact that quantum coherences have thermodynamic value. The natural question arises, whether other genuine quantum properties such as entanglement can also be exploited to extract thermodynamic work. In the present analysis, we show that the ergotropy can be expressed as a function of the quantum mutual information, which demonstrates the contributions to the extractable work from classical and quantum correlations. More specifically, we analyze bipartite quantum systems with locally thermal states, such that the only contribution to the ergotropy originates in the correlations. Our findings are illustrated for a two-qubit system collectively coupled to a thermal bath.
Quantum many-body scars break ergodicity and evade thermalization, resulting in sub-volume law entanglement entropy even with high energy density. While their quantum correlations and entanglement have been elaborated previously, their capacity in storing extractable energy, quantified by the notion ergotropy, remains an open question. Here we focus on the representative PXP model, and unveil the extensive ergotropy scaling of a family of states interpolating between quantum many-body scars and thermal states, the latter of which are known to be passive with vanishing ergotropy in the thermodynamic limit. A phenomenological relation between ergotropy and entanglement is uncovered, which generalizes the existing free fermion integrable results to an interacting scenario. The ergotropy in a dynamical protocol shows that a reset with a global uniform coherent rotation can inject extractable energy, as a proof of principle way to charge a quantum "battery". Our protocol is tailored for near term Rydberg neutral atoms array, while also being feasible for other quantum processors. Our results establish that quantum many-body scars, despite the tiny fraction of the Hilbert space they occupy, can be efficiently exploited for storing extractable energy, and "scarring" a many-body system as a promising route for engineering quantum many-body battery.
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.