Ergotropy as Permutation Work: A First-Principles Unification of Quantum Battery Scaling Laws and a Universal Spectral Bound This research establishes a fundamental reinterpretation of ergotropy—the central figure of merit for quantum battery performance—as a combinatorial sorting distance derived from the Hardy-Littlewood-Polya Rearrangement Inequality. By shifting the foundation from traditional thermodynamics to spectral graph theory, the author proves a Universal Ergotropy Bound: \Delta(T) = J[\lambda_{max}(A_T) - \lambda_{min}(A_T)] where A_T is the adjacency matrix of the battery's coupling topology T. This single spectral theorem provides a unified framework that recovers all three major scaling laws previously reported in literature: O(1) for ring topologies, O(N) for complete graphs, and O(\sqrt{N}) for wheel graphs. Key Contributions: New Scaling Regime: Predicts and proves an O(\log N) scaling for hypercube topologies, a regime previously unidentified in quantum battery research. Decoherence Dynamics: Derives the exact ergotropy evolution equation under graph-structured dephasing using the graph Laplacian. Optimal Topology: Establishes the Ergotropy-Decoherence Tradeoff and proves that the hypercube is the uniquely optimal topology for the energy-coherence product. AI Inference Conjecture: Proposes a formal mapping between neural network layers and quantum batteries, providing a first-principles justification for why sparse architectures (e.g., Transformers, MoE) are thermodynamically mandated for energy efficiency. All results are proven from first principles, connecting significant mathematical constants (such as 4J, 2, and \sqrt{N}) to deep structures in condensed matter physics, quantum information, and information theory
Ergotropy as Permutation Work: A First-Principles Unification of Quantum Battery Scaling Laws and a Universal Spectral Bound This research establishes a fundamental reinterpretation of ergotropy—the central figure of merit for quantum battery performance—as a combinatorial sorting distance derived from the Hardy-Littlewood-Polya Rearrangement Inequality. By shifting the foundation from traditional thermodynamics to spectral graph theory, the author proves a Universal Ergotropy Bound: \Delta(T) = J[\lambda_{max}(A_T) - \lambda_{min}(A_T)] where A_T is the adjacency matrix of the battery's coupling topology T. This single spectral theorem provides a unified framework that recovers all three major scaling laws previously reported in literature: O(1) for ring topologies, O(N) for complete graphs, and O(\sqrt{N}) for wheel graphs. Key Contributions: New Scaling Regime: Predicts and proves an O(\log N) scaling for hypercube topologies, a regime previously unidentified in quantum battery research. Decoherence Dynamics: Derives the exact ergotropy evolution equation under graph-structured dephasing using the graph Laplacian. Optimal Topology: Establishes the Ergotropy-Decoherence Tradeoff and proves that the hypercube is the uniquely optimal topology for the energy-coherence product. AI Inference Conjecture: Proposes a formal mapping between neural network layers and quantum batteries, providing a first-principles justification for why sparse architectures (e.g., Transformers, MoE) are thermodynamically mandated for energy efficiency. All results are proven from first principles, connecting significant mathematical constants (such as 4J, 2, and \sqrt{N}) to deep structures in condensed matter physics, quantum information, and information theory
The maximum work that can be extracted from a quantum battery is bounded by the ergotropy of the system, which is determined by the spectral properties of the Hamiltonian. In this paper, we employ the formalism of quantum walks to investigate how the topology of the battery and the chirality of the Hamiltonian influence its performance as an energy storage unit. We analyze architectures of battery cells based on ring, complete, and wheel graph structures and analyze their behavior in the presence of noise. Our results show that these structures exhibit distinct ergotropy scaling, with the interplay between chirality and topology providing a tunable mechanism to optimize work extraction and enhance robustness against decoherence. In particular, chirality enhances ergotropy for complete quantum cells, without altering the linear scaling with size, whereas in ring cells, it bridges the performance gap between configurations with odd and even number of units. Additionaly, chirality may be exploited to force degeneracies in the Hamiltonian, a condition that can spare the ergotropy to vanish in the presence of pure dephasing. We conclude that topology and chirality are key resources for improving ergotropy, offering guidelines to optimize quantum energy devices and protocols.
ABSTRACT The maximum work that can be extracted from a quantum battery is bounded by the ergotropy of the system, which is determined by the spectral properties of the Hamiltonian. In this paper, we employ the formalism of quantum walks to investigate how the topology of the battery and the chirality of the Hamiltonian influence its performance as an energy storage unit. We analyze architectures of battery cells based on ring, complete, and wheel graph structures and analyze their behavior in the presence of noise. Our results show that these structures exhibit distinct ergotropy scaling, with the interplay between chirality and topology providing a tunable mechanism to optimize work extraction and enhance robustness against decoherence. In particular, chirality enhances ergotropy for complete quantum cells, without altering the linear scaling with size, whereas in ring cells, it bridges the performance gap between configurations with odd and even number of units. Additionally, chirality may be exploited to force degeneracies in the Hamiltonian, a condition that can spare the ergotropy to vanish in the presence of pure dephasing. We conclude that topology and chirality are key resources for improving ergotropy, offering guidelines to optimize quantum energy devices and protocols.
The realization of scalable quantum battery architectures requires concern not only with how much energy can be stored, but also with how energy is transported, distributed, and converted into extractable work across connected battery nodes. While previous studies mainly focused on collective charging in multi-cell quantum batteries, the topology-dependent transport law and the corresponding work-oriented performance of quantum battery networks remain largely unexplored. In this work, we investigate quantum battery networks with engineered reciprocal and nonreciprocal couplings and compare different connection topologies, including cascaded and parallel architectures, within a unified transport framework. In the nonreciprocal regime, the optimal coupling follows distinct scaling laws for the two connection topologies, namely Jopc∝N for cascaded transport and Jopp∝N−1/2 for parallel charging in the large-N limit. In reciprocal cascaded networks, a parity-dependent spectral response produces an odd-even transport effect that is absent in the nonreciprocal and parallel configurations. We further analyze the role of thermal and squeezed reservoirs and show that thermal noise mainly increases passive energy, whereas squeezing enhances ergotropy and thus the useful fraction of stored energy. These results shift the emphasis from charging enhancement to transport engineering and provide architecture-level design principles for quantum battery networks.
Ergotropy--a key figure of merit for quantum battery (QB) performance--plays a crucial role. However, the dynamics and physical mechanisms governing ergotropy evolution remain open challenges. Here, we investigate the ergotropy of a general QB model and find that the charging process is accompanied by the variation and inversion of the energy level populations. In the absence of population inversion, the ergotropy is fully consistent with coherent ergotropy; in local and global population inversion, it is determined by both coherent and incoherent ergotropy. Via random sampling of quantum states and Hamiltonians, we show that coherence and the participation ratio enhance coherent ergotropy, whereas incoherent ergotropy--whether enhanced, unchanged, or suppressed--depends on diagonal entropy, the participation ratio, and energy level population ordering. We demonstrate that the ergotropy lower bound is incoherent ergotropy, the upper bound is the QB stored energy, and enhanced QB purity suppresses locked energy and boosts charging efficiency. Furthermore, we use the Tavis-Cummings (TC) and Jaynes-Cummings (JC) batteries as paradigms to validate our findings. Our work elucidates ergotropy underlying mechanisms in general QBs and establishes a rigorous framework for optimizing ergotropy and charging efficiency, paving the way for high-performance quantum energy-storage devices.
We establish a universal scaling law for collective charging of quantum batteries, independent of microscopic details. We prove that the ergotropy-to-energy ratio approaches unity at least as fast as ∼N−1 with the number of batteries N, implying generic asymptotic freedom. We further show how the universal 1/N scaling can be overcome: when the battery state becomes asymptotically pure, the convergence can be substantially faster, including ∼N−b with b>1 and even exponential scaling in N2. Rigorous finite-N upper and lower bounds on the ergotropy-to-energy ratio are further derived, providing nonasymptotic guarantees for the universal 1/N scaling.
This study investigates the performance and ergotropy protection of open collective quantum batteries subject to superradiant decay. By employing a passive spectral detuning strategy within an intermediate cavity, an optimal detuning value (Δ∗) is analytically derived and numerically verified to spectrally isolate the system and protect quantum coherence, achieving up to 1088% ergotropy improvement for single qubits and superextensive collective advantage for N≥3. Our analysis resolves a "non-Markovian paradox," revealing that maximizing ergotropy does not strictly require non-Markovian memory; rather, suppressing environmental memory via detuning optimally preserves coherence, which serves as the fundamental resource. Survival maps across different environments demonstrate that thermal noise dissipates coherence more severely than telegraph noise. Finally, we establish that collective amplification of the effective coupling (geff=gN) inevitably drives large qubit arrays into the ultra-strong coupling regime, providing a quantitative ceiling Nmax on the validity of the Tavis-Cummings description and the current ergotropy protection protocol.