Quantum Batteries - Ergotropy

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