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Super-Extensive Charging Power in the Absence of Global Operations

Anupam, Sheryl Mathew, Sibasish Ghosh

Abstract

Quantum batteries have emerged as a platform for investigating whether quantum effects can accelerate energy storage beyond classical limits. Although a variety of charging schemes have reported signatures of quantum advantage, the fundamental physical requirements for achieving superextensive charging power remain insufficiently understood. Here, we show that, in addition to Hamiltonian locality, a key structural property, g-extensiveness, quantifying the distribution of interaction energy across lattice sites places a fundamental bound on charging performance in spin-lattice models. We prove that superextensive power scaling is possible only when the interaction-energy distribution becomes increasingly nonuniform, with the maximal local weight growing with system size. This criterion explains why many previously studied protocols fail to exhibit superextensive power, even when the Hamiltonians involve large participation numbers. We further demonstrate that this condition is realized in an experimentally relevant interacting model, where, despite fixed interaction order, the charging power scales superextensively. Our results establish g-extensiveness as a necessary resource for quantum advantage in direct-charging protocols and provide a systematic framework for identifying and engineering physically feasible quantum batteries capable of outperforming classical counterparts in charging power.

Super-Extensive Charging Power in the Absence of Global Operations

Abstract

Quantum batteries have emerged as a platform for investigating whether quantum effects can accelerate energy storage beyond classical limits. Although a variety of charging schemes have reported signatures of quantum advantage, the fundamental physical requirements for achieving superextensive charging power remain insufficiently understood. Here, we show that, in addition to Hamiltonian locality, a key structural property, g-extensiveness, quantifying the distribution of interaction energy across lattice sites places a fundamental bound on charging performance in spin-lattice models. We prove that superextensive power scaling is possible only when the interaction-energy distribution becomes increasingly nonuniform, with the maximal local weight growing with system size. This criterion explains why many previously studied protocols fail to exhibit superextensive power, even when the Hamiltonians involve large participation numbers. We further demonstrate that this condition is realized in an experimentally relevant interacting model, where, despite fixed interaction order, the charging power scales superextensively. Our results establish g-extensiveness as a necessary resource for quantum advantage in direct-charging protocols and provide a systematic framework for identifying and engineering physically feasible quantum batteries capable of outperforming classical counterparts in charging power.

Paper Structure

This paper contains 8 sections, 1 theorem, 64 equations, 7 figures.

Table of Contents

  1. Bounding the transition matrix element in commuting case
  2. Commutator bound for commuting Hamiltonians
  3. Discretizing the spectrum of the commuting Hamiltonian
  4. Bounding the commutator using transition elements
  5. Examples of the $6 g k \|H\|$ Bound
  6. Commutator Evaluation between $H_B$ and $H_C$
  7. Variance of $H_B$ and $H_C$ for the product state
  8. Power and Rate of Entanglement Growth

Key Result

Lemma 1

Kuwahara2015Thesis For any $g$-extensive commuting Hamiltonian $H=\sum_{X}h_{X}$ where $[h_{X},h_{X'}]=0~~\forall X,X'$ and $g\sim\mathcal{O}(1)$. The transition matrix element of another $q$-local operator $h^{(q)}$ acting on sites $S\subset\lambda$ is bounded as follows,

Figures (7)

  • Figure 1: Energy Transitions. (a) Heatmap of absolute values of transition matrix elements $\| P[E_a] H_C P[E_b] \|$ for an Ising charger in the diagonal basis of a modified central-spin battery. Degenerate levels are grouped as projections onto energy subspaces of the battery Hilbert space. (b) Same as (a), but with the battery and charger Hamiltonians interchanged. Here, total number of sites is seven.
  • Figure 2: Scaling with system size of the commutator operator norm and bounds for an Ising battery and MCS charger
  • Figure 3: Heatmap of absolute values of transition matrix elements $\| P[E_a] H_C P[E_b] \|$ for the MCS charger in the diagonal basis of the MCS battery. Degenerate levels are grouped as projections onto energy subspaces of the battery Hilbert space.
  • Figure 4: Battery Ground State. For the MCS battery quenched with the MCS charger, we plot: (a) Evolution of the initial ground state with time in the eigenbasis of the Battery, (b) The instantaneous power plotted against time, (c) The maximum instantaneous power versus total number of sites.
  • Figure 5: Product State. For the MCS battery quenched with the MCS charger, we plot: (a) Evolution of the initial product state with time in the eigenbasis of the Battery, (b) The instantaneous power plotted against time, (c) The maximum instantaneous power versus total number of sites.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Lemma 1
  • proof