Quantum Batteries: Capacity and Power

• Quantum batteries are energy storage devices governed by quantum mechanics, leveraging coherence, entanglement, and quantum correlations.
• They utilize entropy-preserving operations to optimize both energy capacity and charging power through nonlocal variance and Fisher information.
• Models like the Dicke and LMG systems illustrate the balance between enhanced quantum charging performance and practical limitations from decoherence.

Quantum batteries are energy storage devices in which the storage, transfer, and release of energy are governed by quantum mechanical principles rather than classical chemical processes. These systems, typically realized using ensembles of quantum two-level systems (qubits), higher-dimensional quantum units (qutrits), or harmonic oscillators, leverage uniquely quantum resources such as entanglement, coherence, and quantum correlations to achieve potentially enhanced charging power and extractable work relative to classical analogs. Quantum batteries can be charged via coherent and, in idealized scenarios, entropy-preserving operations, enabling both fast and reversible energy cycles. The theoretical and experimental paper of quantum batteries has advanced substantially, with the interplay between capacity, charging power, and quantum correlations forming the central axis of this research.

1. Fundamental Limits: Capacity and Power in Quantum Batteries

The central quantities characterizing quantum battery performance are the capacity (the maximal storable/extractable energy) and the charging (or discharging) power (the rate at which this energy can be invested or retrieved). For a collection of quantum cells with fixed local Hamiltonians, the ultimate storage capacity is determined by the energy–entropy diagram. Each state is mapped to a point $(E(\rho),\,S(\rho))$, and the optimal energy that can be achieved at a given entropy $S$ is attained by thermal (completely passive) states. The maximal energy change, or capacity at fixed entropy, is given by

$C(S) = E_{\max}(S) - E_{\min}(S)$

where $E_{\max}(S)$ and $E_{\min}(S)$ are the maximal and minimal energies compatible with entropy $S$, respectively. Under entropy-preserving (e.g., unitary) operations, an initial state $\rho_0$ with entropy $S(\rho_0)$ can only reach these boundaries via horizontal moves in the energy–entropy diagram. Accordingly, the energy storable (or extractable) under ideal charging (or discharging) is bounded by:

$$E^s_{\rho_0} \leq E_{\max}(S(\rho_0)) - E(\rho_0), \quad E^e_{\rho_0} \leq E(\rho_0) - E_{\min}(S(\rho_0))$$

The entropy-preserving assumption ensures that these bounds define the optimal performance under coherent, isolated conditions. Any additional entropy produced, for example by decoherence, reduces attainable capacity by constraining movement within the diagram.

2. Geometric Bounds on Power and Role of Quantum Correlations

The charging power, defined as the instantaneous rate of mean energy change $P(t)=d\langle H_B \rangle/dt$, is bounded by a geometric relation that intimately involves quantum correlations in the battery state. The bound reads:

$P(t)^2 \leq \Delta H_B(t)^2 I_E(t)$

where $\Delta H_B(t)^2$ is the variance of the battery Hamiltonian, and $I_E(t)$ is a Fisher information that quantifies the "speed" of evolution in the energy eigenbasis:

$$I_E(t) = \sum_k \left( \frac{d}{dt} \log p_k(t) \right)^2 p_k(t)$$

with $p_k(t)$ the population in the $k$-th energy level.

The energy variance,

$$\Delta (H_B)^2_\rho = \sum_i \Delta (h_i)^2 + \sum_{i \neq j} \left[ \langle h_i h_j \rangle - \langle h_i \rangle \langle h_j \rangle \right],$$

includes local contributions and nonlocal (correlation) terms. In product states (uncorrelated), the cross terms vanish. Only with quantum correlations—most notably entanglement—can these terms be nonzero, thus increasing the energy variance. The geometric bound asserts that higher nonlocal variance (possible only with quantum correlations) can, in principle, lead to enhanced charging power, provided the Fisher information is also large.

3. Quantum Entanglement as a Resource and Fundamental Bound on Power

For batteries built from qubits (spin-$1/2$ systems), multipartite entanglement directly impacts the upper bound of charging power. Using a $k$-producibility hierarchy, it is shown that for $N$ qubits governed by $H_B=(1/2)\sum_{j=0}^{N-1}\sigma_z^j$, if the state is at most $k$-producible, then

$4\Delta (H_B)^2_\rho \leq r k^2 + (N - r k)^2$

where $r$ is the integer part of $N/k$. This constrains the possible variance for states with limited entanglement. Highly entangled states (e.g., Greenberger–Horne–Zeilinger (GHZ) states, $k=N$) reach the maximal variance scaling, directly maximizing power. Substitution into the geometric bound yields:

$$P(t)^2 \leq \frac{1}{4} [r k^2 + (N - r k)^2] I_E(t)$$

demonstrating that quantum entanglement is both a necessary resource for achieving non-classical power scaling and a fundamental limiting mechanism.

4. Entropy-Preserving Dynamics and Operational Implications

A critical operational regime is entropy-preserving dynamics, commonly associated with closed, unitary evolution or specific entropy-preserving protocols in the many-copy limit. Entropy preservation is not only key for reaching the energy–entropy boundaries but also ensures well-defined work extraction—establishing a clear distinction between stored energy and usable work (ergotropy). Entropy increase (e.g., due to undesired environmental coupling) restricts the allowed trajectories in the energy–entropy plane, limiting both capacity and accessible work. Efficient quantum batteries therefore require engineering of both the Hamiltonian landscape and the protocol to suppress entropy production during the charging/discharging cycle.

5. Representative Many-Body Quantum Battery Models

Several paradigmatic models illuminate the above principles in concrete quantum systems:

Model	Key Features	Power Scaling
Integrable Spin Chain	1D spin-$1/2$ (Jordan–Wigner solvable), dominant local terms, weak quantum advantage	$\propto N$
Lipkin–Meshkov–Glick (LMG)	Infinite-range spin interactions, strong multipartite entanglement possible	linear in $N$
Dicke Model	$N$ spins interact with a single cavity, can realize strong entanglement in strong-coupling, but ultimate power scaling limited by speed in energy space	linear in $N$ (except narrow regimes)

In integrable spin chains, despite the presence of interactions, the effective local structure in momentum space restricts potential quantum advantage to near-classical (linear) scaling. The LMG model, though generating near quadratic scaling of energy variance via strong entanglement, does not yield a superextensive power scaling because the Fisher information $I_E(t)$ does not scale superextensively. The Dicke model with strong collective coupling can temporarily realize enhanced variance but, again, Fisher information bottlenecks lead to power scaling that is at best linear in $N$ for realistic parameters.

6. Quantum Advantage: Subtleties and Physical Interpretation

While quantum batteries are expected to outperform classical analogs due to the presence of quantum correlations, the quantum advantage is nuanced. The ultimate storage capacity does not require quantum correlations for noninteracting cell Hamiltonians—classically accessible states (product thermal/passive) suffice. Instead, quantum speed-up derives from exploiting entanglement and other forms of correlation to enhance energy variance, which, together with rapid energy-space evolution (high $I_E$), can boost charging power. However, most physical models studied so far reveal that, in practical regimes, the enhancement is often modest, and overall power typically scales at most linearly with the number of battery cells. The quantum advantage, accordingly, is a balance between achievable nonlocal variance and the actual attainable speed in the energy eigenspace.

7. Perspectives and Research Directions

This theoretical framework dictates the criteria for quantum battery optimization:

• Entropy-preserving charging and discharging protocols are essential for maximizing capacity and achieving reversible work extraction.
• Engineering multipartite entanglement can amplify energy variance and theoretically enhance power, but must be supported by protocols that also permit high Fisher information in energy space.
• Realistic models such as Dicke, LMG, and integrable spin chains set practical benchmarks, demonstrating that while quantum resources are necessary for outperforming classical batteries, scaling laws and decoherence constraints can impose significant operational limits.

Future research is directed toward identifying models and protocols in which the collective interplay between quantum correlations and nonlocal excitation transfer can yield genuine, robust quantum advantage for both capacity and power, and bridging to experimental platforms capable of realizing such entangled, entropy-preserving evolutions (Julia-Farre et al., 2018).