Fundamental Limitations on the Reliabilities of Power and Work in Quantum Batteries

TL;DR

The paper quantifies reliability in quantum batteries by defining noise-to-signal ratios (NSRs) for work and power via full counting statistics, and derives universal lower bounds that tie NSRs to charging speed and time correlations. It further uncovers a Schrödinger–Robertson-type trade-off that prevents simultaneous minimization of both NSRs, implying a fundamental incompatibility between high reliability of work and power. Through paradigmatic, Ising-like, and hybrid many-body models, it shows that stronger entanglement (higher power) generally degrades power reliability, while work reliability can improve, yielding a universal cluster scaling $\mathcal{N}^{W}_{t}\mathcal{N}^{P}_{t}=\frac{k^{2}}{N^{2}}$ in certain regimes and favoring hybrid charging schemes. These insights inform practical design of reliable, high-performance quantum batteries, suggesting that optimal operation lies in balanced, intermediate-range interactions rather than purely parallel or fully collective charging.

Abstract

Quantum batteries, microscopic devices designed to address energy demands in quantum technologies, promise high power during charging and discharging processes. Yet their practical usefulness and performance depend critically on reliability, quantified by the noise-to-signal ratios (NSRs), i.e., normalized fluctuations of work and power, where reliability decreases inversely with increasing NSR. We establish fundamental limits to this reliability: both work and power NSRs are universally bounded from below by a function of charging speed, imposing a reliability limit inherent to any quantum battery. More strikingly, we find that a quantum mechanical uncertainty relation forbids the simultaneous suppression of work and power fluctuations, revealing a fundamental trade-off that also limits the reliability of quantum batteries. We analyze the trade-off and limits, as well as their scaling behavior, across parallel (local), collective {(fully non-local)}, and hybrid (semi-local) charging schemes for many-body quantum batteries, finding that increasing power by exploiting stronger entanglement comes at the cost of diminished reliability of power. Similar trends are also observed in the charging of quantum batteries utilizing transverse Ising-like interactions. These suggest that achieving both high power and reliability require neither parallel nor collective charging, but a hybrid charging scheme with an intermediate range of interactions. Therefore, our analysis shapes the practical and efficient design of reliable and high-performance quantum batteries.

Figures (3)

• Figure 1: The Plots (a)–(c) illustrate the charging dynamics of a quantum battery governed by a $k$-body charging Hamiltonian given in Eq. \ref{['eq:HcManyBody']}, with $N = 12$, $\Omega_0 = 1$, and $\Omega_k = \tfrac{k}{N}\Omega_0$. In all Plots, $k =2, \ 3, \hbox{and}$ 4 represented by purple, red, and blue curves, respectively. Plot (a) shows the average work $\langle \mathcal{W}_t^{(k)} \rangle = N\omega_0\sin^2(\Omega_k t)$ (solid lines) and average power $\langle \mathcal{P}_t^{(k)} \rangle = k\omega_0\Omega_0\sin(2\Omega_k t)$ (dashed lines). Plot (b) depicts the corresponding fluctuations $(\Delta \mathcal{W}_t^{(k)})^2 = Nk\omega_0^2\sin^2(2\Omega_k t)/4$ (solid) and $(\Delta \mathcal{P}_t^{(k)})^2 = 4Nk^3(\omega_0\Omega_0)^2\sin^4(\Omega_k t)$ (dashed). (c) depicts the NSRs of work and power, $\mathcal{N}_t^{\mathcal{W},(k)} = \tfrac{k}{N}\cot^2(\Omega_k t)$ (solid) and $\mathcal{N}_t^{\mathcal{P},(k)} = \tfrac{k}{N}\tan^2(\Omega_k t)$ (dashed), shows their trade-off relation.
• Figure 2: Panels (a)–(c) illustrate the charging dynamics of a quantum battery governed by an $s$-body charging Hamiltonian given in Eq. \ref{['XX']}, for $N = 10$, $\Omega_s = 1$, and $\omega_0 = 2$. All plots are generated for $s = 2, 3,$ and $4$, corresponding to two-, three-, and four-body interactions, and are represented by blue, orange, and green curves, respectively. Panel (a) shows the power and the NSR of power, panel (b) presents the average work and the NSR of work, and panel (c) displays the product of the NSRs of work and power, $\mathcal{N}_t^{\mathcal{W},(k)} \mathcal{N}_t^{\mathcal{P},(k)}$. The plots of work and power fluctuations are provided in Appendix \ref{['transverse']}.
• Figure 3: The plots (a)–(c) supplements the Fig. \ref{['XX-Plots']} in the main text. Here, we depict (in plots (a)–(c)) that fluctuations in work and power along with fidelity (between initial and time evolved state) during charging process. Here, we have considered, total spin $N=10$, $\Omega_s = 1$ and $\omega_0=2.$