Quantum Batteries Perform Best When Entanglement Is Minimal

Yiding Wang and colleagues at Hainan Normal University found that battery capacity decreases as quantum entanglement, steering, Bell nonlocality, and coherence increase, reaching a maximum when these resources are absent. The study reveals a positive correlation between residual battery capacity and entanglement, alongside a complex relationship with quantum imaginarity and state texture. These findings, independent of specific system parameters, offer key insights into the dynamic evolution of quantum batteries and contribute to the advancement of quantum energy storage technologies.

Quantum battery performance linked to entanglement, coherence and system modelling

A rapidly evolving field, quantum thermodynamics, focuses on understanding quantum systems at the nanoscale. The concept of quantum batteries represents a significant development, utilising quantum effects to enhance performance characteristics during charging. Initially proposed by Alicki and Fannes, the concept explored the role of quantum resources, including entanglement and coherence, in enabling efficient work extraction from quantum systems. Subsequent work has broadly focused on two areas: the interaction between quantum battery performance and quantum resources, revealing links between figures of merit like ergotropy and charging power with quantum state properties, and the impact of specific models on the charging process.

Spin-chain, Tavis-Cummings, harmonic oscillator, Dicke, three-level, and cavity-optomechanical implementations are among the models explored. These efforts collectively demonstrate how quantum mechanics enhances battery performance, paving the way for nanoscale energy devices. Recent findings have emerged in the field of quantum batteries, particularly concerning dissipative charging, given the inevitable interaction of any quantum system with its environment.

Andolina et al. proposed a quantum battery model saturating the quantum speed limit, comprising two harmonic oscillators coupled through nonlinear interaction during nonequilibrium charging. Hu et al. investigated a wireless charging scheme for quantum batteries utilising multiple charger units, discovering that these units improve charging performance in weak and moderate coupling regimes, but show reduced efficiency in strong coupling regimes. Song et al. proposed a quantum battery scheme utilising the nitrogen-vacancy centre in diamond, revealing that resistance to self-discharge can be improved by increasing coherent ergotropy.

In 2023, Yang et al. introduced a new figure of merit, quantum battery capacity, and recently experimentally verified it using an optical platform. Determining the relationship between quantum resources such as entanglement and coherence, and battery capacity remains an important question, driving ongoing work. Clarifying this connection will advance our understanding of how quantum resources influence quantum energy storage systems, and provide theoretical guidance for designing nanoscale energy devices with quantum advantages.

Relationships have been established between battery state capacity and various quantum resources, including entanglement, steering, Bell nonlocality, coherence, negativity, and quantum state texture, within a system comprising a battery spin and a charger spin, during its dynamical evolution. These findings reveal a trade-off relationship between battery state capacity and all resources except quantum state texture. Established under a specific model of mutual coupling, these relationships are independent of system parameters.

Entanglement is indicated by the mixedness of a reduced density matrix derived from a pure state, quantified using concurrence: E(ρ) = q 2[1 −Try(ρ2 b)]. Quantum steering, stronger than entanglement but weaker than Bell nonlocality, is verified through steering inequalities, such as the linear steering inequality: F3(ρAB, μ) = 1 √ 3| 3X k=1 ⟨AkBk⟩| ≤1. The CHSH inequality, with a maximum violation of Bmax(ρ) = 2√ι1 + ι2, quantifies Bell nonlocality, where ι1 and ι2 are the largest eigenvalues of the correlation matrix. Quantum coherence, the ability to maintain phase relationships, is quantified using the l1 norm: C1(ρ) = min δ∈I ∥ρ −δ∥l1 = X i=j |ρij|. Quantum imaginarity, quantified using the l1-norm, is defined as I(ρ) = X i=j |I’m(ρij)|0. Quantum state texture (QST), a novel resource linked to quantum coherence, is quantified using the trace distance measure: Ttr(ρ) = 1 Try|ρ −f1|, where Try|A| = Try(AA†) 1 2 is the trace norm of A. The battery capacity is defined as the energy difference between the active and passive states: C(ρ; H) = Try[ρ↑H] −Try[ρ↓H], where ρ↑ and ρ↓ are the active and passive states, respectively. An analytical expression for the battery capacity exists, based on unitary invariance, in terms of the spectrum of the battery state and the energy levels of the Hamiltonian: C(ρ; H) = d−1X i=0 ǫi(λi −λd−1−i), where {λi} and {ǫi} represent the energy levels of ρ and the Hamiltonian, respectively.

The Hamiltonian for a two-qubit quantum battery system is defined as H = Hb + Hc + HI, where Hb = −ωbσb z, Hc = −ωcσc z, and HI = J1(σb +σc −+ σb −σc +) + Job zσc z. Here, σb z and σc z represent the spin operators, ωb and ωc denote the effective external magnetic field strengths, σb(c) x,y,z are the standard Pauli operators, and σb(c) ± = (σb(c) x ± iσb(c) y )/2 are the up and down operators. J1 and J2 are the spin flip-flop coupling constants. Our understanding of quantum batteries, devices which promise to revolutionise energy storage at the nanoscale, is steadily being refined.

A curious tension exists; maximising quantum entanglement appears beneficial for optimising energy transfer within the system, evidenced by the positive correlation with residual capacity, yet it simultaneously diminishes the overall amount of energy the battery can actually hold. This suggests a fundamental trade-off between efficient charging and storage potential, a challenge not encountered in conventional battery technology. Recognising this apparent conflict between energy transfer and storage capacity is vital for refining quantum battery design.

This trade-off allows investigation of optimising battery architecture; hybrid systems may balance quantum entanglement, steering, Bell nonlocality, and coherence for rapid charging with materials that enhance storage potential, ultimately delivering both speed and longevity in quantum energy storage. Analysis demonstrates a decreasing relationship between battery capacity and quantum entanglement, steering, Bell nonlocality, and coherence within a two-qubit system, peaking when these four quantum resources are minimal, indicating a need for careful consideration of these factors. Unlike entanglement, steering, Bell nonlocality, and coherence, quantum state texture showed a positive correlation with capacity, suggesting a potential route to optimise energy storage. Establishing these connections, independent of specific system settings, advances the theoretical foundations of quantum batteries, devices harnessing quantum mechanics to improve performance and offering a promising avenue for future energy storage solutions.

The research revealed that battery capacity in a two-qubit system decreases as quantum entanglement, steering, Bell nonlocality, and coherence increase, reaching a maximum when these resources are absent. This finding highlights a trade-off between the speed of energy transfer and the amount of energy storable within the quantum battery. While maximising entanglement aids transfer, it simultaneously reduces overall capacity, a phenomenon not observed in conventional batteries. The study also identified a positive correlation between quantum state texture and battery capacity, offering a potential avenue for optimisation, and contributes to the theoretical understanding of quantum energy storage systems.