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Enhancing the Charging Power of Quantum Batteries
Open accessAbstract
Can collective quantum effects make a difference in a meaningful thermodynamic operation? Focusing on energy storage and batteries, we demonstrate that quantum mechanics can lead to an enhancement in the amount of work deposited per unit time, i.e., the charging power, when N batteries are charged collectively. We first derive analytic upper bounds for the collective quantum advantage in charging power for two choices of constraints on the charging Hamiltonian. We then demonstrate that even in the absence of quantum entanglement this advantage can be extensive. For our main result, we provide an upper bound to the achievable quantum advantage when the interaction order is restricted; i.e., at most k batteries are interacting. This constitutes a fundamental limit on the advantage offered by quantum technologies over their classical counterparts.
Overview
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TL;DR
The paper defines a collective quantum advantage for charging power when N quantum batteries are charged jointly rather than in parallel, and derives upper bounds on this advantage under energetic constraints. It shows that an extensive advantage is possible even without entanglement, but for physically realisable at-most-k-body interactions the advantage cannot scale with N and is bounded by a function quadratic in k.
Key findings
- For two constraints on the collective charging Hamiltonian, the quantum advantage is upper bounded as Γ_C1 ≤ √N and Γ_C2 ≤ N, with β = T‖/T_QSL^(1) quantifying inability to saturate the quantum speed limit in parallel charging.
- An extensive advantage can occur even for highly mixed states inside the separable ball: for N two-level thermal batteries, Γ_C1 = √N, Γ_C2 = N, and Γ_C0 = N, while the joint state need not be entangled during charging.
- For circuit-based charging with interaction order at most k, Theorem 2 gives Γ_C0 < γk, where γ does not scale with N; for ground-to-maximally-excited states γ = π/2.
- For generic time-dependent charging with interaction order k and maximum participation number m, Theorem 3 gives Γ_C0 < γ(k²(m−1)+k). Achievable comparisons are Γ_C1 = √k and Γ_C2 = k when N/k is an integer, and Γ_C0 = k in that same situation.
Methods
- Theoretical study of unitary charging of N quantum batteries, defining the quantum advantage as Γ = P♯/P‖ = T‖/T♯, where P♯ is collective charging power and P‖ is parallel charging power.
- Analytic derivation using the quantum speed limit, Bures angle L_N, Uhlmann fidelity, and three constraints C0, C1, and C2 on the collective Hamiltonian’s energy, standard deviation, or operator norm.
- Supplementary proofs use a piecewise unitary circuit model, Trotter-Suzuki decomposition, and a numerical search over Haar-random charging Hamiltonians with sample size ν = 10⁵ for (N,k) = (3,2), (4,2), (4,3), and (6,2).
Limitations
- The paper states that its bounds are not tight; Theorem 2’s strict inequality is only different by a constant factor from an achievable bound in a special case.
- Conjecture 4, extending Theorem 2 to any time-dependent k-body interaction Hamiltonian, is not proven; the numerical search found no counterexamples but “does not represent a proof,” and measure-zero counterexamples cannot be excluded.
- The paper states that k-body interactions are hard to engineer and that higher interaction orders are generally difficult; the exact radius of the separable ball is not known, though it has been bounded.
- Evident from the text: the work is theoretical and numerical, with no experimental demonstration of the predicted charging advantage.
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25 Own interpretation · 18 No basis given · 10 Not checked yet · 7 Own result
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“In contrast, the joint charging of N batteries is achieved in T ♯ = T ‖ /α ♯ using the global Hamiltonian H ♯ = α ♯ H ⊗ N , where the positive constant α ♯ is introduced to satisfy the chosen constraint.”
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“Proposition 1 An extensive quantum advantage can be attained even for highly mixed states, including those confined to the separable ball throughout the charging procedure.”
- Also stated in: § Enhancing the charging power of quantum batteries · p. 3 · plain
“no matter how mixed the battery is a quantum advantage that scales with the number of batteries involved is always achievable.”
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“We then go on to analytically prove that, for charging fields with finite interaction order, i.e. , involving at most k -body interaction terms, the quantum advantage is upper bounded by a quadratic function of k and cannot scale with the total number of batteries.”
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“The Hamiltonian used in the example above, and in Ref. [15], to saturate the bound for quantum advantage involves N -body interactions.”
What the sources say
- Not checked yetcites [15] F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, New J. Phys. 17 , 075015 (2015). source
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“Theorem 2 For a circuit based charging procedure with interaction order of at most k , the achievable quantum advantage is upper bounded as Γ C0 < γk , where γ is a constant that does not scale with the number N of batteries.”
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“The optimal local charging scheme is achieved in time T ‖ = π/ 2 by applying Hamiltonian H = | E 0 〉〈 E 1 | + | E 1 〉〈 E 0 | to each battery.”
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“We then demonstrate that even in the absence of quantum entanglement this advantage can be extensive.”
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“In the important case where ρ and σ are the ground and maximally excited states respectively, γ = π/ 2 .”
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“In this sense, it is a stricter constraint than C1 or C2.”
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“We first derive analytic upper bounds for the collective quantum advantage in charging power for two choices of constraints on the charging Hamiltonian.”
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“Firstly, for orthogonal pure initial and final states, the QSL can be saturated and β = 1 .”
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“In both cases (local and global) the deposited work is identical; thus, the quantum advantage is simply the ratio of T ‖ to T ♯ : Γ = α ♯ , which can be evaluated for the choice of constraint.”
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“Constraint C0 guarantees that both the time-averaged standard deviation and the time-averaged energy are bounded from above, as shown in Sec. A of SM.”
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“The two bounds in Eq. (4) are independent from each other, and constraint C1 is stronger than C2, as it leads to a stricter bound on the quantum advantage.”
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“The two bounds in Eq. (4) are independent from each other, and constraint C1 is stronger than C2, as it leads to a stricter bound on the quantum advantage.”
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“There, we show that E ♯ upper bounds both E ♯ / 2 and ∆ E ♯ .”
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“We find Γ C1 = √ N and Γ C2 = N (also Γ C0 = N for C0 given in Eq. (5)).”
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“Remarkably, neither T ‖ nor T ♯ depend on ϵ , while the total work done does.”
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“Here we show that ∆ E ♯ ≤ E ♯ and E ♯ ≤ 2 E ♯ by direct computation.”
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“We have proven two fundamental upper bounds for the quantum advantage, each corresponding to a different constraint on the charging Hamiltonian.”
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“Next, we show that attaining a quantum advantage requires entangling operations, but not entanglement itself.”
- Also stated in: § Enhancing the charging power of quantum batteries · p. 3 · plain
“Proposition 1 implies that, while a quantum advantage requires entangling operations, the joint state of N -batteries does not have to be entangled during the charging process.”
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“Since the distance from the maximally mixed state cannot change under unitary evolution, for a small enough choice of ϵ , the joint state of N batteries will lie within this ball throughout the evolution; yet, the quantum advantage remains extensive.”
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“Though the quantum advantage for power could be larger in other cases, including where the battery states are mixed [26], the improvement cannot grow with the number of batteries; i.e. , β is a constant function of N .”
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“For comparison, it has been shown elsewhere that Γ C1 = √ k and Γ C2 = k are achievable if the total number of batteries N can be divided by k , i.e. , if N/k = s ∈ N [15, 36].”
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“This numerical evidence does not represent a proof of our conjecture since there could be a measure zero set of Hamiltonians for which P > 1 .”
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In the paper
“The quantum advantage has been interpreted as the result of rapid evolution through the space of high-dimensional quantum states, typically obtained by means of global operations [15].”
What the sources say
- Not checked yetcites [15] F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, New J. Phys. 17 , 075015 (2015). source
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“Note that these bounds are not tight”
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In the paper
“There, an enhancement in sensing is still available for highly mixed states lying inside the separable ball [43].”
What the sources say
- Not checked yetcites [43] K. Modi, H. Cable, M. Williamson, and V. Vedral, Phys. Rev. X 1 , 021022 (2011). source
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In the paper
“While, in the case of pure states, entanglement is a necessary consequence of these global operations, a fully separable evolution is still accessible for those states that live in the separable ball.”
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In the paper
“Since each of the M terms at each time step must be implemented sequentially, U Trot takes M times longer to run than U , with a corresponding drop in power P ♯ = MP Trot .”
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“This Letter demonstrates that thermodynamic processes can indeed benefit from collective effects when time enters the picture, though physical limitations on the interaction order prevent us from utilising them.”
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In the paper
“For many physical systems, both k and m are limited: 2 or 3-body interactions are the norm for fundamental processes, and higher interaction orders are generally hard to engineer here [37-39].”
What the sources say
- Not checked yetcites [37] H. P. B¨ uchler, A. Micheli, and P. Zoller, Nat. Phys. 3 , 726 (2007). source
- Not checked yetcites [38] S. Boixo, S. T. Flammia, C. M. Caves, and J. Geremia, Phys. Rev. Lett. 98 , 090401 (2007). source
- Not checked yetcites [39] M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. Sewell, and M. W. Mitchell, Nature 471 , 486 (2011). source
plain- Asserted: passes — The authors put it forward as true.
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The authors state this as an assumption.
Each hµ acts on a completely different set of k batteries, i.e., [hµ, hµ'] = 0 for all µ, µ'.
“assuming that each h µ acts on a completely different set of k batteries, i.e. , [ h µ , h µ ′ ] = 0 ∀ µ, µ ′ .”
The authors state this as an assumption.
The reach of the interaction should be limited.
“This restriction is motivated by the idea that the reach of the interaction should be limited.”
The paper relies on this without support.
The effective participation number m tends to be constrained by the spatial arrangement of systems and the fact that interaction strength often drops off with distance.
“The effective participation number, or reach, m tends to be constrained by the spatial arrangement of systems and the fact that interaction strength often drops off with distance.”
The paper relies on this without support.
Collective behaviour has been demonstrated to provide an advantage in performing many information theoretic tasks, but such distinctions from classical behaviour are few and far between in thermodynamics.
“While collective behaviour has been demonstrated to provide an advantage in performing many information theoretic tasks, such distinctions from classical behaviour are few and far between in thermodynamics.”
Claims other papers make about this one
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In the citing paper
“In addition, similar techniques have been employed to optimally charge a quantum battery [110,111].”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [111]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Appendix C: Proof of Theorem 3 · p. 7
“( k = 3 , m = 2) Now the first 3-partition (1 , 2 , 3) is followed by (1 , 4 , 5) , (2 , 4 , 6) and (3 , 5 , 6) .”
§ Enhancing the charging power of quantum batteries · p. 4
“each battery is simultaneously interacting with at most m others.”
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Optimal building block of multipartite quantum battery in the driven-dissipative charging
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Common Environmental Effects on Quantum Thermal Transistor
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Efficient wireless charging of a quantum battery
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Algorithm for solving unconstrained unitary quantum brachistochrone problems
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Quantum Speed Limit in Quantum Sensing
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Metadata
- Authors
- Campaioli, F., Pollock, F. A., Binder, F. C., Céleri, L., Goold, J., Vinjanampathy, S., & Modi, K.
- Affiliations
- Monash University (AU)
- Nanyang Technological University (SG)
- Universidade Federal de Goiás (BR)
- The Abdus Salam International Centre for Theoretical Physics (ICTP) (IT)
- Centre for Quantum Technologies (SG)
- ORCID
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2017-04-12
- Volume
- 118
- Issue
- 15
- Pages
- 150601-150601
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2561264561
- Cited by
- 531
- References
- 50
- Citations by year
- 2026: 108 · 2025: 107 · 2024: 77 · 2023: 51 · 2022: 51 · 2021: 43 · 2020: 39 · 2019: 30 · 2018: 23 · 2017: 2
- FWCI
- 13.30
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- PubMed (publishedVersion)
- RMIT Research Repository (RMIT University Library) (submittedVersion)
- DR-NTU (Nanyang Technological University) (submittedVersion)
- DSpace (IIT Bombay) (submittedVersion)
- Figshare (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Research Foundation Singapore (NRF-CRP14-2014-02, NRF-NRFF2016-02)
- European Cooperation in Science and Technology (MP1209)
- Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq: 401230/2014-7, CNPq: 305086/2013-8, CNPq: 445516/2014-3)
- IITB-IRCC (16IRCCSG019)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Engineering › Mechanical Engineering › Advanced Thermodynamic Systems and Engines
- Keywords
- quantum batteries, charging power, quantum advantage, energy storage, work rate, upper bounds, quantum entanglement, interaction order, quantum thermodynamics
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