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Cited by 531Advanced Thermodynamics and Statistical MechanicsQuantum Information and CryptographyAdvanced Thermodynamic Systems and Engines

Enhancing the Charging Power of Quantum Batteries

Open access
Campaioli, Francesco, Pollock, Felix A., Binder, Felix C., Céleri, Lucas, Goold, John, Vinjanampathy, Sai, Modi, Kavan·2017·Physical Review Letters
DOI 10.1103/physrevlett.118.150601
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Abstract

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.

AI snapshot

Written from the paper’s full text.

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.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevlett.118.150601

Cites

48 of 48 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 16 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § Enhancing the charging power of quantum batteries · p. 3

    For the same N two-level batteries, the joint charging of N batteries is achieved in T♯ = T‖/α♯ using the global Hamiltonian H♯ = α♯ H⊗N, where α♯ is a positive constant introduced to satisfy the chosen constraint.

    “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.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Enhancing the charging power of quantum batteries · p. 3

    An extensive quantum advantage can be attained even for highly mixed states, including those confined to the separable ball throughout the charging procedure.

    “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.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Enhancing the charging power of quantum batteries · p. 1

    For charging fields with finite interaction order, involving at most k-body interaction terms, the quantum advantage is upper bounded by a quadratic function of k.

    “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.”
    Quote in the paperSubstantial claim
  • Findingcitation
    § Enhancing the charging power of quantum batteries · p. 3

    The Hamiltonian used in the example above, and in Ref. [15], to saturate the bound for quantum advantage involves N-body interactions.

    “The Hamiltonian used in the example above, and in Ref. [15], to saturate the bound for quantum advantage involves N -body interactions.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [15] F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, New J. Phys. 17 , 075015 (2015). source

      Not checked in the free run.

  • Findingreasoning
    § Enhancing the charging power of quantum batteries · p. 4

    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.

    “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 &lt; γk , where γ is a constant that does not scale with the number N of batteries.”
    Quote in the paperSubstantial claim
  • May combine two claims. 17
  • Classified as probably not a substantial claim. 6

  • May combine two claims.

    Findingnone given
    § Enhancing the charging power of quantum batteries · p. 3

    For orthogonal pure initial and final states, the quantum speed limit can be saturated and β = 1.

    “Firstly, for orthogonal pure initial and final states, the QSL can be saturated and β = 1 .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Enhancing the charging power of quantum batteries · p. 3

    In both local and global charging cases, the deposited work is identical; thus the quantum advantage is simply the ratio of T‖ to T♯: Γ = α♯.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Enhancing the charging power of quantum batteries · p. 4

    Constraint C0 guarantees that both the time-averaged standard deviation and the time-averaged energy are bounded from above.

    “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Findingnone given
    § Enhancing the charging power of quantum batteries · p. 3

    The two bounds in Eq. (4) are independent from each other.

    “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.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § Enhancing the charging power of quantum batteries · p. 3

    Constraint C1 is stronger than C2, leading to a stricter bound on the quantum advantage.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Enhancing the charging power of quantum batteries · p. 4

    E♯ upper bounds both E♯/2 and ∆E♯.

    “There, we show that E ♯ upper bounds both E ♯ / 2 and ∆ E ♯ .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Enhancing the charging power of quantum batteries · p. 3

    For the explicit example, Γ_C1 = √N and Γ_C2 = N, and also Γ_C0 = N for C0 given in Eq. (5).

    “We find Γ C1 = √ N and Γ C2 = N (also Γ C0 = N for C0 given in Eq. (5)).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given
    § Enhancing the charging power of quantum batteries · p. 3

    Neither T‖ nor T♯ depend on ε, while the total work done does.

    “Remarkably, neither T ‖ nor T ♯ depend on ϵ , while the total work done does.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Appendix A: Relation between constraints · p. 6

    ∆E♯ ≤ E♯ and E♯ ≤ 2E♯.

    “Here we show that ∆ E ♯ ≤ E ♯ and E ♯ ≤ 2 E ♯ by direct computation.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given
    § Conjecture 4 Theorem 2 holds for any time-dependent k -body interaction Hamiltonian. · p. 4

    We have proven two fundamental upper bounds for the quantum advantage, each corresponding to a different constraint on the charging Hamiltonian.

    “We have proven two fundamental upper bounds for the quantum advantage, each corresponding to a different constraint on the charging Hamiltonian.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § Enhancing the charging power of quantum batteries · p. 1

    A quantum advantage does not require entanglement itself; the joint state of N batteries need not be entangled during the charging process.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § Enhancing the charging power of quantum batteries · p. 3

    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 the separable ball throughout the evolution; yet the quantum advantage remains extensive.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § Enhancing the charging power of quantum batteries · p. 3

    The quantum advantage for power cannot grow with the number of batteries; β is a constant function of N.

    “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 .”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § Enhancing the charging power of quantum batteries · p. 4

    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.

    “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].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [15] F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, New J. Phys. 17 , 075015 (2015). source

      Not checked in the free run.

    • cites [36] V. Giovannetti, S. Lloyd, and L. Maccone, J. Opt. B: Quant. Sem. Opt. 6 , S807 (2004). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § Appendix D: On Conjecture 4. · p. 8

    This numerical evidence does not represent a proof of the conjecture since there could be a measure zero set of Hamiltonians for which P > 1.

    “This numerical evidence does not represent a proof of our conjecture since there could be a measure zero set of Hamiltonians for which P &gt; 1 .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Conjecture 4 Theorem 2 holds for any time-dependent k -body interaction Hamiltonian. · p. 4

    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.

    “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].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [15] F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, New J. Phys. 17 , 075015 (2015). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given
    § Enhancing the charging power of quantum batteries · p. 4

    These bounds are not tight.

    “Note that these bounds are not tight”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Conjecture 4 Theorem 2 holds for any time-dependent k -body interaction Hamiltonian. · p. 4

    In quantum metrology, an enhancement in sensing is still available for highly mixed states lying inside the separable ball.

    “There, an enhancement in sensing is still available for highly mixed states lying inside the separable ball [43].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [43] K. Modi, H. Cable, M. Williamson, and V. Vedral, Phys. Rev. X 1 , 021022 (2011). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationnone given
    § Conjecture 4 Theorem 2 holds for any time-dependent k -body interaction Hamiltonian. · p. 4

    In the case of pure states, entanglement is a necessary consequence of these global operations, but a fully separable evolution is still accessible for those states that live in the separable ball.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § Appendix C: Proof of Theorem 3 · p. 7

    U_Trot takes M times longer to run than U, with a corresponding drop in power P♯ = M P_Trot.

    “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 .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § Conjecture 4 Theorem 2 holds for any time-dependent k -body interaction Hamiltonian. · p. 4

    Thermodynamic processes can benefit from collective effects when time enters the picture, though physical limitations on the interaction order prevent us from utilising them.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation
    § Enhancing the charging power of quantum batteries · p. 4

    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.

    “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].”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [37] H. P. B¨ uchler, A. Micheli, and P. Zoller, Nat. Phys. 3 , 726 (2007). source

      Not checked in the free run.

    • cites [38] S. Boixo, S. T. Flammia, C. M. Caves, and J. Geremia, Phys. Rev. Lett. 98 , 090401 (2007). source

      Not checked in the free run.

    • cites [39] M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. Sewell, and M. W. Mitchell, Nature 471 , 486 (2011). source

      Not checked in the free run.

22 of 22

  • 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.”
4 of 4

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

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.

  • The k=3, m=2 example exceeds the stated interaction reach

    Minor

    The stated restriction allows each battery to interact with at most two others when m=2. In the example, battery 1 belongs to both (1,2,3) and (1,4,5), so it interacts with four other batteries: 2, 3, 4, and 5.

1 of 1

Cites

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2 of the 50 references couldn’t be matched to a known paper.

Research works printed but not linked: 1

  1. L. Mandelstam, 1945, J. Phys. (Moscow), 9, 1

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Related

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
DOI
10.1103/physrevlett.118.150601
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
Licence
link.aps.org/licenses/aps-default-license (vor)
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 › Computer Science › Artificial Intelligence › 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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