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Quantacell: powerful charging of quantum batteries
Open accessAbstract
We study the problem of charging a quantum battery in finite time. We demonstrate an analytical optimal protocol for the case of a single qubit. Extending this analysis to an array of N qubits, we demonstrate that an N-fold advantage in power per qubit can be achieved when global operations are permitted. The exemplary analytic argument for this quantum advantage in the charging power is backed up by numerical analysis using optimal control techniques. It is demonstrated that the quantum advantage for power holds when, with cyclic operation in mind, initial and final states are required to be separable.
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Written from the paper’s full text.
TL;DR
The paper derives an analytical optimal protocol for charging a single-qubit quantum battery in finite time and extends it to arrays of N qubits. It shows that global entangling operations can yield an N-fold advantage in power per qubit over local parallel charging, even when initial and final states are separable.
Key findings
- For a single qubit, the power-optimal driving is along a geodesic at constant angular speed \(E_{\max}\), with no driving along the reference Hamiltonian direction.
- For an array of N qubits, global driving with the same maximum energy gap \(E_{\max}^{(N)}=N E_{\max}\) achieves an N-fold increase in power per qubit compared with parallel local driving.
- The quantum advantage persists under cyclic operation with separable initial and final states; entanglement appears only during the protocol.
- Numerical optimal control for full charging shows the minimum charging time scales as \(1/N\), and for a four-qubit solution entanglement across the 12–34 bipartition vanishes at the start and end.
Methods
- Analytical derivation for a single qubit using the von Neumann equation, Bloch-sphere parametrization, and optimization of a family of objective functions interpolating power and ergotropy under a trace-norm bound on the driving Hamiltonian.
- Extension to N-qubit arrays comparing parallel local driving with global entangling driving, with average work per qubit set to unity in the examples.
- Numerical optimal control for full charging of N qubits and for entanglement dynamics in a four-qubit numerical solution.
Limitations
- Stated: a full optimization over all possible unitary transformations is not analytically feasible, so the N-qubit advantage is demonstrated with a specific evolution.
- Stated: future work is needed to allow for degradation/purifying stages and to generalize the approach to open systems with noise and decoherence.
- Evident: the analysis assumes ideal cyclic unitary dynamics and purity preservation; no experimental implementation or realistic noise model is provided.
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11 No basis given · 5 Own result · 3 Own interpretation · 2 Not checked yet
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In the paper
“Extending this analysis to an array of N qubits, we demonstrate that an N-fold advantage in power per qubit can be achieved when global operations are permitted.”
- Also stated in: § Additional information · plain
“Permitting for the same constraint as in the parallel case, that is E NE N max ( ) max = , an N-fold increase in power per qubit is achieved.”
§ Quantacell: powerful charging of quantum batteries > Abstract
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- Also stated in: § Additional information · plain
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In the paper
“That is, entangling processes, which allow for the creation of quantum correlations, can achieve a higher power than local operations.This is the case even when correlations are only allowed to appear during the process but not in the input and output states.”
§ Quantacell: powerful charging of quantum batteries > 1. Introduction
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In the paper
“We demonstrate an analytical optimal protocol for the case of a single qubit.”
- Also stated in: § 5. Conclusion · boosted
“In this article we have given a full derivation of a qubit protocol that achieves maximum power when charging a quantum battery-a work qubit-under constrained driving.”
§ Quantacell: powerful charging of quantum batteries > Abstract
boosted- Asserted: passes — The authors put it forward as true.
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- Also stated in: § 5. Conclusion · boosted
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In the paper
“This can be regarded as pathological and it is worth pointing out that for 1 α < and θ π < the optimum is always reached for finite T.”
§ Quantacell: powerful charging of quantum batteries > 3. Powerful driving for a single qubit
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“It is demonstrated that the quantum advantage for power holds when, with cyclic operation in mind, initial and final states are required to be separable.”
- Also stated in: § Additional information · hedged
“Interestingly, the advantage from correlations appears even though initial and final states are separable.”
§ Quantacell: powerful charging of quantum batteries > Abstract
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In the paper
“The optimal solution is found for the driving time T m that satisfies: ( ) ( ) E T E T E T cos sin cos . (15 ) m m m 0 m a x max 0 m a x 0 θ α θ θ + - + =”
§ Quantacell: powerful charging of quantum batteries > 3. Powerful driving for a single qubit
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In the paper
“P E N (19 ) N max ( ) π 〈 〉 =”
§ Quantacell: powerful charging of quantum batteries > Additional information
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In the paper
“The solution corresponds, unsurprisingly perhaps, to driving along the geodesic with fixed r and t 0 ϕ ϕ = at constant angular speed E max , that is E t. (11 ) t 0 m a x θ θ = +”
§ Quantacell: powerful charging of quantum batteries > 3. Powerful driving for a single qubit
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The optimal protocol under the above constraint is found for v E v E v sin , cos , 0. (10 ) t x t t y t t z max max ϕ ϕ = - = =”
§ Quantacell: powerful charging of quantum batteries > 3. Powerful driving for a single qubit
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In the paper
“The average work in this process is again W N 〈 〉 = .”
§ Quantacell: powerful charging of quantum batteries > 4. Role of entanglement in charging an array of quantum batteries
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In the paper
“We see that the charging time is commensurate with N 1 behaviour.”
§ Quantacell: powerful charging of quantum batteries > Additional information
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In the paper
“This global driving is in fact optimal for the current scenario.”
§ Quantacell: powerful charging of quantum batteries > 5. Conclusion
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In the paper
“Generally, global driving between any two states a | 〉 and b | 〉 is optimized by evolution with H a b b a ˆd ∝ | 〉〈 | + | 〉〈 |where the proportionality is given by the external constraint.”
§ Quantacell: powerful charging of quantum batteries > 5. Conclusion
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In the paper
“This implies that in the optimal case no driving happens in the direction of the reference Hamiltonian.”
§ Quantacell: powerful charging of quantum batteries > 3. Powerful driving for a single qubit
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In the paper
“Consequently, batteries in a pure state have higher capacity than ones in mixed states.”
§ Quantacell: powerful charging of quantum batteries > 2. Charging a quantum battery
What the sources say
- Not checked yetcites Marshall et al 2011
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The quantum speed-up observed here can be understood in part as a consequence of the shorter distance that has to be traveled through state space when entangling operations are permitted.”
§ Quantacell: powerful charging of quantum batteries > Additional information
plainconditional- Asserted: passes — The authors put it forward as true.
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In the paper
“This exemplifies the role of entanglement during the protocol as being responsible for the quantum speed-up.”
§ Quantacell: powerful charging of quantum batteries > Additional information
plaindirect- Asserted: passes — The authors put it forward as true.
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- It may state two separate claims as one. 3
- Probably not a substantial claim. 1
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The work done is trivially W N 〈 〉 = -that is, the average work per qubit is unity and the driving time is T E par max π = .”
§ Quantacell: powerful charging of quantum batteries > 4. Role of entanglement in charging an array of quantum batteries
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It possesses N 1 + equidistant eigenvalues with the gap between the largest and the lowest eigenvalue given by NE max .”
§ Quantacell: powerful charging of quantum batteries > 4. Role of entanglement in charging an array of quantum batteries
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“We note that at the beginning and at the end of the dynamics, the qubits are disentangled, indicated by vanishing entropies.”
§ Quantacell: powerful charging of quantum batteries > Additional information
hedgedSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“Such infinitely powerful driving, however, would be in conflict with so-called quantum speed limits which bound the evolution time by the inverse mean energy (Margolus and Levitin 1998), or, alternatively, its variance (Mandelstam and Tamm 1945) and apply to time-dependent unitary evolution between any two quantum states (Deffner and Lutz 2013).”
§ Quantacell: powerful charging of quantum batteries > 2. Charging a quantum battery
What the sources say
- Not checked yetcites Margolus and Levitin 1998
- Not checked yetcites Mandelstam and Tamm 1945
- Not checked yetcites Deffner and Lutz 2013 source
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper relies on this without support.
All thermal states are passive.
“Note that all thermal states are passive.”
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?
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 [110]
In the citing paper
“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].”
From Enhancing the Charging Power of Quantum Batteries (2017) · cites this paper as [15]
In the citing paper
“The Hamiltonian used in the example above, and in Ref. [15], to saturate the bound for quantum advantage involves N -body interactions.”
From Enhancing the Charging Power of Quantum Batteries (2017) · cites this paper as [15]
In the citing paper
“The quantum advantage is tight for orthogonal initial and final states, due to the example given in Ref. [15].”
From Enhancing the Charging Power of Quantum Batteries (2017) · cites this paper as [15]
In the citing paper
“In this particular case, such a speed-up can be obtained for pure states using the time-independent Hamiltonian H = √ s ∑ s µ =1 h µ , with h µ = | 1 〉 ⊗ k 〈 d | ⊗ k + h.c. , assuming that each h µ acts on a completely different set of k batteries, i.e. , [ h µ , h µ ′ ] = 0 ∀ µ, µ ′ .”
From Enhancing the Charging Power of Quantum Batteries (2017) · cites this paper as [15]
In the citing 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].”
From Enhancing the Charging Power of Quantum Batteries (2017) · cites this paper as [15]
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
§ Quantacell: powerful charging of quantum batteries > Additional information
“This is commensurate with the demand that all batteries be available for individual use at the final time with loss of purity.”
§ Quantacell: powerful charging of quantum batteries > 4. Role of entanglement in charging an array of quantum batteries
“We define this-the condition that there is no reduction in the purity of the marginals at the end of the protocol-as the condition of no degradation.”
Citations
Cites
Inequalities: Theory of Majorization and Its Applications
Marshall, A. W., Olkin, I., & Arnold, B. C. (2010) · Springer series in statistics · doi:10.1007/978-0-387-68276-1 · Cited by 2,593
Geometry of Quantum States: An Introduction to Quantum Entanglement
Bengtsson, I., & Zyczkowski, K. (2006) · doi:10.1017/cbo9780511535048 · Cited by 1,357
The maximum speed of dynamical evolution
Margolus, N., & Levitin, L. B. (1998) · Physica D: Nonlinear Phenomena · doi:10.1016/s0167-2789(98)00054-2 · Cited by 887
Entanglement boost for extractable work from ensembles of quantum batteries
Alicki, R., & Fannes, M. (2013) · Physical Review E · doi:10.1103/physreve.87.042123 · Cited by 648
Maximal work extraction from finite quantum systems
ALLAHVERDYAN, A. E., Balian, R., & Nieuwenhuizen, T. M. (2004) · Europhysics Letters (EPL) · doi:10.1209/epl/i2004-10101-2 · Cited by 631
Passive states and KMS states for general quantum systems
Pusz, W., & Woronowicz, S. L. (1978) · Communications in Mathematical Physics · doi:10.1007/bf01614224 · Cited by 530
Entanglement Generation is Not Necessary for Optimal Work Extraction
Hovhannisyan, K. V., Perarnau-Llobet, M., Huber, M., & Acín, A. (2013) · Physical Review Letters · doi:10.1103/physrevlett.111.240401 · Cited by 351
Thermodynamical proof of the Gibbs formula for elementary quantum systems
Lenard, A. (1978) · Journal of Statistical Physics · doi:10.1007/bf01011769 · Cited by 337
More bang for your buck: super-adiabatic quantum engines
Campo, A. D., Goold, J., & Paternostro, M. (2014) · Scientific Reports · doi:10.1038/srep06208 · Cited by 306
Self-citationTime-Optimal Quantum Evolution
Carlini, A., Hosoya, A., Koike, T., & Okudaira, Y. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.060503 · Cited by 301
Research works printed but not linked: 3
- 10.1103/physreve.91.032118 · DOI
- Mandelstam L, 1945, J. Phys. (USSR), 9, 249
- 10.1103/physrevlett.111.260501 · DOI · Paper page
Cited by
Sorted by citations
Vinjanampathy, S., & Anders, J. (2016) · Contemporary Physics · doi:10.1080/00107514.2016.1201896 · Cited by 940
Self-citationThe role of quantum information in thermodynamics—a topical review
Goold, J., Huber, M., Riera, A., del Rio, L., & Skrzypczyk, P. (2016) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/49/14/143001 · Cited by 973
Self-citationQuantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control
Deffner, S., & Campbell, S. (2017) · Journal of Physics A: Mathematical and Theoretical · doi:10.1088/1751-8121/aa86c6 · Cited by 585
Enhancing the Charging Power of Quantum Batteries
Campaioli, F., Pollock, F. A., Binder, F. C., Céleri, L., Goold, J., Vinjanampathy, S., & Modi, K. (2017) · Physical Review Letters · doi:10.1103/physrevlett.118.150601 · Cited by 531
Self-citationHigh-Power Collective Charging of a Solid-State Quantum Battery
Ferraro, D., Campisi, M., Andolina, G. M., Pellegrini, V., & Polini, M. (2018) · Physical Review Letters · doi:10.1103/physrevlett.120.117702 · Cited by 543
Metadata
- Authors
- Binder, F. C., Vinjanampathy, S., Modi, K., & Goold, J.
- Affiliations
- University of Oxford (GB)
- Centre for Quantum Technologies (SG)
- Monash University (AU)
- The Abdus Salam International Centre for Theoretical Physics (ICTP) (IT)
- Venue
- New Journal of Physics
- Publisher
- IOP Publishing
- Published
- 2015-07-22
- Volume
- 17
- Issue
- 7
- Pages
- 075015
- ISSN
- 1367-2630 (ISSN-L)
- OpenAlex
- W1526062388
- Cited by
- 475
- References
- 17
- Citations by year
- 2026: 100 · 2025: 83 · 2024: 69 · 2023: 41 · 2022: 50 · 2021: 33 · 2020: 36 · 2019: 25 · 2018: 21 · 2017: 6 · 2016: 9 · 2015: 2
- FWCI
- 7.97
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- New Journal of Physics (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- Monash University Research Portal (Monash University) (submittedVersion, cc-by)
- National University of Singapore (submittedVersion, cc-by)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Research Foundation
- National Research Foundation Singapore
- Centre for Quantum Technologies
- 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, battery charging, optimal control, qubit arrays, global operations, charging power, separable states, cyclic operation, quantum advantage
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