Concentration of ergotropy in many-body systems

Janet Anders; Karen V. Hovhannisyan; Rick P. A. Simon

REVIEW 1 major objections 7 minor 90 references

Concentration of ergotropy in many-body systems

T0 review · 1 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that the maximal unitarily extractable work from a large many-body quantum system is exponentially concentrated around its average for almost all states, and that this average is macroscopic.

desk verdict Solid new concentration theorem for ergotropy, but the macroscopic-average conclusion leans on an unproven assumption that fails for some perfectly local Hamiltonians. read the letter →

arxiv 2412.19801 v1 pith:TQUXEB54 submitted 2024-12-27 quant-ph cond-mat.stat-mech

Karen V. Hovhannisyan , Rick P. A. Simon , Janet Anders This is my paper · ORCID

classification quant-phcond-mat.stat-mech MSC 15B5260B2081P45 PACS 03.65.-w05.30.-d

keywords ergotropyquantumbatteriesmeasureconcentrationLevy'slemmaBuresHilbert-SchmidtvonNeumannentropymany-bodysystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Ergotropy is the maximal work extractable from a quantum state by cyclic unitary operations, and it is the standard way to quantify the charge of a quantum battery. This paper proves that for large many-body systems whose Hamiltonian has at most k-body interactions, ergotropy concentrates: almost all states sampled from the Hilbert-Schmidt measure have ergotropy within an exponentially small window of the ensemble average. The average is itself macroscopic, scaling with the Hamiltonian's norm rather than decaying with system size, and the same concentration holds for von Neumann entropy. The proof rests on a Lipschitz-continuity estimate for ergotropy in the Bures distance and Levy's concentration lemma. For the Bures measure, the least-informative prior, no analytic bound is available, but the paper's numerics show the same concentration.

What carries the argument

The load-bearing object is the Lipschitz continuity of ergotropy with respect to the Bures distance, with constant L_E^B ≤ 2||H||_op, proved in Theorem 1 using the Lidskii-Wielandt theorem. Because purification maps Hilbert-Schmidt-sampled states to Haar-uniform points on the sphere $S^{{2d²−1}}$, this continuity combined with Levy's concentration lemma turns Bures-distance control into exponential tail bounds. For entropy, the same route uses the known Lipschitz bound L_S^B ≤ π ln d / ln 2 (equivalently L_E^S ≤ √(8 ln d)/ln 2).

What would settle it

Compute ⟨Ê⟩_HS for nGUE Hamiltonians at dimensions far beyond d ≈ 8200, for example d = $10^{5}$ or larger, using Hilbert-Schmidt state sampling; if the average normalized ergotropy decreases monotonically to zero rather than saturating at a positive constant, the macroscopic-charge conclusion fails, although concentration itself would survive.

Watch

Extended reading notes

Core claim

The paper's central claim is that ergotropy obeys a concentration of measure: for a many-body Hamiltonian with local terms bounded by a constant and at most k-body interactions, and for states drawn from the Hilbert-Schmidt measure, the deviation of ergotropy from its ensemble average decays doubly exponentially. Concretely, Prob(|E(ρ,H) − ⟨E⟩_HS| > ℓ) ≤ 3 $e^{{−ℓ²/Υ_E²}}$ with Υ_E = O(N^k/$e^{{κN}}$) when N ≫ k and ln d ∝ N. The average is not a small residue: numerical evaluation for random nGUE Hamiltonians gives ⟨Ê⟩_HS ≳ 0.23, so ⟨E⟩_HS ∝ ||H||_op, under the macroscopic-energy condition that the infinite-temperature average energy is extensive. The same machinery bounds the concentration of von Neumann entropy, with width Υ_S = O(ln d / d). For the Bures measure, no analytic concentration bound is proven; the paper supplies numerical evidence that both quantities concentrate there as well.

Load-bearing premise

The argument needs the average normalized ergotropy ⟨Ê⟩_HS to stay above some fixed positive value ε for all Hilbert-space dimensions; the paper proves concentration regardless, but without this assumption the concentrated value could be zero, so the battery charge would be trivial.

Editorial extensions

If this is right

Typical states of a large quantum battery store a macroscopic amount of extractable work; no fine-tuning of the initial state is needed for high charge.
The charge level is noise-robust: small perturbations move the state within the typical set, so close-to-average ergotropy is stable.
The first moment of extracted work is predictable, yet fluctuations of work itself are not suppressed: the noise-to-signal ratio saturates near 1.29.
The same concentration applies numerically to the least-informative Bures measure, suggesting typicality extends beyond Hilbert-Schmidt sampling.
For Hamiltonians satisfying the macroscopic-energy condition, typical ergotropy is proportional to ||H||_op, so the typical battery charge grows with the system's energy scale N^k.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the macroscopic-average condition ⟨Ê⟩_HS ≥ ε fails for some Hamiltonian family, concentration still holds but converges to a vanishing charge; separating which Hamiltonians satisfy this condition is a concrete open problem the paper leaves open.
The result suggests a typicality principle for quantum batteries analogous to canonical typicality for entanglement: for mesoscopic systems that randomize enough, unknown states generically carry near-optimal ergotropy, which may simplify protocols for work extraction from unknown quantum sources.
A testable extension would be to compute the concentration width numerically for structured spin-chain Hamiltonians, such as transverse-field Ising or Heisenberg models, rather than GUE/nGUE spectra, to see how the O(N^k/e^{κN}) bound behaves when k is fixed and N grows.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The real result here is Theorem 1 plus the Levy argument: ergotropy is Lipschitz in the Bures, trace, and Hilbert–Schmidt distances, with clean bounds, and therefore concentrates exponentially under the Hilbert–Schmidt measure for k-local Hamiltonians. Appendix A is rigorous and standard—Lidskii–Wielandt, Fuchs–van de Graaf, all accounted for. That is a new and useful contribution, and the paper is honest about what it proves versus what it conjectures, at least in the main text.

What I would push back on is the leap from concentration to macroscopic ergotropy. The claim that _HS >= epsilon for all d is Eq. (39), and the paper itself admits it is unproven. The stress-test example is not exotic: H = sum_i |1><1|_i on N qubits has k=1, satisfies Eq. (38), and yet the normalized spectrum has width O(1/sqrt N), so the HS-averaged passive energy tends to the flat average and _HS -> 0. That kills Eq. (39) for a standard one-body spin Hamiltonian. Footnote 53 tries to exclude this by assuming the number of distinct eigensubspaces is proportional to d, but that assumption is not part of the theorem statement and is not proven for the local Hamiltonians the paper claims to cover. So the concentration theorem stands, but the "most states store macroscopic ergotropy" conclusion does not, unless an explicit non-degeneracy condition is added and Eq. (39) is actually proved.

Also worth noting: the Bures-measure section fits a hypothesized tail law to numerics and then says "we have therefore shown" exponential concentration. The abstract is more careful, but the text overstates. That is a framing issue, not a fatal one. Minor slips: Eq. (27) drops a 1/ln 2 factor, and there are broken citation placeholders after Eq. (A8). No code or data shipped, though the numerics are simple enough to reproduce.

The entropy half is a compilation of known results with the Bures Lipschitz bound adapted from Sekatski et al. Fine, but not new.

Bottom line: this is a serious paper with a clean new theorem and an overreach in the interpretation. A referee should engage with it, but the macroscopic claim needs either a proof under stated assumptions or an explicit spectral-condition hypothesis. I would send it to review, expecting major revision. If the authors fix the scope of the macroscopic claim, the concentration theorem alone justifies publication.

Referee Report

1 major / 7 minor

Summary. The paper proves that ergotropy, as a function of the quantum state, is Lipschitz continuous with respect to trace, Bures, and Hilbert-Schmidt distances (Theorem 1, Appendix A). Using the purification correspondence between Hilbert-Schmidt-random states and Haar-random pure states on a larger space, the authors apply Levy's lemma to show that for large Hilbert-space dimension d the ergotropy of a state drawn from the Hilbert-Schmidt measure is exponentially concentrated around its ensemble average (Eqs. (24), (25), (29), (30)). For k-body Hamiltonians with ln d proportional to the particle number N, the absolute width of the ergotropy distribution is O(N^k/e^{kappa N}), so the concentration is exponentially strong in N (Eq. (35)). The paper further argues that the average ergotropy is macroscopic, proportional to ||H||_op (Eq. (42)), and provides numerical evidence for analogous concentration when states are sampled from the Bures measure (Section VII).

Significance. If the macroscopic-average claim were established under the stated assumptions, the paper would be a significant contribution to quantum thermodynamics: it would show that typical highly mixed states of many-body batteries contain a macroscopic amount of unitarily extractable work and that this amount is stable under small perturbations. The technical core of the paper, Theorem 1 and the concentration argument in Section IV, is sound and clearly presented; the Lipschitz proof via Lidskii-Wielandt and Fuchs-van de Graaf is clean. The numerical Bures-measure section is honestly labeled as evidence rather than proof. The main weakness is that the macroscopic-ergotropy conclusion relies on an unproved and, as stated, false lower bound on the average normalized ergotropy, so the advertised claim is broader than what is actually demonstrated.

major comments (1)

[Sec. V, Eqs. (37)-(42), footnote 53] The claim that <E_hat>_HS >= epsilon for all d (Eq. (39)) is load-bearing for the macroscopic-ergotropy conclusion (Eq. (42)) and for the discussion in Section VIII, but it is not proved and it is false for Hamiltonians satisfying the stated assumptions of Section I. Consider H = sum_{i=1}^N |1><1|_i on N qubits, with ||H||_op = N, k=1, and ln d = N ln 2. This Hamiltonian satisfies Eq. (38) with h = 1/2. Its normalized spectrum has eigenvalues m/N with binomial degeneracies, i.e., a Gaussian density of states of width O(N^{-1/2}) centered at 1/2. For a Hilbert-Schmidt-random state rho, the passive energy Tr(rho_down H_hat) tends to the average energy Tr(H_hat)/d = 1/2, so <E_hat>_HS tends to zero as N grows and Eq. (39) fails. Footnote 53 excludes this case by assuming that the number of distinct eigensubspaces is proportional to d, but that assumption is not part of the hypotheses in Section I or of Theorem 1, and no proof of Eq. (39) is given under it. The numerical support uses nGUE Hamiltonians, whose normalized spectrum has O(1) width, and thus does not probe the O(1/sqrt N) spectral width typical of local Hamiltonians. The concentration theorem in Eqs. (24) and (35) survives, but the macroscopic-ergotropy result must either be restricted to Hamiltonians with an explicit spectral non-degeneracy condition or be rephrased as a numerical observation for the ensembles tested.

minor comments (7)

[Sec. IV, Eq. (27)] Equation (27) drops the 1/ln 2 factor from Eq. (19): with L_E^S <= sqrt(8 ln d)/ln 2, one obtains Upsilon_S <= sqrt(100 pi ln d)/(ln 2 d), not sqrt(100 pi ln d)/d. This is a constant-factor error and does not change the qualitative concentration statement, but the displayed inequality as written is too strong.
[Sec. V, footnote 53] The assumption that the number of distinct eigensubspaces is proportional to d is stated only in a footnote; since it is needed for the macroscopic-average argument, it should appear in the main text as an explicit hypothesis in Section I and in the theorem statements.
[Appendix A, after Eq. (A8)] The reference for the lower bound in Eq. (A8) contains unresolved placeholder citations "[? ? ]"; these references need to be completed.
[Sec. V, paragraph on Ginibre matrices] There is a typo: "independent, identically identically distributed" should read "independent and identically distributed."
[Appendix C] The word "noniformativeness" should be "noninformativeness."
[Sec. VI, Eq. (44)] The statement that Var_{rho,H}(W) concentrates for Hilbert-Schmidt-random rho is asserted without proof or derivation. Since this is a side remark rather than a central claim, it should be either proved briefly or explicitly labeled as a conjecture.
[Sec. V, Eq. (35)] The notation in Eq. (35) appears to refer to the absolute width Upsilon_E, but it is written as a ratio Upsilon_E/E; please clarify whether the displayed asymptotic is for the absolute width or for the relative fluctuation, since the two differ by a factor ||H||_op when Eq. (42) holds.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the concentration theorem is derived from independent Lipschitz and Levy bounds; the macroscopic-average step rests on an explicitly labeled unproven hypothesis, which is a correctness risk rather than circular reasoning.

full rationale

The paper's central derivation is self-contained and does not reduce to its own inputs. Theorem 1 (Lipschitz continuity of ergotropy) is proven in Appendix A using the Lidskii–Wielandt theorem, the triangle inequality, and standard norm relations; these are independent external results. The subsequent concentration bound applies Levy's lemma from Watrous to the purification map, with the Fubini–Study/Hilbert–Schmidt correspondence taken from Zyczkowski–Sommers; neither step assumes the target concentration result. The entropy concentration results are explicitly compiled from known external references (Sekatski et al., Hayden–Leung–Winter), not presented as new derivations. The macroscopic-ergotropy conclusion does rely on Eq. (39), the existence of a uniform epsilon lower bound on the averaged normalized ergotropy, which the paper explicitly labels as a hypothesis supported only by 'good reasons' and numerics. This is an unproven and potentially false assumption for some local Hamiltonians (e.g., the number operator), but it is not circular: Eq. (39) is not derived from, nor equivalent to, the concentration theorem, and the paper does not rename a fitted parameter as a prediction. The Bures-measure section fits a hypothesized tail law to numerical data; that is empirical fitting, and the paper openly states that no analytic exponential-concentration bound is available there. Self-citations appear only in background and discussion contexts (e.g., work-fluctuation schemes) and are not load-bearing for the main theorems. Overall, no step exhibits self-definition, fitted-input-as-prediction, or a self-citation chain that forces the result.

Assumptions & free parameters 5 free parameters · 10 assumptions · 1 invented entities

The analytic concentration results rest on standard matrix-analysis and measure-concentration theorems plus explicitly stated many-body Hamiltonian assumptions. The macroscopic-average claim depends on unproven Eq. (39) supported by numerics, and the Bures claim depends on a fitted ansatz. No invented physical entities appear.

free parameters (5)

Bures concentration exponent x_e = about 1.24 (Fig. 2b)
Fitted slope of ln(-ln P[l]) versus ln l for ergotropy under the Bures measure in the ansatz Eq. (50).
Bures concentration exponent y_e = about 1.01 (Fig. 2a)
Fitted slope of ln(-ln P[l]) versus ln d for ergotropy under the Bures measure in Eq. (50).
Bures concentration exponent x_s = about 1.52 (Fig. 2b)
Fitted slope for the normalized entropy S/ln d under the Bures measure in Eq. (51).
Bures concentration exponent y_s = about 1.53 (Fig. 2a)
Fitted slope versus ln d for the normalized entropy under the Bures measure in Eq. (51).
Bures tail constants xi_e, theta_e, xi_s, theta_s = not reported
Constant prefactors in the hypothesized concentration laws Eqs. (50)-(51); estimated implicitly by assuming xi near 1 in Eq. (52).

assumptions (10)

standard math Lidskii-Wielandt theorem for eigenvalue differences under trace norm, Eq. (A12).
Used in Appendix A to bound the passive-state contribution to ergotropy differences.
standard math Levy's measure concentration lemma on the sphere, Eq. (20), with alpha = 1/(25 pi).
The engine of the concentration bounds; the constant is taken from Ref. [33].
standard math Uhlmann fidelity and purification bound, Eq. (11).
Connects Bures distance to Euclidean distance between purifications, used for the sphere map.
standard math Von Neumann entropy Lipschitz bounds from Refs. [41,50], Eqs. (17)-(19).
Compiled and adapted for the entropy concentration statements.
domain assumption Hamiltonian has at most k-body interactions with bounded local terms, ||h_alpha||_op <= c, so ||H||_op = O((ln d)^k).
Defines the many-body class; stated in Sec. I and used for Eq. (35).
domain assumption Hilbert-Schmidt sampling of rho corresponds to Haar sampling of purifications on S^{2d^2-1}.
Invoked in Sec. IV, citing Ref. [52]; needed to apply Levy's lemma to density matrices.
domain assumption Bures-distributed states are generated by Eq. (46) from Ginibre plus Haar unitary.
Numerical sampling recipe for the Bures measure, cited from Ref. [74].
ad hoc to paper Hamiltonian lacks extreme degeneracies: the number of distinct eigensubspaces is proportional to d.
Footnote [53] implicitly assumes this so that coherences can contribute positive ergotropy; not stated among the main assumptions.
ad hoc to paper Macroscopic energy at infinite temperature, Eq. (38): (1/d) Tr H_hat >= h > 0 for all d.
Necessary condition for nonzero macroscopic average ergotropy; assumed without proof for the many-body conclusion.
ad hoc to paper Numerical ansatz Eq. (50)-(51) for Bures-measure concentration: P[l] <= xi exp(-theta l^x d^y).
Used to claim exponential concentration for the Bures measure; validated only by fitting Fig. 2.

invented entities (1)

None
purpose: No new physical entity is introduced.
NSR is a derived diagnostic quantity, not a postulated entity; no new particles, forces, dimensions, or conserved quantities appear.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Concentration of ergotropy in many-body systems." pith.science (2026). https://pith.science/paper/TQUXEB54

@misc{pith2026241219801,
  author       = {Pith},
  title        = {Pith review of: Concentration of ergotropy in many-body systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQUXEB54}},
  note         = {Machine review of arXiv:2412.19801}
}

read the original abstract

Ergotropy -- the maximal amount of unitarily extractable work -- measures the ``charge level'' of quantum batteries. We prove that in large many-body batteries ergotropy exhibits a concentration of measure phenomenon. Namely, the ergotropy of such systems is almost constant for almost all states sampled from the Hilbert--Schmidt measure. We establish this by first proving that ergotropy, as a function of the state, is Lipschitz-continuous with respect to the Bures distance, and then applying Levy's measure concentration lemma. In parallel, we showcase the analogous properties of von Neumann entropy, compiling and adapting known results about its continuity and concentration properties. Furthermore, we consider the situation with the least amount of prior information about the state. This corresponds to the quantum version of the Jeffreys prior distribution -- the Bures measure. In this case, there exist no analytical bounds guaranteeing exponential concentration of measure. Nonetheless, we provide numerical evidence that ergotropy, as well as von Neumann entropy, concentrate also in this case.

Figures

Figures reproduced from arXiv: 2412.19801 by the authors.

**Figure 2.** The concentration properties of ergotropy and en [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

90 extracted references · 57 canonical work pages

[1]

Lett67, 565 (2004)

A.E.Allahverdyan, R.Balian,andT.M.Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhys. Lett67, 565 (2004)

2004
[2]

Pusz and S

W. Pusz and S. L. Woronowicz, Passive states and KMS states for general quantum systems, Commun. Math. Phys. 58, 273 (1978)

1978
[3]

Lenard, Thermodynamical proof of the Gibbs formula for elementary quantum systems, J

A. Lenard, Thermodynamical proof of the Gibbs formula for elementary quantum systems, J. Stat. Phys.19, 575 (1978)

1978
[4]

Alicki and M

R. Alicki and M. Fannes, Entanglement boost for ex- tractable work from ensembles of quantum batteries, Phys. Rev. E87, 042123 (2013)

2013
[5]

Sparaciari, D

C. Sparaciari, D. Jennings, and J. Oppenheim, Ener- getic instability of passive states in thermodynamics, Nat. Commun.8, 1895 (2017)

2017
[6]

Lett.103, 60005 (2013)

D.Gelbwaser-Klimovsky, R.Alicki,andG.Kurizki,Work and energy gain of heat-pumped quantized amplifiers, Europhys. Lett.103, 60005 (2013)

2013
[7]

Acín, Entanglement generation is not necessary for optimal work extraction, Phys

K.V.Hovhannisyan, M.Perarnau-Llobet, M.Huber,and A. Acín, Entanglement generation is not necessary for optimal work extraction, Phys. Rev. Lett.111, 240401 (2013)

2013
[8]

Perarnau-Llobet, K

M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Acín, Extractable work from correlations, Phys. Rev. X5, 041011 (2015)

2015

Show all 90 references

[9]

F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, Quantacell: powerfulchargingofquantumbatteries,New J. Phys.17, 075015 (2015)

2015
[10]

Campaioli, F

F. Campaioli, F. A. Pollock, F. C. Binder, L. Céleri, J.Goold, S.Vinjanampathy,andK.Modi,Enhancingthe charging power of quantum batteries, Phys. Rev. Lett. 118, 150601 (2017)

2017
[11]

Ferraro, M

D. Ferraro, M. Campisi, G. M. Andolina, V. Pellegrini, and M. Polini, High-power collective charging of a solid- state quantum battery, Phys. Rev. Lett. 120, 117702 (2018)

2018
[12]

Barra, Dissipative charging of a quantum battery, Phys

F. Barra, Dissipative charging of a quantum battery, Phys. Rev. Lett.122, 210601 (2019)

2019
[13]

Farina, G

D. Farina, G. M. Andolina, A. Mari, M. Polini, and V. Giovannetti, Charger-mediated energy transfer for quantum batteries: An open-system approach, Phys. Rev. B99, 035421 (2019)

2019
[14]

G. M. Andolina, M. Keck, A. Mari, M. Campisi, V. Gio- vannetti, and M. Polini, Extractable work, the role of correlations, and asymptotic freedom in quantum bat- teries, Phys. Rev. Lett.122, 047702 (2019)

2019
[15]

Niedenzu, M

W. Niedenzu, M. Huber, and E. Boukobza, Concepts of work in autonomous quantum heat engines, Quantum3, 195 (2019)

2019
[16]

S.Gherardini, F.Campaioli, F.Caruso,andF.C.Binder, Stabilizing open quantum batteries by sequential mea- surements, Phys. Rev. Res.2, 013095 (2020)

2020
[17]

K. V. Hovhannisyan, F. Barra, and A. Imparato, Charg- ing assisted by thermalization, Phys. Rev. Res.2, 033413 (2020)

2020
[18]

Çakmak, Ergotropy from coherences in an open quan- tum system, Phys

B. Çakmak, Ergotropy from coherences in an open quan- tum system, Phys. Rev. E102, 042111 (2020)

2020
[19]

S. Seah, M. Perarnau-Llobet, G. Haack, N. Brunner, and 11 S. Nimmrichter, Quantum speed-up in collisional battery charging, Phys. Rev. Lett.127, 100601 (2021)

2021
[20]

Tirone, R

S. Tirone, R. Salvia, and V. Giovannetti, Quantum en- ergy lines and the optimal output ergotropy problem, Phys. Rev. Lett.127, 210601 (2021)

2021
[21]

Barra, K

F. Barra, K. V. Hovhannisyan, and A. Imparato, Quan- tum batteries at the verge of a phase transition, New J. Phys. 24, 015003 (2022)

2022
[22]

J.-Y. Gyhm, D. Šafránek, and D. Rosa, Quantum charg- ing advantage cannot be extensive without global opera- tions, Phys. Rev. Lett.128, 140501 (2022)

2022
[23]

Tirone, R

S. Tirone, R. Salvia, S. Chessa, and V. Giovannetti, Quantum work capacitances (2022), arXiv:2211.02685 [quant-ph]

2022 arXiv
[24]

Łobejko, P

M. Łobejko, P. Mazurek, and M. Horodecki, The asymp- totic emergence of the second law for a repeated charging process (2022), arXiv:2209.05339 [quant-ph]

2022 arXiv
[25]

Yang, Y.-H

X. Yang, Y.-H. Yang, M. Alimuddin, R. Salvia, S.-M. Fei, L.-M. Zhao, S. Nimmrichter, and M.-X. Luo, Battery capacity of energy-storing quantum systems, Phys. Rev. Lett. 131, 030402 (2023)

2023
[26]

Campaioli, S

F. Campaioli, S. Gherardini, J. Q. Quach, M. Polini, and G. M. Andolina, Colloquium: Quantum batteries (2023), arXiv:2308.02277 [quant-ph]

2023 arXiv
[27]

Song, H.-B

W.-L. Song, H.-B. Liu, B. Zhou, W.-L. Yang, and J.-H. An,Remotecharginganddegradationsuppressionforthe quantum battery, Phys. Rev. Lett.132, 090401 (2024)

2024
[28]

Feliú and F

D. Feliú and F. Barra, System-bath correlations and finite-time operation enhance the efficiency of a dissipa- tive quantum battery (2024), arXiv:2403.08573 [quant- ph]

2024 arXiv
[29]

R. P. A. Simon, J. Anders, and K. V. Hovhannisyan, Correlations enable lossless ergotropy transport (2024), arXiv:2406.10468 [quant-ph]

2024 arXiv
[30]

24, 1 (1996)

M.Talagrand,Anewlookatindependence,Ann.Probab. 24, 1 (1996)

1996
[31]

V. D. Milman and G. Schechtman,Asymptotic Theory of Finite Dimensional Normed Spaces , Lecture Notes in Mathematics, Vol. 1200 (Springer, Berlin, 2001)

2001
[32]

Bengtsson and K

I. Bengtsson and K. Życzkowski, Geometry of Quan- tum States: An Introduction to Quantum Entanglement (Cambridge University Press, New York, 2006)

2006
[33]

Watrous,The Theory of Quantum Information (Cam- bridge University Press, Cambridge, 2018)

J. Watrous,The Theory of Quantum Information (Cam- bridge University Press, Cambridge, 2018)

2018
[34]

E. T. Jaynes, Prior probabilities, IEEE Trans. Syst. Sci. Cybern. 4, 227 (1968)

1968
[35]

Amari and H

S.-i. Amari and H. Nagaoka,Methods of Information Ge- ometry (Oxford University Press, New York, 2000)

2000
[36]

J. I. Cirac and F. Verstraete, Renormalization and tensor product states in spin chains and lattices, J. Phys. A42, 504004 (2009)

2009
[37]

P. B. Slater, Quantum Fisher–Bures information of two- level systems and a three-level extension, J. Phys. A29, L271 (1996)

1996
[38]

M.J.W.Hall,Randomquantumcorrelationsanddensity operator distributions, Phys. Lett. A242, 123 (1998)

1998
[39]

P. B. Slater, Comparative noninformativities of quantum priors based on monotone metrics, Phys. Lett. A247, 1 (1998)

1998
[40]

Bures, An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinitew∗- algebras, Trans

D. Bures, An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinitew∗- algebras, Trans. Am. Math. Soc.135, 199 (1969)

1969
[41]

Sekatski, J.-D

P. Sekatski, J.-D. Bancal, X. Valcarce, E. Y.-Z. Tan, R. Renner, and N. Sangouard, Device-independent quan- tumkeydistributionfromgeneralizedCHSHinequalities, Quantum 5, 444 (2021)

2021
[42]

P. B. Slater, Silver mean conjectures for 15-dimensional volumes and 14-dimensional hyperareas of the separable two-qubit systems, J. Geom. Phys.53, 74 (2005)

2005
[43]

V. B. Lidskii, On the proper values of a sum and product of symmetric matrices, Dokl. Akad. Nauk SSSR75, 769 (1950)

1950
[44]

Mirsky, Symmetric gauge functions and unitarily in- variant norms, Quart

L. Mirsky, Symmetric gauge functions and unitarily in- variant norms, Quart. J. Math11, 43 (1960)

1960
[45]

Bhatia, Matrix analysis (Springer, New York, 1997)

R. Bhatia, Matrix analysis (Springer, New York, 1997)

1997
[46]

transition probability

A. Uhlmann, The “transition probability” in the state space of a∗-algebra, Rep. Math. Phys.9, 273 (1976)

1976
[47]

Fannes, A continuity property of the entropy density for spin lattice systems, Commun

M. Fannes, A continuity property of the entropy density for spin lattice systems, Commun. Math. Phys.31, 291 (1973)

1973
[48]

K. M. R. Audenaert, A sharp continuity estimate for the von Neumann entropy, J. Phys. A40, 8127 (2007)

2007
[49]

Petz, Quantum Information Theory and Quantum Statistics (Springer, Berlin, 2008)

D. Petz, Quantum Information Theory and Quantum Statistics (Springer, Berlin, 2008)

2008
[50]

Hayden, D

P. Hayden, D. W. Leung, and A. Winter, Aspects of generic entanglement, Commun. Math. Phys. 265, 95 (2006)

2006
[51]

(20) holds for someα, then it automat- ically holds for∀α′ ≤ α

Note that if Eq. (20) holds for someα, then it automat- ically holds for∀α′ ≤ α
[52]

Życzkowski and H.-J

K. Życzkowski and H.-J. Sommers, Induced measures in the space of mixed quantum states, J. Phys. A.34, 7111 (2001)

2001
[53]

We implicitly assume that the Hamiltonian does not have extreme degeneracies, in the sense that the number of distinct eigensubspaces is∝ d
[54]

Mezzadri, How to generate random matrices from the classical compact groups, Not

F. Mezzadri, How to generate random matrices from the classical compact groups, Not. Am. Math. Soc.54, 592 (2007)

2007
[55]

Edelman and N

A. Edelman and N. R. Rao, Random matrix theory, Acta Numerica 14, 233 (2005)

2005
[56]

There is no preferred basis in the problem, so this is a desirable property
[57]

Ledoux,The Concentration of Measure Phenomenon , Mathematical surveys and monographs, Vol

M. Ledoux,The Concentration of Measure Phenomenon , Mathematical surveys and monographs, Vol. 89 (Ameri- can Mathematical Society, Providence, 2001)

2001
[58]

Kurchan, A quantum fluctuation theorem (2000), arXiv:cond-mat/0007360

J. Kurchan, A quantum fluctuation theorem (2000), arXiv:cond-mat/0007360

2000 arXiv
[59]

Tasaki, Jarzynski relations for quantum systems and some applications (2000), arXiv:cond-mat/0009244

H. Tasaki, Jarzynski relations for quantum systems and some applications (2000), arXiv:cond-mat/0009244

2000 arXiv
[60]

A. E. Allahverdyan and T. M. Nieuwenhuizen, Fluctu- ations of work from quantum subensembles: The case against quantum work-fluctuation theorems, Phys. Rev. E 71, 066102 (2005)

2005
[61]

Talkner and P

P. Talkner and P. Hänggi, Aspects of quantum work, Phys. Rev. E93, 022131 (2016)

2016
[62]

Perarnau-Llobet, E

M. Perarnau-Llobet, E. Bäumer, K. V. Hovhannisyan, M.Huber,andA.Acín,No-gotheoremforthecharacteri- zation of work fluctuations in coherent quantum systems, Phys. Rev. Lett.118, 070601 (2017)

2017
[63]

K. V. Hovhannisyan and A. Imparato, Energy conserva- tion and fluctuation theorem are incompatible for quan- tum work, Quantum8, 1336 (2024)

2024
[64]

Bäumer, M

E. Bäumer, M. Lostaglio, M. Perarnau-Llobet, and R. Sampaio, Fluctuating work in coherent quantum sys- tems: Proposals and limitations, inThermodynamics in the Quantum Regime: Fundamental Aspects and New Di- rections, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, a...

2018
[65]

Pei, J.-F

J.-H. Pei, J.-F. Chen, and H. T. Quan, Explor- ing quasiprobability approaches to quantum work in the presence of initial coherence: Advantages of the Margenau–Hill distribution, Phys. Rev. E108, 054109 (2023)

2023
[66]

Lostaglio, A

M. Lostaglio, A. Belenchia, A. Levy, S. Hernández- Gómez, N. Fabbri, and S. Gherardini, Kirkwood–Dirac quasiprobability approach to the statistics of incompati- ble observables, Quantum7, 1128 (2023)

2023
[67]

A. E. Allahverdyan, Nonequilibrium quantum fluctua- tions of work, Phys. Rev. E90, 032137 (2014)

2014
[68]

Petz and C

D. Petz and C. Sudár, Geometries of quantum states, J. Math. Phys.37, 2662 (1996)

1996
[69]

Sommers and K

H.-J. Sommers and K. Życzkowski, Bures volume of the setofmixedquantumstates,J.Phys.A 36,10083(2003)

2003
[70]

Dittmann, On the Riemannian metric on the space of density matrices, Rep

J. Dittmann, On the Riemannian metric on the space of density matrices, Rep. Math. Phys.36, 309 (1995)

1995
[71]

Dittmann, The scalar curvature of the Bures metric on the space of density matrices, J

J. Dittmann, The scalar curvature of the Bures metric on the space of density matrices, J. Geom. Phys.31, 16 (1999)

1999
[72]

P. B. Slater, A priori probability that two qubits are un- entangled, Quantum Inf. Process.1, 397 (2002)

2002
[73]

P. B. Slater, A priori probability that a qubit–qutrit pair is separable, J. Optics B5, S651 (2003)

2003
[74]

V. A. Osipov, H.-J. Sommers, and K. Życzkowski, Ran- dom Bures mixed states and the distribution of their pu- rity, J. Phys. A43, 055302 (2010)

2010
[75]

Gemmer, M

J. Gemmer, M. Michel, and G. Mahler, Quantum Thermodynamics, Vol. 657 (Lecture Notes in Physics, Springer, Berlin, 2004)

2004
[76]

Gogolin and J

C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems, Rep. Prog. Phys.79, 056001 (2016)

2016
[77]

T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, Ther- malization and prethermalization in isolated quantum systems: a theoretical overview, J. Phys. B51, 112001 (2018)

2018
[78]

K. V. Hovhannisyan, S. Nemati, C. Henkel, and J. An- ders, Long-time equilibration can determine transient thermality, PRX Quantum4, 030321 (2023)

2023
[79]

Bertoni, C

C. Bertoni, C. Wassner, G. Guarnieri, and J. Eisert, Typ- ical thermalization of low-entanglement states (2024), arXiv:2403.18007 [quant-ph]

2024 arXiv
[80]

Šafránek, D

D. Šafránek, D. Rosa, and F. C. Binder, Work extraction from unknown quantum sources, Phys. Rev. Lett.130, 210401 (2023)

2023
[81]

Watanabe and R

K. Watanabe and R. Takagi, Black box work ex- traction and composite hypothesis testing (2024), arXiv:2407.03400 [quant-ph]

2024 arXiv
[82]

C. A. Fuchs and J. van de Graaf, Cryptographic dis- tinguishability measures for quantum-mechanical states, IEEE Trans. Inf. Theory45, 1216 (1999)

1999
[83]

(A8) was first proven in Ref

The lower bound in Eq. (A8) was first proven in Ref. [40]. See also [? ? ]
[84]

P. J. Coles, M. Cerezo, and L. Cincio, Strong bound be- tween trace distance and Hilbert–Schmidt distance for low-rank states, Phys. Rev. A100, 022103 (2019)

2019
[85]

Winter, Tight uniform continuity bounds for quantum entropies: Conditional entropy, relative entropy distance and energy constraints, Commun

A. Winter, Tight uniform continuity bounds for quantum entropies: Conditional entropy, relative entropy distance and energy constraints, Commun. Math. Phys.347, 291 (2016)

2016
[86]

W. K. Wootters, Statistical distance and Hilbert space, Phys. Rev. D23, 357 (1981)

1981
[87]

S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994)

1994
[88]

Petz, Monotone metrics on matrix spaces, Linear Al- gebra Appl.244, 81 (1996)

D. Petz, Monotone metrics on matrix spaces, Linear Al- gebra Appl.244, 81 (1996)

1996
[89]

Pérez-García, M

D. Pérez-García, M. M. Wolf, D. Petz, and M. B. Ruskai, Contractivity of positive and trace-preserving maps un- der lp norms, J. Math. Phys.47, 083506 (2006)

2006
[90]

Ozawa, Entanglement measures and the Hilbert– Schmidt distance, Phys

M. Ozawa, Entanglement measures and the Hilbert– Schmidt distance, Phys. Lett. A268, 158 (2000)

2000

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Lett67, 565 (2004)

A.E.Allahverdyan, R.Balian,andT.M.Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhys. Lett67, 565 (2004)

2004

[2] [2]

Pusz and S

W. Pusz and S. L. Woronowicz, Passive states and KMS states for general quantum systems, Commun. Math. Phys. 58, 273 (1978)

1978

[3] [3]

Lenard, Thermodynamical proof of the Gibbs formula for elementary quantum systems, J

A. Lenard, Thermodynamical proof of the Gibbs formula for elementary quantum systems, J. Stat. Phys.19, 575 (1978)

1978

[4] [4]

Alicki and M

R. Alicki and M. Fannes, Entanglement boost for ex- tractable work from ensembles of quantum batteries, Phys. Rev. E87, 042123 (2013)

2013

[5] [5]

Sparaciari, D

C. Sparaciari, D. Jennings, and J. Oppenheim, Ener- getic instability of passive states in thermodynamics, Nat. Commun.8, 1895 (2017)

2017

[6] [6]

Lett.103, 60005 (2013)

D.Gelbwaser-Klimovsky, R.Alicki,andG.Kurizki,Work and energy gain of heat-pumped quantized amplifiers, Europhys. Lett.103, 60005 (2013)

2013

[7] [7]

Acín, Entanglement generation is not necessary for optimal work extraction, Phys

K.V.Hovhannisyan, M.Perarnau-Llobet, M.Huber,and A. Acín, Entanglement generation is not necessary for optimal work extraction, Phys. Rev. Lett.111, 240401 (2013)

2013

[8] [8]

Perarnau-Llobet, K

M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Acín, Extractable work from correlations, Phys. Rev. X5, 041011 (2015)

2015