Pith. sign in

REVIEW 4 minor 59 references

For autonomous quantum energy pumps whose terminals are ideal translation clocks, every ordered moment of the terminal-work observables is exactly represented by pump-only operators, making the phase-derivative current a genuine observable

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-04 01:30 UTC pith:DIPAV4YB

load-bearing objection A carefully scoped, mathematically explicit framework for terminal-resolved work statistics in quantum pumps; the ideal-clock limitation is real but honestly quantified, and the paper deserves serious peer review.

arxiv 2608.00638 v1 pith:DIPAV4YB submitted 2026-08-01 quant-ph cond-mat.stat-mech

Work Statistics of Autonomous Quantum Energy Pumps

Yang Peng This is my paper
classification quant-ph cond-mat.stat-mech
keywords quantum energy pumpterminal-resolved workideal quantum clocksphase-derivative currentFloquet quantum metricnoise matchingwork-variance gap
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

An autonomous quantum energy pump is a central pump coupled to terminals that supply and receive energy. The paper's goal is to define what 'work done by a terminal' means at the level of full quantum statistics, not just mean current. It argues that when each terminal is an ideal translation clock, the exact terminal-work observables on the full Hilbert space reduce blockwise to operators on the pump alone, and that every ordered moment of terminal work is reproduced by this pump-space representation. This reduction gives the conventional phase-derivative current a microscopic justification, separates transported work from energy accumulated in the pump, and ties finite-cycle fluctuations to the Floquet quantum metric. A reader should care because it turns work statistics of autonomous energy converters from a bookkeeping convention into an observable with computable moments and bounds.

Core claim

The central claim is Theorem 2: for ideal translation-clock terminals with the pump–clock coupling diagonal in clock coordinates, the full terminal-work observable Ŵ_i(t) is diagonal in the initial clock-coordinate basis, and its pump-space block is i ω_i Û_s†(t) ∂_{s_i} Û_s(t). Equation (16) then states that, for any initial pump–clock state and any ordered index string, the expectation of the product of terminal works equals the trace over the pump of the coordinate-diagonal density with the product of pump-space blocks. This makes the phase-derivative current the exact continuity current of an explicitly modeled terminal, and it extends to repeated cycles, where the mean work in a Floquet

What carries the argument

The carrying mechanism is the coordinate-block decomposability of the autonomous propagator: with free clocks Ĥ_i = -iω_i ∂_{s_i} and a coupling diagonal in |s⟩⟨s|, the propagator becomes Û(t)=∫d^D s Û_s(t)⊗|s+ωt⟩⟨s|, so every terminal-work observable is block diagonal in the initial clock coordinates. The block Ŵ_i^{(s)}(t)=iω_i Û_s†(t)∂_{s_i}Û_s(t) is the central object; it simultaneously acts as the exact work operator, the generator of ordered mixed moments, and the integrand of the phase-derivative current.

Load-bearing premise

The exact reduction rests on each terminal being an ideal translation clock whose Hamiltonian is -iω_i ∂_{s_i} and whose coupling to the pump is diagonal in the clock-coordinate basis; the paper's own Section VII shows that physical terminals only approximate this, producing the work-variance gap.

What would settle it

Measure the full terminal-work distribution for a physical terminal that is not an ideal clock (for instance, a finite-ladder battery or a coherent cavity at low occupation) and compare the exact second moment of Ŵ_q(t) with the pump-only prediction; if the positive work-variance gap Ĝ_q(t) develops negative eigenvalues, or if the variance decomposition Eq. (101) fails to reproduce the exactly computed variance, the construction is falsified. A clean numerical check at finite occupation would settle whether the gap closes exactly in the large-occupation limit.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The phase-derivative current used throughout driven-pump theory is not a convention: it is the expectation of an exact, Hermitian terminal-work observable on the pump space for ideal-clock terminals.
  • For commensurate drives, a Floquet eigenstate transports energy with a mean that grows linearly in the number of cycles while each directional variance remains bounded by four times the Floquet quantum metric; fluctuations do not accumulate.
  • At long times the rescaled terminal-work operators converge in operator norm to a commuting family of asymptotic currents, so the pump becomes increasingly catalytic and the terminal currents acquire a common spectral description.
  • In the exactly solvable two-clock qubit, balanced couplings make transport work perfectly sharp at every cycle even though individual terminal energies and receiver gain remain noisy — a noise-matching regime distinct from deterministic delivery.
  • For physical terminals, the work-variance gap Ĝ_q(t) is positive and measures what a pump-only description misses; the coherent-cavity benchmark shows the mean transport current can be within 0.25% of the ideal-clock value while 7.3% of the transport variance is still missing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same operator-inequality argument that produces the variance gap should yield higher-moment gaps, so the framework suggests a practical hierarchy test: a pump-only model that reproduces the variance also reproduces all polynomial moments only when the gap vanishes.
  • The correlation-assisted gain, which is already classical, invites a resource viewpoint: for a fixed phase distribution, the gap between product and correlated optima measures how much directional energy transfer is unlocked by classical pump–clock correlations, and one could ask whether a mutual-information-like bound governs this gain.
  • The noise-matching condition suggests a design rule for transducers: if the source and receiver couplings are symmetric, common-mode terminal fluctuations decouple from the transport channel, potentially sharpening transferred energy without requiring quiet terminals — a testable engineering target in superconducting or photonic implementations.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper develops a terminal-resolved theory of work statistics for autonomous quantum energy pumps, in which the energy source and sink are modeled as explicit quantum terminals and the work supplied by each terminal is defined as the decrease of that terminal's own Hamiltonian. For ideal translation clocks with coordinate-diagonal coupling to the pump, the central Theorem 2 gives an exact pump-space representation of all ordered moments of the terminal-work observables, recovering the conventional phase-derivative currents of driven pumping. The framework handles arbitrary initial pump-terminal correlations and shows that classical correlations suffice for the identified correlation-assisted pumping advantage. For commensurate periodic drives, the paper derives exact finite-cycle Floquet work statistics: the mean grows linearly with the number of cycles, the variance is bounded by the Floquet quantum metric, and the long-time cycle-averaged currents become mutually commuting. An exactly solvable two-clock qubit model illustrates a noise-matching effect in which balanced couplings make transport sharp through cancellation of common-mode terminal fluctuations. For physical terminals beyond the ideal-clock limit, the paper introduces a positive work-variance gap and, in a coherent-cavity benchmark, demonstrates convergence toward the ideal-clock regime with increasing occupation, while emphasizing that agreement of the mean current alone does not guarantee accurate work stati

Significance. If the results hold, this is a significant contribution to the theory of autonomous quantum energy conversion. The exact moment representation in Eq. (16) provides a microscopic, observable-level foundation for the phase-derivative currents used throughout driven quantum pumping, including their fluctuations and cross-correlations. The Floquet variance bound in Eq. (64) connects finite-cycle work fluctuations to quantum geometry in a falsifiable way, and the noise-matching effect in Sec. VI is a clean example of fluctuations being controlled independently of the mean current. The paper is also strong methodologically: the central derivations are explicit, with proofs in Appendices A-D; the exactly solvable two-clock qubit provides a nontrivial check of the general formulas; and the coherent-cavity benchmark uses a specified truncation with reported convergence, including omitted Poisson weight and fitted power-law decays. No parameters are fitted to data; the main formulas are derived analytically from the autonomous Hamiltonian. The ideal-clock limitation is not a hidden assumption: it is stated in Eq. (2), and Sec. VII quantifies its breakdown for physical terminals through a pos

minor comments (4)
  1. [Sec. IV and Sec. VII] The symbol \hat W_i(t) denotes the full-space observable in Secs. II-III and again in Sec. VII, but in Secs. IV-VI it denotes the pump-space block after localization. The switch is explicit when it happens, but a one-sentence reminder at the start of each section would help the reader avoid confusion.
  2. [Ref. [26]] The DOI for Ref. [26], '10.1103/fnv6-yqmk', appears malformed or placeholder-like. Please verify and replace it with the correct DOI, or use the arXiv identifier if the DOI is not yet assigned.
  3. [Eq. (110)] The truncation expression is written as ⌈n̄_b + 10√n̄_b+1 + 30⌉; adding parentheses, √(n̄_b+1), would remove ambiguity about which term is under the square root.
  4. [Sec. VII / Eq. (106)] The quantity ζ_q is introduced with the caveat that it is understood only when the exact variance is nonzero. This caveat is stated once near Eq. (102), but it would be helpful to repeat it where ζ_tr is defined in Eq. (106), since the denominator there could in principle vanish.

Circularity Check

0 steps flagged

No load-bearing circularity: the pump-space work reduction is derived from the autonomous Hamiltonian; self-citations are contextual and not inputs to the main theorems.

full rationale

The central result, Theorem 2 / Eqs. (13)-(16), is derived rather than assumed: the terminal-work observable is defined in Eq. (9) as the negative change of the terminal Hamiltonian, and Appendix B obtains the pump-space block W_i^(s)(t)=iω_i U_s^†(t)∂_{s_i}U_s(t) from the propagator identity Eq. (4) and the coordinate-diagonal coupling Eq. (2). The phase-derivative current is therefore a consequence of the autonomous model, not an input to it. Eq. (18) is derived before the citation to [22], so that self-citation is a pointer and is not load-bearing. The Floquet variance bounds, asymptotic commutativity, and noise-matching effect are analytic consequences of the exactly solvable model, with no fitted parameters renamed as predictions. The coherent-cavity benchmark uses a distinct physical Hamiltonian (Eq. (103)) and exact numerical evolution; the scaling λ_i=g_i/√n̄ is a controlled classical limit, and Sec. VII explicitly introduces a positive work-variance gap via Kadison's inequality as a derived diagnostic rather than a circular rescue. Self-citations [11,16,21,22] appear only in contextual sentences. No specific circular step can be exhibited, and the paper's own limitation statements appropriately scope the exactness to ideal translation clocks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

No fitted parameters appear in the derivations; all results are analytical and parameter-free. The model-specific couplings and phases used in examples are inputs, not fitted quantities. The numerical truncation N_b in Eq. (110) is a convergence parameter, not a free parameter of the theory.

axioms (3)
  • domain assumption Terminals are ideal translation clocks with \hat H_i = -i\omega_i \partial_{s_i} on L^2(R).
    Introduced in Sec. II around Eq. (1). This rigid-translation property is essential for the coordinate-block diagonalization and the exact pump-space representation in Theorem 2.
  • domain assumption Pump-clock coupling is diagonal in the clock-coordinate basis.
    Eq. (2) assumes coupling \int d^Ds \hat H_P(s) \otimes |s\rangle\langle s|. This makes work observables diagonal in the initial clock coordinates and is crucial for Eqs. (13)-(16).
  • standard math Standard quantum mechanics results (Trotter product formula, Kadison's inequality, Choi's multiplicative-domain theorem).
    Used in Appendices A and D to prove the propagator identity and the positivity of the work-variance gap.

pith-pipeline@v1.3.0-daily-deepseek · 242 in / 7886 out tokens · 118868 ms · 2026-08-04T01:30:15.644768+00:00 · methodology

0 comments
read the original abstract

We develop a terminal-resolved theory of work statistics for autonomous quantum energy pumps. By modeling the systems that supply and receive energy as explicit quantum terminals, the energy exchanged with each terminal is defined directly from its Hamiltonian change. When the terminals are modeled by ideal clocks, these full-space observables admit exact representations on the pump Hilbert space and recover the conventional phase-derivative currents of periodically and quasiperiodically driven systems. The framework resolves transported work from energy accumulated in the pump. It also incorporates arbitrary initial pump--terminal correlations and identifies when correlations can enhance directional energy transfer under uncertain driving phases. For periodic pumps, we derive finite-cycle work statistics in Floquet eigenstates, relate their fluctuations to Floquet quantum geometry, and show that the long-time terminal currents become mutually compatible. An exactly solvable two-terminal qubit exhibits noise matching, in which transport becomes sharp through cancellation of common-mode terminal fluctuations even though the individual terminal energies remain noisy. Finally, for physical terminals beyond the ideal-clock limit, we introduce a positive work-variance gap that quantifies the fluctuations missed by a pump-only description. A coherent-cavity benchmark shows systematic convergence toward the ideal-clock regime with increasing occupation while demonstrating that agreement of the mean current alone does not guarantee accurate work statistics.

Figures

Figures reproduced from arXiv: 2608.00638 by Yang Peng.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of a quantum energy pump coupled to [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Phase dependence of the two-clock qubit pump. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Correlation-assisted one-cycle transport under [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Convergence of two coherent cavity terminals toward [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

59 extracted references · 3 linked inside Pith

  1. [1]

    Equation (4) acts as ˆU(t)Ψ (s) = ˆUs−ωt(t)|Ψ(s−ωt)⟩.(B1) Using ˆHi =−iω i∂si, ˆHi ˆU(t)Ψ (s) =−iω i h ∂si ˆUs−ωt(t) i |Ψ(s−ωt)⟩ + ˆUs−ωt(t) ˆHi|Ψ(s−ωt)⟩

    Terminal-work block identity In the clock-coordinate representation, a joint state is a pump-valued wave function|Ψ(s)⟩. Equation (4) acts as ˆU(t)Ψ (s) = ˆUs−ωt(t)|Ψ(s−ωt)⟩.(B1) Using ˆHi =−iω i∂si, ˆHi ˆU(t)Ψ (s) =−iω i h ∂si ˆUs−ωt(t) i |Ψ(s−ωt)⟩ + ˆUs−ωt(t) ˆHi|Ψ(s−ωt)⟩. Applying ˆU†(t)and returning to the initial clock coordi- nate gives ˆU†(t) ˆHi ˆ...

  2. [2]

    Its moments entering the cavity-energy changes are evaluated analytically

    Only the bright mode couples to the qubit, with couplingg/√¯n, while the dark mode evolves freely. Its moments entering the cavity-energy changes are evaluated analytically. The remaining qubit–bright-mode evolution is obtained from numerically converged sparse evolution of the truncated Hamiltonian. The bright-mode Fock space is truncated at Nb = ¯nb + 1...

  3. [3]

    15 Adding and subtracting R dDsp (s)m2 q(s, t)in D ˆW 2 q (t) E − D ˆWq(t) E2 gives the law of total variance in Eq

    Pump–clock correlations Using ˆσP (s) = p(s)˜σP (s), the first two moments of the directional work are D ˆWq(t) E = Z dDsp(s)m q(s, t), D ˆW 2 q (t) E = Z dDsp(s) TrP h ˜σP (s) ˆW (s) q (t) 2i . 15 Adding and subtracting R dDsp (s)m2 q(s, t)in D ˆW 2 q (t) E − D ˆWq(t) E2 gives the law of total variance in Eq. (23). For a product preparation,˜σP (s) = ˆρP...

  4. [4]

    Commutativity of the coordinate derivatives, together withˆU † ˆU=I, gives ∂si ˆBj −∂ sj ˆBi + h ˆBi, ˆBj i = 0

    Maurer–Cartan identity At fixed t, suppress the coordinate and time labels and define ˆBi = ˆU †∂si ˆU = −i ˆAi. Commutativity of the coordinate derivatives, together withˆU † ˆU=I, gives ∂si ˆBj −∂ sj ˆBi + h ˆBi, ˆBj i = 0. Substituting ˆBi =−i ˆAi yields ∂si ˆAj −∂ sj ˆAi = i h ˆAi, ˆAj i , which is Eq. (32)

  5. [5]

    Expanding aroundδ= 0gives ˆU † − ˆU+ =I+ 1 2 h ˆU †∂δ ˆU− ∂δ ˆU † ˆU i + 1 8 h ˆU †∂2 δ ˆU+ ∂2 δ ˆU † ˆU i − 1 4 ∂δ ˆU † ∂δ ˆU +O(∥δ∥ 3)

    Expansion of the PRFT counting amplitude Define ∂δ = P i δi∂ϕi and ˆU± = ˆUϕ±δ/2(t). Expanding aroundδ= 0gives ˆU † − ˆU+ =I+ 1 2 h ˆU †∂δ ˆU− ∂δ ˆU † ˆU i + 1 8 h ˆU †∂2 δ ˆU+ ∂2 δ ˆU † ˆU i − 1 4 ∂δ ˆU † ∂δ ˆU +O(∥δ∥ 3). Using the first and second derivatives ofˆU † ˆU = I reduces this expression to ˆU † − ˆU+ =I−i ˆAδ − 1 2 ˆA2 δ +O(∥δ∥ 3), ˆAδ = i ˆU ...

  6. [6]

    Let ˆUϕ(t2, t1)denote the corresponding propagator from t1 to t2

    Time-origin covariance For a localized initial clock configurationϕ, the driven pump Hamiltonian satisfies ˆH (ϕ+ωτ) P (t) = ˆH (ϕ) P (t + τ ). Let ˆUϕ(t2, t1)denote the corresponding propagator from t1 to t2. The Floquet operator for the shifted initial phases is ˆF (ϕ + ωτ ) = ˆUϕ(T + τ, τ). Periodicity and the composition law give ˆF(ϕ+ωτ) = ˆUϕ(T+τ, τ...

  7. [7]

    Repeated-cycle work statistics Recall ∂q = PD i=1 qiωi∂ϕi. The corresponding one-cycle directional work is ˆAq = i ˆF †∂q ˆF= ˆWq(T).(C1) Differentiating ˆF n gives ˆWq(nT) = i( ˆF n)†∂q ˆF n = n−1X r=0 ˆF −r ˆAq ˆF r.(C2) 16 Let ˆF|α⟩ = e −iθα |α⟩. Differentiating this eigenvalue equation and projecting onto|α⟩gives ⟨α| ˆAq|α⟩=∂ qθα =T wq,α.(C3) Forβ̸=α,...

  8. [8]

    The mean ergodic theorem gives lim n→∞ 1 n n−1X r=0 ˆF −r ˆA ˆF r − PF ( ˆA) = 0(C6) for every pump operator ˆA

    Asymptotic current operators For a finite-dimensional Floquet operator with non- degenerate eigenvalues, define its ergodic projection by PF ( ˆA) = P α |α⟩⟨α| ˆA|α⟩⟨α|. The mean ergodic theorem gives lim n→∞ 1 n n−1X r=0 ˆF −r ˆA ˆF r − PF ( ˆA) = 0(C6) for every pump operator ˆA. Applying this result to ωi ˆAi/Tand using(ω i/T)⟨α| ˆAi|α⟩= wi,α gives lim...

  9. [9]

    H. J. Kimble, The quantum internet, Nature453, 1023 (2008)

    2008

  10. [10]

    N. Lauk, N. Sinclair, S. Barzanjeh, J. P. Covey, M. Saffman, M. Spiropulu, and C. Simon, Perspectives on quantum transduction, Quantum Sci. Technol.5, 020501 (2020)

    2020

  11. [11]

    Niedenzu, M

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

    2019

  12. [12]

    Elouard and C

    C. Elouard and C. L. Latune, Extending the laws of ther- modynamics for arbitrary autonomous quantum systems, PRX Quantum4, 020309 (2023)

    2023

  13. [13]

    Kurman, K

    Y. Kurman, K. Hymas, A. Fedorov, W. J. Munro, and J. Quach, Powering quantum computation with quantum batteries, Phys. Rev. X16, 011016 (2026)

    2026

  14. [14]

    Auffèves, Quantum technologies need a quantum en- ergy initiative, PRX Quantum3, 020101 (2022)

    A. Auffèves, Quantum technologies need a quantum en- ergy initiative, PRX Quantum3, 020101 (2022)

    2022

  15. [15]

    Campaioli, S

    F. Campaioli, S. Gherardini, J. Q. Quach, M. Polini, and G. M. Andolina, Colloquium: Quantum batteries, Rev. Mod. Phys.96, 031001 (2024)

    2024

  16. [16]

    D. J. Thouless, Quantization of particle transport, Phys. Rev. B27, 6083 (1983)

    1983

  17. [17]

    P. W. Brouwer, Scattering approach to parametric pump- ing, Phys. Rev. B58, R10135 (1998)

    1998

  18. [18]

    Martin, G

    I. Martin, G. Refael, and B. Halperin, Topological fre- quency conversion in strongly driven quantum systems, Phys. Rev. X7, 041008 (2017)

    2017

  19. [19]

    Peng and G

    Y. Peng and G. Refael, Topological energy conversion through the bulk or the boundary of driven systems, Phys. Rev. B97, 134303 (2018)

    2018

  20. [20]

    M. H. Kolodrubetz, F. Nathan, S. Gazit, T. Morimoto, and J. E. Moore, Topological floquet-thouless energy pump, Phys. Rev. Lett.120, 150601 (2018)

    2018

  21. [21]

    F.Nathan, I.Martin,andG.Refael,Topologicalfrequency conversion in a driven dissipative quantum cavity, Phys. Rev. B99, 094311 (2019)

    2019

  22. [22]

    P. J. D. Crowley, I. Martin, and A. Chandran, Half-integer quantized topological response in quasiperiodically driven quantum systems, Phys. Rev. Lett.125, 100601 (2020)

    2020

  23. [23]

    D. M. Long, P. J. D. Crowley, and A. Chandran, Nonadia- batic topological energy pumps with quasiperiodic driving, Phys. Rev. Lett.126, 106805 (2021)

    2021

  24. [24]

    Z. Qi, G. Refael, and Y. Peng, Universal nonadiabatic energy pumping in a quasiperiodically driven extended system, Phys. Rev. B104, 224301 (2021)

    2021

  25. [25]

    D. M. Long, P. J. D. Crowley, A. J. Kollár, and A. Chan- dran, Boosting the quantum state of a cavity with floquet driving, Phys. Rev. Lett.128, 183602 (2022)

    2022

  26. [26]

    F.Nathan, I.Martin,andG.Refael,Topologicalfrequency conversion in weyl semimetals, Phys. Rev. Research4, 043060 (2022)

    2022

  27. [27]

    Malz and A

    D. Malz and A. Smith, Topological two-dimensional flo- quet lattice on a single superconducting qubit, Phys. Rev. Lett.126, 163602 (2021)

    2021

  28. [28]

    Psaroudaki and G

    C. Psaroudaki and G. Refael, Energy transfer in random- matrix ensembles of floquet hamiltonians, Phys. Rev. B 108, 064301 (2023)

    2023

  29. [29]

    Peng, Non-abelian geometric quantum energy pump, Phys

    Y. Peng, Non-abelian geometric quantum energy pump, Phys. Rev. B114, L020304 (2026)

    2026

  30. [30]

    Schmid, F

    H. Schmid, F. von Oppen, G. Refael, and Y. Peng, Su- perextensive charging speeds in a correlated quantum charger, Phys. Rev. B114, 014313 (2026)

    2026

  31. [31]

    Friis and M

    N. Friis and M. Huber, Precision and work fluctuations in gaussian battery charging, Quantum2, 61 (2018)

    2018

  32. [32]

    L. P. García-Pintos, A. Hamma, and A. del Campo, Fluc- tuations in extractable work bound the charging power of quantum batteries, Phys. Rev. Lett.125, 040601 (2020)

    2020

  33. [33]

    Bakhshinezhad, B

    P. Bakhshinezhad, B. R. Jablonski, F. C. Binder, and N. Friis, Trade-offs between precision and fluctuations in charging finite-dimensional quantum batteries, Phys. Rev. E109, 014131 (2024)

    2024

  34. [34]

    Mohan, T

    B. Mohan, T. Pandit, M. Lewenstein, and M. N. Bera, Fundamental limitations on the reliabilities of power and work in quantum batteries, PRX Quantum 10.1103/fnv6- yqmk (2026), accepted, arXiv:2601.05315 [quant-ph]

    arXiv 2026

  35. [35]

    L.S.Levitov, H.Lee,andG.B.Lesovik,Electroncounting statistics and coherent states of electric current, J. Math. Phys.37, 4845 (1996)

    1996

  36. [36]

    G. T. Landi, M. J. Kewming, M. T. Mitchison, and P. P. Potts, Current fluctuations in open quantum sys- tems: Bridging the gap between quantum continuous measurements and full counting statistics, PRX Quantum 5, 020201 (2024)

    2024

  37. [37]

    A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhys. Lett.67, 565 (2004)

    2004

  38. [38]

    G. M. Andolina, D. Farina, A. Mari, V. Pellegrini, V. Gio- vannetti, and M. Polini, Charger-mediated energy transfer in exactly solvable models for quantum batteries, Phys. Rev. B98, 205423 (2018)

    2018

  39. [39]

    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 batter- ies, Phys. Rev. Lett.122, 047702 (2019)

    2019

  40. [40]

    Cepollaro, D

    C. Cepollaro, D. Chughtai, A. Spalvieri, M. Huber, and A. Rolandi, Autonomization of quantum systems and the emergence of the work operator, arXiv:2607.20602 (2026), arXiv:2607.20602 [quant-ph]

    Pith/arXiv arXiv 2026

  41. [41]

    C. Han, N. Katz, and E. Sela, Measuring work in quan- tum many-body systems using a dynamical work agent, Phys. Rev. B112, 035416 (2025), arXiv:2503.20729 [cond- mat.mes-hall]

    Pith/arXiv arXiv 2025

  42. [42]

    Zhao and H

    X.-H. Zhao and H. T. Quan, Fluctuation theorems for au- tonomous work in the quantum regime, arXiv:2607.24690 (2026), arXiv:2607.24690 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2026

  43. [43]

    Engelhardt, S

    G. Engelhardt, S. Choudhury, and W. V. Liu, Unified light-matter floquet theory and its application to quantum communication, Phys. Rev. Research6, 013116 (2024)

    2024

  44. [44]

    Engelhardt, J

    G. Engelhardt, J. Luo, V. M. Bastidas, and G. Platero, Photon-resolved floquet theory. i. full counting statistics of the driving field in floquet systems, Phys. Rev. A110, 063707 (2024)

    2024

  45. [45]

    Engelhardt, J

    G. Engelhardt, J. Luo, V. M. Bastidas, and G. Platero, Photon-resolved floquet theory. ii. open quantum systems, Phys. Rev. A110, 063708 (2024)

    2024

  46. [46]

    Seibold, O

    K. Seibold, O. Ameye, and O. Zilberberg, Floquet expan- sion by counting pump photons, Phys. Rev. Lett.134, 060401 (2025)

    2025

  47. [47]

    H. P. Robertson, The uncertainty principle, Phys. Rev. 34, 163 (1929)

    1929

  48. [48]

    Schrödinger, Zum heisenbergschen unschärfeprinzip, Sitzungsberichte der Preußischen Akademie der Wis- 18 senschaften, Physikalisch-Mathematische Klasse , 296 (1930)

    E. Schrödinger, Zum heisenbergschen unschärfeprinzip, Sitzungsberichte der Preußischen Akademie der Wis- 18 senschaften, Physikalisch-Mathematische Klasse , 296 (1930)

    1930

  49. [49]

    Kolodrubetz, D

    M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Geometry and non-adiabatic response in quantum and classical systems, Phys. Rep.697, 1 (2017)

    2017

  50. [50]

    R. V. Kadison, A generalized schwarz inequality and algebraic invariants for operator algebras, Ann. Math.56, 494 (1952)

    1952

  51. [51]

    Manfredi and M

    G. Manfredi and M. R. Feix, Entropy and wigner func- tions, Phys. Rev. E62, 4665 (2000)

    2000

  52. [52]

    Rungta, V

    P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, Universal state inversion and concurrence in arbitrary dimensions, Phys. Rev. A64, 042315 (2001)

    2001

  53. [53]

    R. J. Glauber, Coherent and incoherent states of the radiation field, Phys. Rev.131, 2766 (1963)

    1963

  54. [54]

    M. P. Woods, R. Silva, and J. Oppenheim, Autonomous quantum machines and finite-sized clocks, Ann. Henri Poincaré20, 125 (2019)

    2019

  55. [55]

    Roulet, S

    A. Roulet, S. Nimmrichter, J. M. Arrazola, S. Seah, and V. Scarani, Autonomous rotor heat engine, Phys. Rev. E 95, 062131 (2017)

    2017

  56. [56]

    S. Seah, S. Nimmrichter, and V. Scarani, Work production of quantum rotor engines, New J. Phys.20, 043045 (2018)

    2018

  57. [57]

    von Lindenfels, O

    D. von Lindenfels, O. Gräb, C. T. Schmiegelow, V. Kaushal, J. Schulz, M. T. Mitchison, J. Goold, F. Schmidt-Kaler, and U. G. Poschinger, Spin heat engine coupled to a harmonic-oscillator flywheel, Phys. Rev. Lett. 123, 080602 (2019)

    2019

  58. [58]

    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

  59. [59]

    Choi, A schwarz inequality for positive linear maps onc ∗-algebras, Illinois J

    M.-D. Choi, A schwarz inequality for positive linear maps onc ∗-algebras, Illinois J. Math.18, 565 (1974)

    1974