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Anomalous discharging of quantum batteries: the ergotropic Mpemba effect

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Pith's one-line read The ergotropy of a single-mode Gaussian quantum battery equals a phase-space relative entropy, and squeezed states discharge faster than displaced ones even at equal initial charge.

desk verdict Solid analytic result: ergotropy as Wigner relative entropy gives exact discharge curves and a clean Mpemba example, but the resource claim rests on the explicitly stated instantaneous-extraction assumption. read the letter →

arxiv 2412.13259 v2 pith:STGZWSGC submitted 2024-12-17 quant-ph

Ivan Medina , Oisín Culhane , Felix C. Binder , Gabriel T. Landi , John Goold This is my paper · ORCID
classification quant-ph
keywords quantumbatteriesergotropyMpembaeffectGaussianstatesWignerrelativeentropyopensystemsworkextractionbosonicmode
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

This paper establishes that for a single bosonic mode charged by Gaussian operations and weakly coupled to a thermal bath, the extractable energy (ergotropy) equals a phase-space relative entropy, which makes the discharge curves analytically solvable. In this setting, a squeezed thermal state loses its stored charge faster than a displaced thermal state, even when both start with the same ergotropy. This is an ergotropic Mpemba effect: a more highly charged battery can discharge faster than a less charged one. The authors conclude that charging with displacement is a better resource than charging with squeezing, because the ergotropy dissipates more slowly.

What carries the argument

The central object is the Wigner relative entropy, $K[W_1||W_2] = \int d^2\alpha\, W_1 \ln(W_1/W_2)$, together with the identity $E(W) = \omega f(\beta_\pi) K[W||W_\pi]$ that ties ergotropy to a phase-space divergence. The companion constraint $f(\beta_\pi) = \sqrt{|\Theta|}$, which follows from the unitary connection between the state and its passive version, reduces the problem to the covariance matrix and mean vector, splitting the ergotropy into a displacement part $E_d = \omega|\mu|^2$ and a squeezing part $E_s = \omega f(\beta_\pi)[\cosh(2r)-1]$. The analytic solution of the Lyapunov equation for the covariance matrix then yields explicit time-dependent $f(\beta_t)$ and $r_t$, exposing the non-monotonic passive-state energy of the squeezed state that drives the effect.

What would settle it

Prepare a single bosonic mode in a squeezed thermal state and a displaced thermal state with equal initial ergotropy, couple each weakly to the same thermal bath, and measure the maximum work extractable over time; the claim predicts the squeezed state's ergotropy falls below the displaced state's at the analytic crossing time $\tau_c$ of Eq. (15).

Watch

Extended reading notes

Core claim

For a single Gaussian bosonic mode with Hamiltonian $\hat{H} = \omega(\hat{a}^\dagger\hat{a} + 1/2)$ weakly coupled to a thermal bath, the paper shows that the ergotropy at any time is $E(W) = \omega f(\beta_\pi) K[W||W_\pi]$, where $W_\pi$ is the thermal passive state unitarily connected to $W$ and $K$ is the Wigner relative entropy. Because the passive state of the squeezed state carries a positive energy contribution from the squeezing parameter, its ergotropy relaxes faster than that of a displaced state, and the two curves can cross even when the initial ergotropies are identical. The central claim is that displacement charging is therefore a better resource than squeezing charging, since it makes the extractable charge dissipate more slowly.

Load-bearing premise

The results assume that the unitary operations used to extract the stored work can be performed almost instantly compared with the dissipative timescale; if extraction is slow, the computed ergotropy curves are not the physically achievable discharge profiles.

Editorial extensions

If this is right

  • The ergotropy of a Gaussian state is exactly a Wigner relative entropy, so discharge curves for Gaussianity-preserving dissipative dynamics can be computed in closed form.
  • A squeezed thermal state can discharge faster than a displaced thermal state even when the initial ergotropies are equal, because squeezing adds a positive, non-monotonic contribution to the passive-state energy.
  • The ergotropic Mpemba crossing time is given by a closed-form expression (Eq. 15) that depends only on the charging parameters and the bath temperature.
  • Charging with a displacement operation is a better resource than charging with squeezing, since the ergotropy dissipates more slowly in the first case.
  • The phase-space formulation makes the effect directly testable in quantum optical and mesoscopic platforms where Wigner functions are measured.

Reading between the lines

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

  • The same relative-entropy identity likely extends to multimode Gaussian states, where intermodal correlations could also raise the passive-state energy and produce analogous ergotropic crossings.
  • If the extraction unitaries cannot be performed much faster than the dissipative dynamics, the true achievable discharge profile will lag the computed ergotropy, and the Mpemba crossing may be delayed or hidden.
  • The analytic crossing time could be used to design charging operations that deliberately maximize charge retention by keeping the passive-state energy monotonic.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The manuscript studies a single bosonic-mode Gaussian quantum battery weakly coupled to a thermal bath. Its central results are: (i) an exact rewriting of ergotropy as ω f(βπ) K[W||Wπ] for Gaussian states; (ii) an analytic solution of the dissipative dynamics for displaced and squeezed thermal states, expressed through Eqs. (11)-(14); (iii) the identification of an ergotropic Mpemba effect in which a squeezed-charged battery with higher initial ergotropy discharges faster than a displaced-charged one; and (iv) a proof, via Eq. (16), that if the two batteries start with the same ergotropy, the squeezed battery loses its charge faster at all later times. The appendices contain complete derivations of the Gaussian relative entropy, the passive-state relation f(βπ)=sqrt(|Θ|), and the Lyapunov solution.

Significance. The work is a valuable analytic contribution to the quantum-battery and Mpemba-effect literature. The derivation is parameter-free: no constants are fitted, and the crossing time and the equal-ergotropy comparison are derived consequences of the Lindblad dynamics rather than inputs. The phase-space formulation connects battery physics to experimentally accessible Wigner distributions and provides a rare exactly solvable Mpemba setting. Within its stated scope, the mathematics is internally consistent and the appendices are self-contained. The main caveat is scope: single-mode Gaussian states and instantaneous work extraction; the resource-theoretic conclusion requires an explicit operational qualification.

major comments (1)
  1. [After Eq. (10) and in the Conclusion] The abstract and the concluding sentence state that charging by displacement is "a better resource" than charging by squeezing. This operational claim is only established for the instantaneous ergotropy E(t), as the assumption stated after Eq. (10) makes explicit. If work extraction takes a finite time comparable to 1/γ, the state continues to dissipate during the extraction protocol, and the achievable extracted work is no longer the instantaneous ergotropy but the solution of a finite-time control problem; the ordering of E_s(t) and E_d(t) need not carry over to that setting. Since this is load-bearing for the central resource-theoretic conclusion, please either restrict all operational statements to the instantaneous-extraction limit or provide a quantitative discussion, and ideally numerical evidence, of the finite-duration extraction case.
minor comments (4)
  1. [Discussion of Fig. 3] The phrase "the higher the temperature βπ is" is inconsistent with βπ being an inverse temperature; it should read "the higher the temperature (i.e., the larger \bar n_π and the smaller βπ)".
  2. [Eq. (15)] The crossing-time formula is presented without derivation; please include a short derivation in an appendix or state explicitly that it follows from solving E_s(τc)=E_d(τc) using Eqs. (E6) and (E16).
  3. [Paragraph after Eq. (16)] The equal-initial-ergotropy claim would be much clearer with the displayed inequality E_s(t)=E_d(t)+ω(Δβ−f(βt))≤E_d(t), which follows from f(βt)≥Δβ; the current verbal argument hides the essential step.
  4. [Throughout] There are minor typographical issues: "Reyni-2" should be "Rényi-2", the heading of Appendix C has a stray space in "V ector", and reference [43] has "Inroductory" instead of "Introductory".

Circularity Check

0 steps flagged · score 0.0 of 10

No circular steps found: the ergotropy identity and the displacement-versus-squeezing ordering are derived consequences of the Gaussian dynamics, not fitted inputs.

full rationale

The manuscript's derivation chain is self-contained. Equation (5) is obtained in Appendix B from the Wigner relative-entropy identity Eq. (B4) together with the standard fact that the passive state of a single-mode Gaussian state is a thermal state; it is a mathematical rewriting of the ergotropy, not an input used to construct the dissipative dynamics. Equations (11)-(12) follow directly from the mean-vector and covariance-matrix solutions of the Lyapunov equation in Appendix E. The equal-ergotropy calibration Eq. (16) is obtained algebraically from Eqs. (11)-(12), and the crossing time Eq. (15) is the solution of E_s(t)=E_d(t); neither quantity is fitted to data or equivalent to the conclusion. The displacement-versus-squeezing ordering is a consequence of the explicit time-dependent expressions f(beta_t), cosh(2r_t) and the exponential decay of the displacement parameter. The instantaneous-extraction assumption after Eq. (10) is an explicitly stated validity condition on the operational interpretation, not a circular step. Citations to the authors' prior work appear only as background context and are not load-bearing for any of the paper's claims. No parameter is fitted, no prediction reduces by construction to its input, and no load-bearing claim depends on an unverified self-citation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The derivation is self-contained and parameter-free: all quantities in Eqs. (5)-(16) are derived from Gaussian integrals and the Lindblad/Lyapunov dynamics, with no numbers fitted to data. The cost is paid upstream through standard input assumptions: the weak-coupling Markovian master equation, the symplectic passive-state structure of single-mode Gaussian states, and the identification of the Wigner (Rényi-2) entropy as the entropy preserved by the unitary connection. Numerical values in the figures (r, µ, ¯n, ¯nπ) are illustrative inputs, not fitted parameters.

assumptions (4)
  • domain assumption Markovian weak-coupling Lindblad master equation (Eq. 8) with linear dissipators; Gaussianity-preserving dynamics with the thermal state as fixed point.
    Invoked in the 'Gaussianity preserving dynamics' section and Eq. (9); standard for damped quantum oscillators (refs. [41,42]); it is the backbone of the analytic Lyapunov solution.
  • standard math For a single-mode Gaussian state, the passive state with respect to H = ω(a†a + 1/2) is a thermal state with f(βπ) = √|Θ|.
    Used in Appendix B to obtain Eq. (7); follows from passive-state theory of Gaussian states and preservation of det Θ under symplectic unitaries; it carries the entire ergotropy decomposition.
  • standard math Unitarily connected states share the same Wigner entropy S(W) = −∫W ln W (the Rényi-2 entropy), so S(W) = S(Wπ).
    Used in Eqs. (6)-(7) and Appendix B; true because S(W) depends only on det Θ, which is preserved under the unitary (symplectic) connection.
  • domain assumption Work extraction unitaries act on a timescale much shorter than the dissipative dynamics, so the instantaneous ergotropy is the achievable extractable charge.
    Stated explicitly in the main text after Eq. (10); if false, the ergotropic Mpemba comparison is not the operational discharge profile.

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Cite this review

Pith. "Pith review of Anomalous discharging of quantum batteries: the ergotropic Mpemba effect." pith.science (2026). https://pith.science/paper/STGZWSGC

@misc{pith2026241213259,
  author       = {Pith},
  title        = {Pith review of: Anomalous discharging of quantum batteries: the ergotropic Mpemba effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/STGZWSGC}},
  note         = {Machine review of arXiv:2412.13259}
}
read the original abstract

Anomalous thermal relaxation is ubiquitous in nonequilibrium statistical mechanics. An emblematic example of this is the Mpemba effect, where an initially ``hot'' system cools faster than an initially ``cooler'' one. This effect has recently been studied in a variety of different classical and quantum settings. In this Letter, we find a novel signature of the Mpemba effect in the context of quantum batteries. We identify situations where batteries in higher charge states can discharge faster than less charged states. Specifically, we consider a quantum battery encoded in a single bosonic mode that is charged using unitary Gaussian operations. We show that the ergotropy, used here as a dynamical indicator of the energy stored in the battery, can be recast as a phase space relative entropy between the system's state and the unitarily connected passive state, at each time. Our formalism allows us to compute the ergotropy analytically under dissipative dynamics and allows us to understand the conditions which give rise to a Mpemba effect. We also find situations where two batteries charged to the same value using different operations can discharge at different rates.

Figures

Figures reproduced from arXiv: 2412.13259 by the authors.

Figure 1
Figure 1. (a) Single mode thermal-state quantum oscillator with no [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Energetic and ergotropic time evolution of a squeezed [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Ergotropy as a function of time for a squeezed thermal [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Two-time weak measurement protocol for ergotropy protection in open quantum batteries

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    A two-time weak measurement protocol can slow the self-discharge of open quantum batteries and yield a positive ergotropy gain with zero net measurement-induced energy and ergotropy shifts.

  2. The quantum Mpemba effects

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    A review of the quantum Mpemba effect covering open and isolated quantum systems, key theories, experiments, and open questions.

Reference graph

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    1 2 p |Θ| Tr{Θ} −1 # . (C4) We see that the ergotropy can then be split as E(W ) = Ed(⃗ v) + Es(Θ), Ed(⃗ v) = ω|⃗ v|2, (C5) Es(Θ) = ω p |Θ|

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