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REVIEW 2 major objections 5 minor 76 references

Dynamical Landauer principle: Thermodynamic criteria of transmitting classical information

T0 review · 2 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Transmitting n bits of classical information is equivalent to transmitting n×k_BT ln2 of work-like energy.

desk verdict Solid one-shot capacity bounds, but the advertised exact n-bit/energy equivalence rests on an unjustified per-message step and dropped error terms. read the letter →

arxiv 2502.03603 v1 pith:RLGTPXD2 submitted 2025-02-05 quant-ph cs.ITmath-phmath.ITmath.MP

Chung-Yun Hsieh This is my paper · ORCID
classification quant-phcs.ITmath-phmath.ITmath.MP MSC 81P4594A40
keywords quantumchannelcapacityone-shotclassicalLandauer'sprinciplethermodynamicsworkextractioninformationalnon-equilibriumHolevo-Schumacher-Westmorelandtheoremcommunication
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 tries to prove a quantitative equivalence between transmitting classical information and transmitting energy through a quantum channel. The central result bounds every one-shot classical capacity between two energy-transmission figures of merit, so that sending $n$ bits of classical information is equivalent, up to one-shot error terms, to transmitting $n\times k_BT\ln 2$ of work-like energy. The author calls this a dynamical version of Landauer's principle because it concerns transmitting information rather than erasing it. In the asymptotic limit, the same bounds reproduce the Holevo-Schumacher-Westmoreland theorem and imply strong-converse statements about generating informational non-equilibrium. A reader should care because the result gives a concrete, quantitative sense in which communication and thermodynamics are the same task.

What carries the argument

The central object is the one-shot energy transmission task, whose figure of merit is the extractable work from the receiver's output bipartite correlation, $W^{\epsilon}_{\mathrm{corr}|\Theta,(1)}(\mathcal{N})$ (and its simplified version $W^{\epsilon}_{\Phi|\Theta,(1)}(\mathcal{N})$). The machinery converts the smoothed relative Rényi $0$-entropy $D^{\epsilon}_0$ into units of $k_BT\ln2$ via a one-shot work-extraction theorem, and uses the hypothesis-testing relative entropy $D^{\epsilon}_h$ to bound the classical capacity. Because $D^{\epsilon}_0$ measures both correlation maintenance and extractable work, it is the bridge that makes the two tasks coincide.

What would settle it

One could refute the equivalence by exhibiting a channel that transmits $n$ bits of classical information one-shot but whose physically transmitted energy, measured by a direct accounting of energy changes in sender and receiver with non-degenerate Hamiltonians or coherent inputs, is strictly below $n\times k_BT\ln2$; the paper's energy-transmission definition would then miss an aspect of real energy transfer.

Watch

Extended reading notes

Core claim

The paper's central claim is that one-shot classical information transmission and one-shot work-like energy transmission are the same task up to error terms. The precise statement is Theorem 4: for a set of superchannels $\Theta$, a channel $\mathcal{N}$, and errors $0<\delta\le\omega<\epsilon\le 1-1/\sqrt{2}$, the capacity satisfies $$$W^{{\omega}}$_{\mathrm{corr}|\Theta,(1)}(\mathcal{N}) - k_BT \ln\frac{4\epsilon}{(\epsilon-\omega)^2(1-\omega)} \le (k_BT \ln 2) $C^{{\epsilon}}$_{\Theta,(1)}(\mathcal{N}) \le $W^{{\epsilon+\delta}}$_{\Phi|\Theta,(1)}(\mathcal{N}).$$ Corollary 1 phrases this as the equivalence between transmitting $n$ bits and transmitting $n\times k_BT\ln2$ energy. The paper then argues that this equivalence is genuinely dynamical: in an explicit multi-trial protocol the same channel operation transmits the information and the work-like energy simultaneously, which it calls a dynamical Landauer principle.

Load-bearing premise

The load-bearing premise is that 'genuinely transmitted energy' is correctly quantified by the work extractable from the receiver's output bipartite correlation under fully degenerate Hamiltonians and energy-incoherent states; the paper states this identification in Section IV C rather than deriving it from a more direct definition of energy transmitted by a channel.

Editorial extensions

If this is right

  • A channel's one-shot classical capacity can be bounded from above and below by how much work-like energy it can transmit, so energy-transmission experiments become capacity witnesses.
  • Landauer's principle acquires a dynamical counterpart: erasing a bit costs at least $k_BT\ln2$, and transmitting $n$ bits must carry at least $n\times k_BT\ln2$ of work-like energy with it.
  • In the asymptotic limit the bounds reproduce the HSW theorem, giving that theorem a thermodynamic interpretation.
  • Strong-converse and no-go results follow: allowing encoders and decoders to generate informational non-equilibrium cannot increase the asymptotic classical capacity.

Reading between the lines

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

  • If the equivalence is taken physically, a systematic mismatch between correlation-extractable work and direct energy accounting in an experiment would force a revision of the definition of transmitted energy, not just of the theorem.
  • A natural extension would be to ask whether sending quantum information, such as entanglement or quantum messages, also implies the same per-bit work-like energy transmission; the paper only states the classical case.
  • The one-shot formulation means the equivalence is not just an asymptotic statement, so it could be probed in few-qubit devices where finite-size error terms are non-negligible.
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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

2 major / 5 minor

Summary. This paper proposes a one-shot thermodynamic characterization of classical communication through quantum channels. It defines Θ-assisted classical capacities, proves entropic bounds (Theorems 1 and 2), introduces correlation-based work-extraction quantities as energy transmission tasks (Section IV), and shows that these bound the capacity up to one-shot error terms (Theorem 4). The authors interpret this as a dynamical Landauer principle, namely that transmitting n bits is accompanied by transmitting n k_B T ln 2 of work-like energy, and they derive the HSW theorem plus strong-converse/no-go statements in the asymptotic limit (Proposition 1, Corollaries 3 and 4). The technical proofs are detailed and follow standard one-shot information-theoretic techniques.

Significance. If the central equivalence survives the concerns below, the paper provides a quantitative bridge between classical information transmission and work extraction from correlations, and it gives an interesting thermodynamic reading of the HSW theorem. The proofs of Theorems 1, 2, and 4 are thorough and generally sound at the level of standard one-shot capacity bounds; the asymptotic results and strong-converse statements are nontrivial and go beyond a simple restatement of known capacity formulas. The main caveats are that the strongest advertised corollary relies on a per-message guarantee that is not implied by the average-success definition, and that the "energy transmitted" figure of merit is constructed from the same entropic quantities used in the capacity bound, so the physical content of the equivalence depends on accepting that operational definition.

major comments (2)
  1. [§V B, Eqs. (81)–(82)] Fact 1 (Eq. (7)) defines the capacity via the average success probability over uniformly distributed messages. The proof of Corollary 2 asserts, for every m, that ∥Π_M(N)(|m⟩⟨m|)−|m⟩⟨m|∥_1 = O(ε), and then derives Eq. (82) from this per-message statement. This does not follow from average success: a code can have a √ε fraction of messages failing with probability close to 1 while the average error is still ε. A standard expurgation gives a subcode of size M(1−√ε) with per-message error O(√ε), so the proof establishes at best n−O(√ε) bits accompanied by (n−O(√ε)) k_B T ln 2 of extractable work, not the exact n stated in Corollaries 1 and 2. The advertised dynamical Landauer principle therefore needs either an expurgation step or a restatement of the corollaries at the corrected precision.
  2. [§IV C, Eqs. (74)–(76); Theorem 4] The quantity W^ε_{Φ|Θ,(1)} is defined using the same maximally correlated state Φ_MM′, the same trace-norm constraint ∥Π_M(N)(I/M)−I/M∥_1<2ε, and the same smoothed Rényi 0-entropy D^ε_0 that appears in Theorem 1's capacity upper bound. Via Åberg's Theorem 3, the upper bound in Theorem 4 is therefore a direct translation of Theorem 1's bound into work units. The physical claim that this quantity represents "genuinely transmitted energy" rests on the stipulation in Section IV C that transmitted energy should be identified with extractable work from the output bipartite correlation under fully degenerate Hamiltonians. This is an internally consistent operational definition, but it is not derived from or compared with a pre-existing notion of energy transmission, and the one-shot equivalence between W_corr and W_Φ themselves is asserted rather than proved. The paper should either justify this identification more explicitly or qualify the claim that information and energy transmission are equivalent.
minor comments (5)
  1. [Corollary 1] The statement 'The ability to transmit n bits of information is equivalent to the ability to transmit n×(k_B T ln 2) energy' drops the one-shot error terms that appear in Theorem 4; please state the ε-dependence explicitly or add 'up to one-shot error terms' in the corollary statement.
  2. [§V C, Eq. (105)] The inequality as printed, C^ε_{(1)|θ-equi}(N^{⊗k}) ≤ C^{3ε}_{(1)}(N)+O(log ε), is dimensionally inconsistent because the right-hand side does not depend on k; presumably it should be C^{3ε}_{(1)}(N^{⊗k}).
  3. [§V C, before Eq. (86)] The text reads 'Hovelo information'; this should be 'Holevo information'.
  4. [Theorem 1, Eq. (21)] The placement of the trace-norm constraint inside the supremum is hard to parse; a displayed constraint set would improve readability.
  5. [§IV C, Eq. (72)] The data-processing inequality for D^ε_0 via Fact 6 requires ε to be no larger than the minimal positive eigenvalues of the relevant states; Eq. (72) states this only as 'for a small enough ε' and would benefit from stating the condition explicitly.

Circularity Check

2 steps flagged · score 6.0 of 10

Partially circular: the energy-transmission figures of merit are the capacity bound's smoothed Rényi-0 expression relabeled as work, and Corollary 2 identifies transmitted energy with the very correlation that defines information transmission.

  1. renaming known result [Section IV C, Eq. (76); compare with Theorem 1 bound Eq. (21) and Eq. (70)]
    "𝑊𝜖 Φ|Θ,(1)(N) B sup ... 𝑊𝜖 corr,(1)[(Π𝑀(N)⊗I 𝑀′)(Φ𝑀𝑀′)] ... 0≤𝑊𝜖 corr,(1)(𝜌𝐴𝐵)−( 𝑘𝐵𝑇 ln 2)𝐷𝜖 0(𝜌𝐴𝐵∥𝜌𝐴⊗𝜌𝐵)≤ ... 𝐶𝜖 Θ,(1)(N)≤ sup ... 𝐷𝜖+𝛿 0(...)."

    Equation (76) defines W_Phi as a supremum of W_corr over the same classical versions Pi_M and the same trace-norm constraint appearing in Theorem 1's upper bound. Equation (70) states that W_corr(rho_AB) is, up to the Aberg correction, k_B T ln2 times D0(rho_AB || rho_A tensor rho_B). Hence the D0 functional in Theorem 1's capacity bound is the same functional that is later called genuinely transmitted energy. The lower-bound side of Theorem 4 uses D0 <= D_h and Eq. (70), again the same functional. Thus Theorem 4 restates the entropic capacity bounds of Theorem 1 in work units; the claimed information-energy equivalence is built from the same smoothed min-relative entropy that constitutes the capacity bound, rather than from an independently measured energy transmission value.

  2. self definitional [Section V B, Corollary 2 proof, Eqs. (81)-(82) and Observations C-D]
    "Using Eq. (70), extractable work from Φ𝑀𝑀′’s correlation is 𝑊𝜖 corr,(1)(Φ𝑀𝑀′)=𝑛×( 𝑘𝐵𝑇 ln 2)+ 𝑂(𝜖). Due to the continuity of 𝐷𝜖 0 ... the extractable work from [Π𝑀(N)⊗I 𝑀′](Φ𝑀𝑀′)’s correlation is 𝑛×( 𝑘𝐵𝑇 ln 2), up to an error of the order 𝑂(𝜖)."

    In Corollary 2, transmitting n bits is operationalized by Eq. (81), i.e. Pi_M(N)(|m><m|) ≈ |m><m|, which gives Eq. (82), closeness of the channel output on Phi_MM' to Phi_MM'. The transmitted energy is then not measured as a separate physical quantity; it is identified with the work extractable from that same output correlation, which by Eq. (70) is k_B T ln2 times D0 of the output relative to its marginals. Since D0(Phi || I/M ⊗ I/M) = log M = n and D0 is continuous, the energy value follows directly from the D0 expression used to define W_corr.

full rationale

The derivation is mathematically valid, and the paper is not relying on a load-bearing self-citation: the companion-paper citation [21] is not used as a proof, and the one-shot capacity bounds are proved in the text. The circularity concern is structural. Section III proves that the one-shot capacity is sandwiched by smoothed relative Rényi-0 and hypothesis-testing entropies of the states (Pi_M(N)⊗I)(Phi_MM') and Pi_M(N)(I/M)⊗I/M. Section IV then defines genuinely transmitted work-like energy as Aberg extractable work from exactly those output-correlation states; by Eq. (70) this is k_B T ln2 times the same D0 functional used in Theorem 1. Consequently Theorem 4's equivalence is largely a restatement of Theorem 1 in thermodynamic units, with Aberg's theorem supplying the physical meaning of D0 as work. Corollary 2 uses the same identification: the energy transmitted is the extractable work from the correlation whose preservation is the definition of information transmission in that setting. The separate concern in the proof of Corollary 2 that Eq. (81) requires per-message success while Fact 1 only guarantees average success is a correctness issue, not a circularity, so it is not counted in the score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central results rest on Aberg's work extraction theorem (external), the standard machinery of one-shot information theory (Hayashi-Nagaoka, Stein's lemma, data processing), and a modeling choice that energy transmission is work extraction from output correlation under fully degenerate, diagonal Hamiltonians. No free parameters are fitted to data; the error parameters are of the standard one-shot kind.

assumptions (6)
  • domain assumption Aberg's one-shot work extraction theorem and its corollary Eq. (70) relating extractable work from correlation to kBT ln2 times D_0^epsilon.
    Invoked in Section IV B and used in the proof of Theorem 4; holds only for energy-incoherent states under Aberg's allowed operations (level transformations and thermalisation).
  • domain assumption Finite-dimensional systems, fixed background temperature T, and energy-incoherent states diagonal in the chosen computational/energy eigenbasis; fully degenerate Hamiltonians in the main energy transmission task (Section IV C).
    Restricts the equivalence to classical (incoherent) communication; coherence-based communication is outside the scope.
  • standard math Theta-assisted classical versions Theta_M and Fact 2: C^epsilon_{Theta,(1)}(N) = sup_Pi C^epsilon_(1)[Pi(N)].
    Used throughout, e.g., in the proof of Theorem 1.
  • standard math Hayashi-Nagaoka inequality (Lemma 1) and Quantum Stein's Lemma (Lemma 2).
    Used in the proof of Theorem 2 lower bound and Proposition 1.
  • standard math Data-processing inequalities for D_0^epsilon and D_h^epsilon and the estimate D_h^epsilon(rho||sigma) <= (S(rho||sigma)+H_b(epsilon))/(1-epsilon) from Ref [16].
    Used in Appendix A and the asymptotic proof of Proposition 1.
  • domain assumption The referee can choose local Hamiltonians so that the bipartite input state is locally thermal (Perarnau-Llobet et al., Ref [32]), so extractable work can be attributed to correlation.
    Underlies the definition of work extraction from correlation, Eq. (70).

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Cite this review

Pith. "Pith review of Dynamical Landauer principle: Thermodynamic criteria of transmitting classical information." pith.science (2026). https://pith.science/paper/RLGTPXD2

@misc{pith2026250203603,
  author       = {Pith},
  title        = {Pith review of: Dynamical Landauer principle: Thermodynamic criteria of transmitting classical information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLGTPXD2}},
  note         = {Machine review of arXiv:2502.03603}
}
abstract

Transmitting energy and information are two essential aspects of nature. Recent findings suggest they are closely related, while a quantitative equivalence between them is still unknown. This thus motivates us to ask: Can information transmission tasks equal certain energy transmission tasks? We answer this question positively by bounding various one-shot classical capacities via different energy transmission tasks. Such bounds provide the physical implication that, in the one-shot regime, transmitting $n$ bits of classical information is equivalent to $n\times k_BT\ln2$ transmitted energy. Unexpectedly, these bounds further uncover a dynamical version of Landauer's principle, showing the strong link between "transmitting" (rather than "erasing") information and energy. Finally, in the asymptotic regime, our findings further provide thermodynamic meanings for Holevo-Schumacher-Westmoreland Theorem and a series of strong converse properties as well as no-go theorems.

Figures

Figures reproduced from arXiv: 2502.03603 by the authors.

Figure 1
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