Landauer Principle and Thermodynamics of Computation

Avijit Misra; Goutam Paul; Pritam Chattopadhyay; Tanmoy Pandit

REVIEW 2 major objections 4 minor 300 references

Landauer Principle and Thermodynamics of Computation

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The Landauer bound is the organizing principle of the thermodynamics of computation, and a broad review maps how it changes under finite-time, finite-size, non-Markovian, and nonequilibrium conditions.

desk verdict A broad, useful Landauer review that would benefit from fixing a real misattribution in Sec III.A; no new results, but a serious referee could make it trustworthy. read the letter →

arxiv 2506.10876 v2 pith:WNCCQH5J submitted 2025-06-12 quant-ph cond-mat.stat-mechcs.CCcs.FL

Pritam Chattopadhyay , Avijit Misra , Tanmoy Pandit , Goutam Paul This is my paper · ORCID

classification quant-phcond-mat.stat-mechcs.CCcs.FL

keywords Landauerprincipleboundthermodynamicsofcomputationinformationerasurenon-Markoviandynamicsfinite-timequantumerrorcorrection

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 review sets out to show that the Landauer bound, the minimum heat that must be dissipated to erase one bit of information at temperature $T$, remains the organizing principle for the thermodynamics of computation even when idealized slow, equilibrium erasure is abandoned. It synthesizes recent results that quantify exactly how the bound gains correction terms under finite-time driving, finite-size heat baths, non-Markovian memory effects, and initial system–bath correlations, and it collects the experiments that have approached or tested the bound in both classical and quantum platforms. The review also extends the same thermodynamic logic upward from single-bit erasure to whole models of computation and to error correction, arguing that the energy cost of computation can be expressed in terms of logical and algorithmic information. A reader comes away with a map of which corrections are established, which limits recover the simple $k_{\mathrm{B}}T\ln 2$ floor, and which apparent violations are resolved by counting correlations explicitly.

What carries the argument

The load-bearing object is the sharpened equality form of the Landauer principle, written as $\beta Q = \Delta S + I(S':R') + S(\rho'_R \| \rho_R)$, where $\Delta S$ is the decrease in von Neumann entropy of the erased system, $I(S':R')$ is the mutual information between the final system and reservoir states, and $S(\rho'_R \| \rho_R)$ is the quantum relative entropy measuring the free-energy increase of the bath. This identity carries the argument because it turns the scalar bound $Q \geq k_{\mathrm{B}}T\ln 2$ into a bookkeeping relation: every correction from finite-size baths, correlations, coherence, or non-Markovian dynamics appears as a separately identifiable term. The review then uses this same accounting machinery to derive finite-time bounds, nonequilibrium heat-fluctuation relations, non-Markovian local-violation conditions, and thermodynamic costs of computational models and error-correcting codes.

What would settle it

A single careful experiment or derivation that contradicted the sharpened equality for a finite-size bath, for example erasure heat below $\beta Q = \Delta S + I(S':R') + S(\rho'_R \| \rho_R)$ with an initially thermal, uncorrelated reservoir under unitary joint evolution, would falsify the review's central claim; one could look for this by measuring dissipated heat in a trapped-ion or superconducting erasure protocol while independently measuring system entropy change, system–bath mutual information, and bath relative entropy.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Landauer principle is not a single inequality but a family of exact and approximate statements whose common core is the entropy cost of discarding information. On its own terms, the review argues that the basic bound $Q \geq k_{\mathrm{B}}T\ln 2$ is the infinite-time, infinite-bath limit of a sharper equality in which dissipated heat is decomposed into the system's entropy decrease plus the mutual information built up between system and reservoir plus the relative-entropy free-energy increase of the bath. Every departure from the ideal case adds non-negative corrections, except when initial correlations or non-Markovian memory effects are specifically counted, in which case the apparent bound can be locally lowered or transiently violated. The same framework is carried from erasure to finite-time and nonequilibrium protocols, to computational models such as finite automata and Turing machines, and to classical and quantum error correction, so that the thermodynamic cost of computation is tied to physically meaningful information-theoretic quantities.

Load-bearing premise

The whole review rests on trusting the published results it brings together, especially the equality that puts finite-size corrections on the Landauer bound and the experiments that claim to approach it; if any of those were wrong or mischaracterized, the review's map of where the Landauer principle stands would be misleading.

Editorial extensions

If this is right

Erasing a bit at finite speed, or when the erased state has coherence, costs strictly more than $k_{\mathrm{B}}T\ln 2$, with the excess set by the accuracy of the final state and the total time available.
Finite-size and non-Markovian environments can make the naive Landauer bound appear to fail locally, but the generalized equality shows that the apparent shortfall is carried by system–bath correlations and memory backflow.
The bound extends from binary logic to $N$-valued logic, finite automata, Turing machines, and error correction, so thermodynamic statements about computing can be derived from logical irreversibility rather than from particular hardware.
Experiments on optical tweezers, trapped ions, nuclear magnetic resonance, and superconducting circuits approach or saturate the bound in both classical and quantum regimes, giving operational meaning to the theoretical limits.
Perfect erasure in finite time is possible in principle only if algorithmic or control complexity is allowed to diverge, which moves the ultimate resource cost from thermodynamics to computational complexity.

Reading between the lines

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

The equality form suggests a tunable trade-off: an erasure protocol could store part of the entropy cost in system–bath correlations instead of heat, a design principle the review identifies but does not fully develop.
The non-Markovian local violations imply that memory effects might be harnessed as a resource for lower-dissipation erasure, which would be a natural next step beyond the review's catalog of conditions.
The zero-temperature improved bound implies a specific, testable prediction: as $T \to 0$, the erasure cost depends on the equilibrium heat capacity of the bath and exceeds the trivial $Q \geq 0$ limit.
The review's thermodynamic analysis of algorithms carries an implicit caution for quantum computing: quantum speedups do not automatically translate into energy advantages, since some quantum algorithms show no energy improvement over classical ones in Brownian-computation models.

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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

Colleague—

Quick take: this is explicitly a review, not a research paper, and its value is organizational. It maps the Landauer bound across finite-time, non-equilibrium, non-Markovian, experimental, computational-model, and error-correction territory. For someone entering the field, the bibliography alone is worth something. The coverage of experiments—optical tweezers, trapped ions, NMR, superconducting flux—is reasonably accurate and the authors are honest about what's not covered (gravity, QFT, many-body, biology). No new result is claimed; the one explicit theorem, Eq. (21), is attributed to Li and Vitanyi.

Where it's good: the review is broad without being shallow in most places. Section II gives a fair account of the Reeb–Wolf equality and the generalized bound. Section V is a useful summary of experimental platforms, and Sections VII–VIII connect computation models and error correction to thermodynamics in a way that is genuinely accessible. The future-directions list at the end is sensible.

Soft spots, in order of seriousness.

(1) The stress-test concern is correct. Section III.A states that Reeb and Wolf 'demonstrated that quantum coherence fundamentally alters the thermodynamic cost of erasure' and that coherence in the energy eigenbasis 'necessarily exceeds the classical LB.' That's not in Reeb–Wolf. Their Eq. (2) is basis-independent and has no coherence term; a pure coherent state can be erased unitarily with zero heat. The coherence-dependent excess dissipation is a finite-time result from Miller et al. [22] and Van Vu–Saito [154]. This is not a tiny citation slip: a reader relying on the review will walk away with the wrong attribution for a central result. It should be fixed.

(2) Editorial. Eq. (20)'s conditional Kolmogorov complexity looks reversed (it defines K(ζ|η) but the condition/output in the set are swapped). There are numerous typos and inconsistent notation throughout, which slows reading but doesn't break the review.

(3) The 'comprehensive' claim can't be independently verified from the text; the authors don't re-derive any cited result. That's normal for a review, so I don't weight it heavily.

Verdict: it's a useful entry point, not a landmark. The core mapping is broadly sound; the coherence attribution is a real flaw that should be caught in review. I'd send it to peer review rather than desk reject—the fix is local, and the survey would be worth having once corrected.

Referee Report

2 major / 4 minor

Summary. The paper is a review of the Landauer principle and the thermodynamics of computation. It surveys the generalized Landauer bound, finite-time and non-equilibrium erasure, non-Markovian corrections, experimental tests, reversible and Brownian computers, the thermodynamic costs of finite automata and Turing machines, error correction, and additional topics such as electronic circuits and algorithmic energy costs. The authors state their scope explicitly and list several areas they do not cover. The central claim is that the review provides an accurate and comprehensive map of recent progress on the Landauer bound and its role in computation.

Significance. If accurate, this review would be a valuable reference work: it connects the core Landauer bound to a broad literature on open quantum systems, finite-time thermodynamics, experiments, computational models, and error correction. The paper is honest about its scope and includes recent experimental demonstrations on optical tweezers, ABEL traps, trapped ions, NMR, and superconducting flux logic. Its value, however, depends on the correctness of attributions and displayed formulas; the issues below affect that reliability. The paper does not present new derivations, so its contribution is synthetic rather than original.

major comments (2)

[Sec. III.A] The paragraph beginning 'Reeb and Wolf's seminal work [116]' states that Reeb and Wolf demonstrated that quantum coherence fundamentally alters the thermodynamic cost of erasure and that coherence in the energy eigenbasis necessarily raises dissipation above the classical Landauer bound. This is inconsistent with Eq. (2), βQ = ΔS + I(S′:R′) + S(ρ′_R||ρ_R), which contains no coherence-dependent term and is invariant under a common unitary change of basis. The Reeb–Wolf bound applies to all initial states, including coherent ones, and the coherence-induced excess dissipation is a property of the finite-time protocols studied in [22] and [154], not of the general Reeb–Wolf equality. Since Sec. III.A is a core technical section of the review, this misattribution should be corrected and the coherence effect should be attributed to the relevant finite-time results.
[Sec. VII.A, Eq. (20)] Equation (20) defines the conditional Kolmogorov complexity K(ζ|η) via the condition ξ_i(\bar p_c, \bar ζ) = η, and the following sentence says the quantity measures the effort required to transform ζ into η. Both the equation and the sentence reverse the input/output roles relative to the stated object, which is 'computing ζ from η'. As written, the condition should involve η as the input and ζ as the output, i.e., ξ_i(\bar p_c, \bar η) = ζ, or the notation K(η|ζ) should be used. This reversal propagates into Theorem 1, Eq. (21), and the Zurek bound, Eq. (22), and should be fixed before those thermodynamic-cost formulas can be evaluated.

minor comments (4)

[Sec. II] The phrase 'Boltzman constant' should read 'Boltzmann constant'.
[Sec. III.C, Eqs. (17)-(18)] Equation (17) has an unbalanced closing parenthesis in the term − S(ρ_En||ρ_th_En)), and the condition in Eq. (18) repeats this issue; please restate the formula exactly.
[Sec. III.C] The text contains 'Ladauer-like principle' for 'Landauer-like principle' and similar spacing/typographical artifacts in headings such as 'V ALIDA TION' and 'F or'; these should be cleaned throughout.
[Sec. V.A] The sentence reporting Bérut et al. [219] says the asymptotic limit is approximately (0.13–0.49)kBT while the full-erasure limit is 0.69kBT; this is unclear without specifying which quantity each number refers to, and should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a survey of external results, with only peripheral self-citations and no fitted parameter renamed as a prediction.

full rationale

This paper is a review article. Its central content is a synthesis of externally published bounds and experiments; it does not fit parameters to data and then predict closely related quantities, and it does not derive a new central result from its own inputs. The only presented theorem, Eq. (21) for thermodynamic cost in terms of Kolmogorov complexity, is explicitly attributed to [279] (Li and Vitanyi), not derived anew by the authors. The self-citations that appear, e.g., [129,130] by Misra and [329,330] by Pandit, are used in background discussions of open quantum dynamics and periodically driven systems; they are not load-bearing for the review's claims about the status of the Landauer bound. The Sec. III.A statement that Reeb and Wolf demonstrated a coherence-dependent correction to the Landauer bound is inconsistent with the basis-independent equality in Eq. (2), but that is a correctness or attribution concern, not a circularity: the review does not define its conclusion into its premises. The experimental and theoretical results surveyed are external and independently checkable, so the review is self-contained as a survey and does not reduce its claims to self-citation or to a fitted input.

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

The paper is a review, so the ledger contains no original fitted parameters or new postulated entities. The summary relies on standard domain assumptions from the cited literature: thermal or uncorrelated reservoirs, CPTP dynamics, fluctuation theorems, and the Li-Vitanyi axiomatic definition of thermodynamic cost.

assumptions (4)

domain assumption The reviewed Landauer-bound results assume an initially thermal, uncorrelated reservoir and unitary joint system-reservoir evolution (Reeb-Wolf assumptions).
Invoked in Section II for Eq. (2) and extended in Section III to finite-time and non-Markovian settings.
domain assumption Thermodynamic cost of a computation is determined by counting irreversibly provided or deleted bits, as formalized by Li and Vitanyi [279] (Axioms 1-4).
Underlies Theorem 1 in Section VII.A and the Kolmogorov-complexity cost formula.
domain assumption Fluctuation theorems and full counting statistics apply to quantum erasure protocols.
Used in Section III.B to derive heat-fluctuation relations, e.g., Eqs. (6)-(8) and (11)-(12).
standard math Relative entropy is non-increasing under CPTP maps.
Used in Section III.C to derive the open-system Landauer inequality Eq. (15).

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Cite this review

Pith. "Pith review of Landauer Principle and Thermodynamics of Computation." pith.science (2026). https://pith.science/paper/WNCCQH5J

@misc{pith2026250610876,
  author       = {Pith},
  title        = {Pith review of: Landauer Principle and Thermodynamics of Computation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WNCCQH5J}},
  note         = {Machine review of arXiv:2506.10876}
}

read the original abstract

According to the Landauer principle, any logically irreversible process accompanies entropy production, which results in heat dissipation in the environment. Erasing of information, one of the primary logically irreversible processes, has a lower bound on heat dissipated into the environment, called the Landauer bound (LB). However, the practical erasure processes dissipate much more heat than the LB. Recently, there have been a few experimental investigations to reach this bound both in the classical and quantum domains. There has also been a spate of activities to enquire about this LB in finite time, with finite-size heat baths, non-Markovian and nonequilibrium environments in the quantum regime, where the effects of fluctuations and correlation of the systems with the bath can no longer be ignored. This article provides a comprehensive review of the recent progress on the Landauer bound, which serves as a fundamental principle in the thermodynamics of computation. We also provide a perspective for future endeavors in these directions. Furthermore, we review the recent explorations toward establishing energetic bounds of a computational process. We also discuss the thermodynamic aspects of error correction, which is an indispensable part of information processing and computations. In doing so, we briefly discuss the basics of these fields to provide a complete picture.

Figures

Figures reproduced from arXiv: 2506.10876 by the authors.

**Figure 2.** FIG. 2. The schematic of the process with a three-level system [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3. The system [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: FIG. 4. The system [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5. Schematic of the qubit model with a Brownian par [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: FIG. 6. The erasure protocol that is considered in the exper [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: FIG. 7. A schematic of a BLC proposed by Toffoli. The key [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: FIG. 8. A schematic of the BWC: a) without energy trap, [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: FIG. 9. Transition diagram of the FA-controlled toll gate. [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: FIG. 10. A schematic representation of a TM includes an [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 11.** Figure 11: FIG. 11. A schematic representation of the domain of the [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

**Figure 12.** Figure 12: FIG. 12. A schematic of the [[ [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]

**Figure 13.** Figure 13: FIG. 13. A schematic of the reversible cycle of the classical [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

**Figure 14.** Figure 14: FIG. 14. A schematic of the thought experiment is shown. [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]

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