The Locality Gap: A Thermodynamic Law for Objective Facts

Maxim V. Churilov

REVIEW 2 major objections 3 minor 46 references

The Locality Gap: A Thermodynamic Law for Objective Facts

T0 review · 2 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Redundant records of an event create an exact thermodynamic work asymmetry between local and global controllers, quantified by the source-loss law…

desk verdict A clean, exact thermodynamic double difference for objective records—real but narrower than the title, with one fixable typo and one load-bearing catalytic-side-information assumption. read the letter →

arxiv 2608.05753 v1 pith:J7LNTWX3 submitted 2026-08-06 quant-ph gr-qc

Maxim V. Churilov This is my paper · ORCID

classification quant-phgr-qc PACS 03.65.Ta05.70.-a

keywords quantumDarwinismobjectivefactsthermodynamicfactualitychargelocalitygaptotalcorrelationsecretsharingno-broadcastingLandauererasure

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 establish that redundant objective records---the same event value imprinted in many environment fragments---are not thermodynamically inert copies but a measurable work asymmetry between controllers with different access to the records. It derives an exact source-loss law: the reversible isothermal erasure penalty of a partition-local controller relative to a global controller, with the event label $X$ supplied as catalytic classical side information, equals $k_{\mathrm{B}}T$ times the excess of summed blockwise mutual informations over the joint mutual information. That single signed quantity, called the thermodynamic factuality charge, ranges from $-k_{\mathrm{B}}T H(X)$ for perfect secret sharing through zero for unique storage to $(|\mathcal{P}|-1)k_{\mathrm{B}}T H(X)$ for perfect broadcast objectivity. If correct, it gives objectivity a thermodynamic meaning---a thermodynamically public fact is a classical variable independently decodable from several record blocks---and yields quantitative Landauer-scale bounds on the erasure cost of redundant records.

What carries the argument

The carrying object is the thermodynamic locality gap: the difference between the minimum work to reset a record cloud by erasing each block independently and the minimum work to reset the whole cloud with global reversible control, $G^{\varnothing}_{\mathcal{P}}=k_{\mathrm{B}}T\,T_{\mathcal{P}}(F)$, where $T_{\mathcal{P}}(F)=\sum_{B\in\mathcal{P}}S(F_B)-S(F)$ is the partition total correlation. The source-loss law is obtained by repeating both resets with the event label $X$ supplied as catalytic classical side information, giving $G^{X}_{\mathcal{P}}=k_{\mathrm{B}}T\,T_{\mathcal{P}}(F|X)$, and subtracting: $\mathcal{F}^{T}_{\mathcal{P}}(X)=G^{\varnothing}_{\mathcal{P}}-G^{X}_{\mathcal{P}}=k_{\mathrm{B}}T[\sum_{B\in\mathcal{P}}I(X{:}F_B)-I(X{:}F)]$. What makes the construction work is the subtraction: it cancels the nonnegative conditional background $T_{\mathcal{P}}(F|X)$ and isolates the signed, event-specific architecture charge $A^{\mathcal{P}}_{X}(F)$, turning a correlation measure into a quantity whose endpoints are broadcast objectivity and secret sharing.

What would settle it

Prepare a three-qubit register in the classical mixture of $|000\rangle$ and $|111\rangle$, and measure the quasistatic isothermal work to reset the register per qubit versus globally, with and without the classical bit $X$ available as side information; the theory predicts the double difference is exactly $2k_{\mathrm{B}}T\ln 2$ for singleton blocks, and any measured dependence on the cost of distributing $X$ to the separate resetters would show that the side-information channel, not the record architecture, is carrying the effect.

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Extended reading notes

Core claim

The central claim is the source-loss law of factuality, Eq. (19): for a classical--quantum record cloud $\rho_{XF}=\sum_x p_x |x\rangle\langle x|\otimes \rho_{F|x}$ and a partition $\mathcal{P}$ into independently controllable blocks, the difference between the unconditioned locality gap $G^{\varnothing}_{\mathcal{P}}$ and the source-assisted gap $G^{X}_{\mathcal{P}}$ is exactly $\mathcal{F}^{T}_{\mathcal{P}}(X)=k_{\mathrm{B}}T[\sum_{B\in\mathcal{P}} I(X{:}F_B)-I(X{:}F)]$. The author proves this from reversible isothermal reset free-energy differences with an additive block Hamiltonian, using the decomposition $T_{\mathcal{P}}(F)=A^{\mathcal{P}}_{X}(F)+T_{\mathcal{P}}(F|X)$ that splits total correlation into event-attributable and conditional-background parts. The charge obeys sharp bounds $-k_{\mathrm{B}}T H(X)\leq \mathcal{F}^{T}_{\mathcal{P}}(X)\leq (|\mathcal{P}|-1)k_{\mathrm{B}}T H(X)$ with endpoint rigidity: near the broadcast ceiling every block must be an almost complete record, while near the secret-sharing floor every block is almost ignorant even though the union is almost complete. The paper also derives Landauer--Darwin bounds from independently decodable records, exact growth and partition-refinement laws, a continuity certificate, and closed formulas for noisy classical and quantum collision models, and it shows that maximal positive factuality requires orthogonal conditional record states, which by no-broadcasting is possible only for a classical commuting pointer variable.

Load-bearing premise

The load-bearing premise is that the event label $X$ is catalytic classical side information whose preparation, copying, and distribution cost is identical for partition-local and global controllers, so that cost cancels in the double difference; the law does not follow if fanning $X$ out to separate blocks is more expensive than handing it to one global controller, or if $X$ cannot be copied exactly because the source is quantum and noncommuting.

Editorial extensions

If this is right

If the law holds, a cloud of $R$ perfect copies of one bit has a locality gap of $(R-1)k_{\mathrm{B}}T\ln 2$, so every independently controllable record after the first costs exactly one Landauer unit of extra reversible work to a partition-local controller.
Independent decodability of $R$ fragments with information deficit $\delta$ forces the factuality charge to be at least $k_{\mathrm{B}}T[R(1-\delta)-1]H(X)$, a quantitative thermodynamic version of Darwinian redundancy.
Perfect secret sharing saturates the lower endpoint at $-k_{\mathrm{B}}TH(X)$ and unique storage gives zero, although zero can also arise from exact cancellation between redundant and synergistic structure.
Maximal positive factuality forces orthogonal conditional record states on every block, so the saturating variable must be classical; by no-broadcasting, an arbitrary noncommuting quantum source cannot supply the catalytic side-information baseline in the same way.
The theory does not modify quantum mechanics or introduce collapse; it predicts a four-work double difference that can be measured in quasistatic isothermal erasure experiments on prepared record clouds.

Reading between the lines

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

An extension the author leaves open: the factuality charge could serve as a thermodynamic witness for the emergence of classicality in mesoscopic devices, since it becomes large only when records acquire orthogonal conditional supports; measuring it across a decoherence crossover may track broadcast structure without full tomography.
The no-broadcasting connection suggests replacing the classical label $X$ by a quantum reference and using Holevo quantities of restricted ensembles; the resulting double difference would quantify the thermodynamic penalty of noncommuting records and might link the factuality charge to quantum discord.
Because the identity relies on identical side-information cost across the two architectures, an experimental protocol should separately calibrate the cost of distributing $X$ to block controllers; if that cost is varied and the measured double difference shifts, the source-loss interpretation rather than the algebraic identity is what is being tested.
The causal-sealing barrier, which the author flags as conjectural, could be tested in lattice systems with Lieb--Robinson cones by checking whether the infimum over causally reachable partitions of $\mathcal{F}^{T}_{\mathcal{P}}(X)$ stays positive; that would give a concrete order parameter for apparent irreversibility.

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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

Short version: this is a real result, but it is narrower than the title suggests, and one of its displayed equations has a typo. The central identity is an elementary double difference of free energies, and the paper is honest that the unconditioned half is known work-deficit. What is new—the source-conditioned subtraction, the endpoint-rigidity identities, the Landauer–Darwin bounds, and the closed-form collision models—is correctly derived and clearly separated from conjecture. I checked Appendix B; the algebra is exact, and the numerical checks are a good guard against implementation errors.

The soft spots are real but not fatal. The load-bearing assumption is in Section III.C: the event label X is supplied catalytically, with the same set of copies to both local and global controllers. The paper asserts this costs the same across comparisons but does not give a physically concrete protocol that equalizes fanout and distribution costs. If a partition-local controller needs m copies while the global one needs one, the double difference picks up an uncancelled (m−1)γ, and Eq. (19) fails. For a classical source this is at least plausibly implementable with a single classical control line, but the paper's wording "must be kept identical" is a condition, not a procedure. The law does not extend to noncommuting quantum sources; the paper's own Proposition 6 says exactly that, so that is not a flaw.

Second, Eq. (28) as printed is wrong. For perfect broadcast with R blocks, the right-hand side should be (R−1)H(X) in nats, not (R−1)+H(X); as written it contradicts Eq. (32). The surrounding text mentions a "positive part," so this is likely a typesetting issue rather than a conceptual one, but it needs fixing. Third, the reproducibility appendix says an archive exists but gives no URL or hash. Minor.

The citation pattern is fair: the paper explicitly cedes Eq. (15) to earlier work-deficit results, and there is no self-citation. No fitted parameters appear anywhere.

Who should read this: quantum-foundations and quantum-thermodynamics people working on quantum Darwinism. It gives a measurable coordinate (the thermodynamic record number) that cleanly separates broadcast, unique storage, and secret sharing. It is not a paradigm shift, but it is a useful diagnostic.

Recommendation: send it to a serious referee. With Eq. (28) fixed and a more careful operational account of the side-information cost, it would be publishable in a good quantum-information journal.

Referee Report

2 major / 3 minor

Summary. The manuscript studies the thermodynamics of redundant records in quantum Darwinism. For a classical event label X with a cq record cloud F and a partition P into independently controllable blocks, it compares four reversible isothermal reset tasks: global versus partition-local erasure, each with and without X supplied as catalytic classical side information. The main result is the source-loss law F^T_P(X)=k_B T [sum_B I(X:F_B)-I(X:F)], obtained by subtracting the source-assisted locality gap from the unassisted one. The paper proves sharp bounds identifying broadcast objectivity, unique storage, and secret sharing as the two endpoints and one interior point of a single signed charge, derives Landauer–Darwin lower bounds, endpoint-rigidity certificates, growth and refinement chain rules, interaction-Hamiltonian corrections with a Pinsker refinement, continuity bounds, closed-form classical and quantum collision models, and an experimental protocol. A claim map in the introduction separates exact results from the explicitly conjectural pointer-selection proposal.

Significance. The central contribution is a clean, parameter-free thermodynamic identity that assigns an exact work asymmetry to the architecture of redundant evidence. If the operational assumptions are accepted, Eq. (19) is a genuine law of the stated reset model, with concrete predictions (Eqs. (42), (45), (47)) and no fitted constants. The paper is unusually careful about the status of its claims: the claim map labels pointer selection and causal sealing as conjectural, the limitations section is explicit about the catalytic side-information idealization, and all algebraic identities were stress-tested numerically with a reproducible archive. The endpoint-rigidity certificate (Theorem 5) and the interaction corrections (Theorems 8–9) go beyond a bare information identity and strengthen the paper. The main open risk is operational rather than algebraic: whether the side-information distribution cost can be made to cancel as the manuscript asserts.

major comments (2)

[Section IV, Eq. (28), and Appendix C] The displayed inequality G^∅_P/k_B T >= [R(1−δ)−1] + H(X) is not implied by the accompanying proof. From T_P(F) = A^P_X(F) + T_P(F|X) with T_P(F|X) >= 0 and T_P(F) >= 0, one obtains only G^∅_P/k_B T >= max{0, [R(1−δ)−1]H(X)}; the argument in Appendix C ('T = A + T(·|X) >= A and T >= 0') never produces the additive H(X) term. The sentence about the positive part suggests the intended bound may be max{0, [R(1−δ)−1]H(X)}. Please correct Eq. (28) and the appendix proof so that the displayed bound follows from the stated inequalities.
[Section III.C, Eqs. (16)–(18), and Section XI] The source-loss law (19) rests on the assumption that the same set of classical control copies is supplied to the partition-local and global controllers and that copying, fanout, or distribution of X carries no differential work cost. The manuscript asserts this but does not supply a protocol that realizes it. If supplying X to m separate block controllers requires m copies while the global controller needs one, and if each copy or distribution step costs gamma, the double difference contains an uncanceled (m−1)gamma term and Eq. (19) fails. Because this is the central claim, please either exhibit a concrete reversible classical fanout protocol with the side-channel cost included and shown to cancel, or state the law with an explicit side-channel correction term and the conditions under which it vanishes.

minor comments (3)

[Throughout] The notation for the main quantity is inconsistent: Eq. (19) writes F^T_P(X:F), while Eq. (56) and much of the text write F^T_P(X). Choose one notation and define it once near Eq. (1).
[Figure 2] The caption uses ΔW_P/(k_B T ln 2) while the text and Theorem 2 use G^∅_P; unify the symbols in the figure and the main text.
[Appendix C] After correcting Eq. (28), the final sentence of the appendix proof should state explicitly how the corrected inequality follows from T = A + T(·|X) and T >= 0, rather than citing the uncorrected equation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central law is a derived consequence of explicit free-energy definitions and an information-theoretic identity, with no fitted parameters or self-citation chain.

full rationale

The derivation chain is self-contained. Equations (13)-(14) define the partition-local and global reset works; subtracting them and using the block-additive Hamiltonian plus product blank gives the locality gap G∅_P = k_B T T_P(F) in Eq. (15), with the energy terms cancelling exactly as shown in Appendix B (Eqs. B1-B3). The source-assisted works in Eqs. (16)-(17) yield G^X_P = k_B T T_P(F|X) in Eq. (18) by the same calculation conditioned on x. Proposition 1 (Eq. 7), proved in Appendix A, is the exact identity T_P(F) = A_X(F) + T_P(F|X), where A_X(F) = sum_B I(X:F_B) - I(X:F). Subtracting Eq. (18) from Eq. (15) and using Eq. (7) gives the source-loss law, Eq. (19). This is a deduction from the stated operational definitions, not a fitted prediction: no free parameter is introduced, no quantity is renamed as a prediction, and no load-bearing result is imported from the author's own prior work. The paper explicitly disclaims novelty for the unconditional gap, attributes it to prior correlation-free-energy results, and marks the conjectural pointer-selection proposal as not used in any proof. The catalytic-side-information assumption in Section III.C and Section XI is an explicitly stated validity condition: the law holds when the cost of supplying the classical label is held fixed between local and global comparisons. That is an assumption about the physical protocol, and the paper itself flags the quantum no-broadcasting limitation in Proposition 6. An unverified or protocol-dependent assumption is a correctness risk, not circularity, and it does not make the algebraic derivation reduce to its own conclusion. All displayed identities were numerically checked against independent classical and quantum ensembles, but the proofs do not depend on those checks. Therefore the central claim has independent content beyond its definitions, and no specific circular step can be quoted.

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

The central law has no fitted constants. The only inputs are the Landauer free-energy principle and the stated control model, both explicit. The numerical stress test guards against implementation errors and is not used as evidence for unproved equalities.

assumptions (6)

domain assumption Reversible isothermal minimum work equals free-energy difference
Used throughout Section III to convert reset tasks into free-energy differences; standard Landauer principle in the quasistatic limit.
ad hoc to paper Block-additive record Hamiltonian and product blank
Eq. (11) and Section III.A make the energy terms cancel, leaving the entropy-only gap. Interacting Hamiltonians are treated separately in Theorem 8.
ad hoc to paper Partition-local reset forbids cross-block gates, communication, shared memory, feed-forward, and hidden shared resources
Section III.A; if relaxed, the work gap interpolates toward the global value and Eq. (19) loses its stated sharpness.
ad hoc to paper Source X is classical, supplied catalytically, with identical preparation cost in local and global comparisons
Section III.C Eqs. (16)-(18); this makes the double difference well defined and is also required for the no-broadcasting-based endpoint argument.
domain assumption Finite-dimensional cq state form rho_XF = sum_x p_x |x><x| tensor rho_F|x
Eq. (2); all information inequalities and models assume this form.
standard math Data processing, Fano, Audenaert-Fannes, and quantum Pinsker inequalities
Used in Section IV, Appendix C, and Theorems 7 and 9; background results in information theory.

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Cite this review

Pith. "Pith review of The Locality Gap: A Thermodynamic Law for Objective Facts." pith.science (2026). https://pith.science/paper/J7LNTWX3

@misc{pith2026260805753,
  author       = {Pith},
  title        = {Pith review of: The Locality Gap: A Thermodynamic Law for Objective Facts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J7LNTWX3}},
  note         = {Machine review of arXiv:2608.05753}
}

abstract

Objective facts in quantum Darwinism are values recorded redundantly in many independently accessible fragments of the environment. Redundancy does not create additional logical information, so its thermodynamic meaning has remained unclear. We show that it creates an exact work asymmetry between controllers with different access architectures. For a record cloud $F$ and a partition $\mathcal{P}$ into blocks that cannot be jointly controlled, the reversible isothermal erasure penalty relative to a global controller is $k_{\mathrm{B}}T$ times the partition total correlation. If the source event $X$ is supplied as catalytic classical side information, subtracting the corresponding penalty gives the source-loss law $\mathcal{F}_{\mathcal{P}}^{T}=k_{\mathrm{B}}T\left[\sum_{B\in\mathcal{P}}I(X{:}F_B)-I(X{:}F)\right]$. We call this the thermodynamic factuality charge. It ranges from $-k_{\mathrm{B}}T H(X)$ for perfect secret sharing to $(|\mathcal{P}|-1)k_{\mathrm{B}}T H(X)$ for perfect broadcast objectivity. Its normalized form defines a thermodynamic record number between $0$ and $|\mathcal{P}|$. We derive Landauer--Darwin bounds, endpoint-rigidity certificates, exact growth and partition-refinement laws, spectrum-broadcast saturation, and closed formulas for noisy classical and quantum collision models. The theory does not modify quantum mechanics or posit objective collapse; it identifies the thermodynamic resource generated by redundant records under restricted control.

Figures

Figures reproduced from arXiv: 2608.05753 by the authors.

**Figure 2.** FIG. 2. Controller-scale flow for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. The normalized factuality charge distinguishes redun [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Exact factuality/locality gap for independent binary [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Exact source-loss charge for the pure-state collision [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Exact residual branch-coherence factor after tracing [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Reference graph

Works this paper leans on

46 extracted references · 41 canonical work pages

[1]

Consider two statistically independent record systems,(X, F)and (Y, G), with product state and the disjoint-union partition P⊔Q

Invariance and additivity Local isometries preserve every entropy appearing in equation (6); hence they preserveAP X and FT P. Consider two statistically independent record systems,(X, F)and (Y, G), with product state and the disjoint-union partition P⊔Q . Mutual information is additive under tensor products, giving AP⊔Q XY (FG) =AP X(F) +AQ Y (G).(A8) Th...

work page
[2]

Binary-symmetric record cloud For R conditionally independent binary-symmetric records, letK= ∑R j=1Fj. With uniformX, P(K=k) = 1 2 (R k )[ ϵk(1−ϵ) R−k + (1−ϵ) kϵR−k ] ,(E1) qk≡P(X= 1|K=k) = [ 1 + ( ϵ 1−ϵ )2k−R]−1 .(E2) Therefore H(X|F 1:R) = R∑ k=0 P(K=k)h 2(qk),(E3) which inserted into equation (45) yields the plotted curves

work page
[3]

A single record therefore has entropyh2[(1+ c)/2], whereas the joint conditional states have overlap cR and the cloud entropy ish2[(1 +cR)/2]

Pure-state collision cloud For two equal-prior pure states|ψ0⟩,|ψ 1⟩ with overlap q, the nonzero eigenvalues of 1 2 (|ψ0⟩⟨ψ0|+|ψ 1⟩⟨ψ1|)(E4) are(1±q)/2. A single record therefore has entropyh2[(1+ c)/2], whereas the joint conditional states have overlap cR and the cloud entropy ish2[(1 +cR)/2]. Substitution into equation (4) proves equation (47)

work page
[4]

Conditioning on the parity fixes one global constraint: S(F|X) = (R− 1) ln 2, while each conditional marginal remains fair

Parity cloud Unconditionally, theR fragments are independent fair bits, so S(Fj) = ln 2, S(F) = Rln 2, andT (F) = 0. Conditioning on the parity fixes one global constraint: S(F|X) = (R− 1) ln 2, while each conditional marginal remains fair. ThusT (F|X) = ln 2and equation (7) gives A=−ln 2. Appendix F: Numerical methods and reproducibility Classical trials...

work page
[5]

W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of Modern Physics75, 715 (2003)

2003
[6]

W. H. Zurek, Quantum darwinism, Nature Physics5, 181 (2009). 13

work page 2009
[7]

Blume-Kohout and W

R. Blume-Kohout and W. H. Zurek, Quantum darwinism: En- tanglement, branches, and the emergent classicality of redun- dantly stored quantum information, Physical Review A73, 062310 (2006)

work page 2006
[8]

Blume-Kohout and W

R. Blume-Kohout and W. H. Zurek, Quantum darwinism in quantum brownian motion, Physical Review Letters101, 240405 (2008)

work page 2008

Show all 46 references

[9]

Zwolak, H

M. Zwolak, H. T. Quan, and W. H. Zurek, Quantum darwinism in a hazy environment, Physical Review Letters103, 110402 (2009)

2009
[10]

Zwolak, C

M. Zwolak, C. J. Riedel, and W. H. Zurek, Amplification, re- dundancy, and quantum chernoff information, Physical Review Letters112, 140406 (2014)

2014
[11]

J. K. Korbicz, P. Horodecki, and R. Horodecki, Objectivity in a noisy photonic environment through quantum state information broadcasting, Physical Review Letters112, 120402 (2014)

2014
[12]

F. G. S. L. Brandão, M. Piani, and P. Horodecki, Generic emergence of classical features in quantum darwinism, Nature Communications6, 7908 (2015)

2015
[13]

T. P. Le and A. Olaya-Castro, Strong quantum darwinism and strong independence are equivalent to spectrum broadcast structure, Physical Review Letters122, 010403 (2019)

2019
[14]

J. K. Korbicz, Roads to objectivity: Quantum darwinism, spec- trum broadcast structures, and strong quantum darwinism—a review, Quantum5, 571 (2021)

2021
[15]

Touil, B

A. Touil, B. Yan, D. Girolami, S. Deffner, and W. H. Zurek, Eavesdropping on the decohering environment: Quantum dar- winism, amplification, and the origin of objective classical reality, Physical Review Letters128, 010401 (2022)

2022
[16]

Touil, F

A. Touil, F. Anza, S. Deffner, and J. P. Crutchfield, Branch- ing states as the emergent structure of a quantum universe, Quantum8, 1494 (2024)

2024
[17]

Doucet and S

E. Doucet and S. Deffner, Classifying two-body hamiltonians for quantum darwinism, Physical Review X14, 041064 (2024)

2024
[18]

Landauer, Irreversibility and heat generation in the com- puting process, IBM Journal of Research and Development5, 183 (1961)

R. Landauer, Irreversibility and heat generation in the com- puting process, IBM Journal of Research and Development5, 183 (1961)

1961
[19]

C. H. Bennett, The thermodynamics of computation—a review, International Journal of Theoretical Physics21, 905 (1982)

1982
[20]

C. H. Bennett, Notes on landauer’s principle, reversible compu- tation, and maxwell’s demon, Studies in History and Philosophy of Modern Physics34, 501 (2003)

2003
[21]

Sagawa and M

T. Sagawa and M. Ueda, Minimal energy cost for thermody- namic information processing: Measurement and information erasure, Physical Review Letters102, 250602 (2009)

2009
[22]

J. M. R. Parrondo, J. M. Horowitz, and T. Sagawa, Thermo- dynamics of information, Nature Physics11, 131 (2015)

2015
[23]

Goold, M

J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, The role of quantum information in thermodynamics—a topical review, Journal of Physics A: Mathematical and Theoretical 49, 143001 (2016)

2016
[24]

Reeb and M

D. Reeb and M. M. Wolf, An improved landauer principle with finite-size corrections, New Journal of Physics16, 103011 (2014)

2014
[25]

del Rio, J

L. del Rio, J. Åberg, R. Renner, O. Dahlsten, and V. Vedral, The thermodynamic meaning of negative entropy, Nature474, 61 (2011)

2011
[26]

Faist, F

P. Faist, F. Dupuis, J. Oppenheim, and R. Renner, The minimal work cost of information processing, Nature Communications 6, 7669 (2015)

2015
[27]

Faist and R

P. Faist and R. Renner, Fundamental work cost of quantum processes, Physical Review X8, 021011 (2018)

2018
[28]

Oppenheim, M

J. Oppenheim, M. Horodecki, P. Horodecki, and R. Horodecki, Thermodynamical approach to quantifying quantum correla- tions, Physical Review Letters89, 180402 (2002)

2002
[29]

Groisman, S

B. Groisman, S. Popescu, and A. Winter, Quantum, classical, and total amount of correlations in a quantum state, Physical Review A72, 032317 (2005)

2005
[30]

Horodecki, P

M. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim, A.Sen(De), U.Sen,andB.Synak-Radtke,Localversusnonlocal information in quantum-information theory: Formalism and phenomena, Physical Review A71, 062307 (2005)

2005
[31]

A. B. Boyd, D. Mandal, and J. P. Crutchfield, Thermodynamics of modularity: Structural costs beyond the landauer bound, Physical Review X8, 031036 (2018)

2018
[32]

R. Guan, J. Liu, and J.-H. Jiang, Global-local duality of ener- getic control cost in multipartite quantum correlated systems, Physical Review Research7, L022054 (2025)

2025
[33]

C. L. Latune and C. Elouard, A thermodynamically consis- tent approach to the energy costs of quantum measurements, Quantum9, 1614 (2025)

2025
[34]

Munson, N

A. Munson, N. B. T. Kothakonda, J. Haferkamp, N. Yunger Halpern, J. Eisert, and P. Faist, Complexity- constrained quantum thermodynamics, PRX Quantum6, 010346 (2025)

2025
[35]

Badhani, D

H. Badhani, D. G. S., S. Choudhary, V. Anand, and S. Das, Era- sure cost of a quantum process: A thermodynamic meaning of the dynamical min-entropy, Quantum Science and Technology 10.1088/2058-9565/ae34e2 (2026), arXiv:2506.05307 [quant-ph]

2026 arXiv
[36]

Watanabe, Information theoretical analysis of multivariate correlation, IBM Journal of Research and Development4, 66 (1960)

S. Watanabe, Information theoretical analysis of multivariate correlation, IBM Journal of Research and Development4, 66 (1960)

1960
[37]

F. E. Rosas, P. A. M. Mediano, M. Gastpar, and H. J. Jensen, Quantifying high-order interdependencies via multivariate ex- tensions of the mutual information, Physical Review E100, 032305 (2019)

2019
[38]

D. A. Chisholm, L. Innocenti, and G. M. Palma, Importance of using the averaged mutual information when quantifying quantum objectivity, Physical Review A110, 012218 (2024)

2024
[39]

T. M. Cover and J. A. Thomas,Elements of Information Theory, 2nd ed. (Wiley, Hoboken, NJ, 2006)

2006
[40]

Barnum, C

H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schu- macher, Noncommuting mixed states cannot be broadcast, Physical Review Letters76, 2818 (1996)

1996
[41]

Bérut, A

A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillen- schneider, and E. Lutz, Experimental verification of landauer’s principle linking information and thermodynamics, Nature483, 187 (2012)

2012
[42]

J. Hong, B. Lambson, S. Dhuey, and J. Bokor, Experimental test of landauer’s principle in single-bit operations on nano- magnetic memory bits, Science Advances2, e1501492 (2016)

2016
[43]

Chiribella, F

G. Chiribella, F. Meng, R. Renner, and M.-H. Yung, The nonequilibrium cost of accurate information processing, Nature Communications13, 7155 (2022)

2022
[44]

Unden, D

T. Unden, D. Louzon, M. Zwolak, W. H. Zurek, and F. Jelezko, Revealing the emergence of classicality using nitrogen-vacancy centers, Physical Review Letters123, 140402 (2019)

2019
[45]

E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Communications in Mathematical Physics28, 251 (1972)

1972
[46]

G. J. Milburn and S. Shrapnel, Classical and quantum causal interventions, Entropy20, 687 (2018)

2018

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Consider two statistically independent record systems,(X, F)and (Y, G), with product state and the disjoint-union partition P⊔Q

Invariance and additivity Local isometries preserve every entropy appearing in equation (6); hence they preserveAP X and FT P. Consider two statistically independent record systems,(X, F)and (Y, G), with product state and the disjoint-union partition P⊔Q . Mutual information is additive under tensor products, giving AP⊔Q XY (FG) =AP X(F) +AQ Y (G).(A8) Th...

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[2] [2]

Binary-symmetric record cloud For R conditionally independent binary-symmetric records, letK= ∑R j=1Fj. With uniformX, P(K=k) = 1 2 (R k )[ ϵk(1−ϵ) R−k + (1−ϵ) kϵR−k ] ,(E1) qk≡P(X= 1|K=k) = [ 1 + ( ϵ 1−ϵ )2k−R]−1 .(E2) Therefore H(X|F 1:R) = R∑ k=0 P(K=k)h 2(qk),(E3) which inserted into equation (45) yields the plotted curves

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[3] [3]

A single record therefore has entropyh2[(1+ c)/2], whereas the joint conditional states have overlap cR and the cloud entropy ish2[(1 +cR)/2]

Pure-state collision cloud For two equal-prior pure states|ψ0⟩,|ψ 1⟩ with overlap q, the nonzero eigenvalues of 1 2 (|ψ0⟩⟨ψ0|+|ψ 1⟩⟨ψ1|)(E4) are(1±q)/2. A single record therefore has entropyh2[(1+ c)/2], whereas the joint conditional states have overlap cR and the cloud entropy ish2[(1 +cR)/2]. Substitution into equation (4) proves equation (47)

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[4] [4]

Conditioning on the parity fixes one global constraint: S(F|X) = (R− 1) ln 2, while each conditional marginal remains fair

Parity cloud Unconditionally, theR fragments are independent fair bits, so S(Fj) = ln 2, S(F) = Rln 2, andT (F) = 0. Conditioning on the parity fixes one global constraint: S(F|X) = (R− 1) ln 2, while each conditional marginal remains fair. ThusT (F|X) = ln 2and equation (7) gives A=−ln 2. Appendix F: Numerical methods and reproducibility Classical trials...

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[5] [5]

W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of Modern Physics75, 715 (2003)

2003

[6] [6]

W. H. Zurek, Quantum darwinism, Nature Physics5, 181 (2009). 13

work page 2009

[7] [7]

Blume-Kohout and W

R. Blume-Kohout and W. H. Zurek, Quantum darwinism: En- tanglement, branches, and the emergent classicality of redun- dantly stored quantum information, Physical Review A73, 062310 (2006)

work page 2006

[8] [8]

Blume-Kohout and W

R. Blume-Kohout and W. H. Zurek, Quantum darwinism in quantum brownian motion, Physical Review Letters101, 240405 (2008)

work page 2008