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arXiv:2608.05753

The Locality Gap: A Thermodynamic Law for Objective Facts

Maxim V. Churilov

quant-phgr-qcSubmitted Aug 6, 2026• Announced Aug 7, 2026• 0 citations
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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.

Comments
12 pages, 6 figures, 42 cited references; source and reproducibility archive included

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•Aug 25, 2026•0 citations
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 FFF and a partition P\mathcal{P}P into blocks that cannot be jointly controlled, the reversible isothermal erasure penalty relative to a global controller is kBTk_{\mathrm{B}}TkB​T times the partition total correlation. If the source event XXX is supplied as catalytic classical side information, subtracting the corresponding penalty gives the source-loss law FPT=kBT[∑B∈PI(X:FB)−I(X:F)]\mathcal{F}_{\mathcal{P}}^{T}=k_{\mathrm{B}}T\left[\sum_{B\in\mathcal{P}}I(X{:}F_B)-I(X{:}F)\right]FPT​=kB​T[∑B∈P​I(X:FB​)−I(X:F)]. We call this the thermodynamic factuality charge. It ranges from −kBTH(X)-k_{\mathrm{B}}T H(X)−kB​TH(X) for perfect secret sharing to (∣P∣−1)kBTH(X)(|\mathcal{P}|-1)k_{\mathrm{B}}T H(X)(∣P∣−1)kB​TH(X) for perfect broadcast objectivity. Its normalized form defines a thermodynamic record number between 000 and ∣P∣|\mathcal{P}|∣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.
Abstract Classical objectivity arises when many independent observers can retrieve the same information about a quantum system without disturbing it. Quantum Darwinism (QD) explains this by treating the environment as a communication channel that redundantly broadcasts information about pointer observables. We develop an operational framework for quantifying this redundancy through the lens of functional information. By introducing a one-sided adequacy criterion based on the Holevo bound, we measure the \emph{abundance} of environment fragments that individually contain nearly all classically accessible pointer information. Using a heterogeneous pure–dephasing model, we sample fragments to extract onset statistics and apply isotonic regression to obtain stable estimates of redundancy and its functional-information counterpart FIQD=log⁡2RδFI_{QD}=\log_2 R_\deltaFIQD​=log2​Rδ​. The results reveal a generic onset–plateau structure: redundancy grows rapidly at early times, then saturates at a capacity-limited ceiling, universally across adequacy tolerances. We further connect FIQDFI_{QD}FIQD​ to thermodynamic bounds, showing that each additional bit of redundancy implies exponentially increasing minimal heat costs for record stabilization. This approach provides a conservative, model-agnostic measure of emergent classicality, suitable for both theoretical analysis and experimental probing.
A fact of the classical world is public property: many agents can learn it without erasing or revising it. Quantum mechanics offers no such notion, since unknown states cannot be copied and measurement disturbs. This review develops quantum Darwinism’s resolution, which locates classicality in the environment as a broadcast medium carrying independent records of an observable. The framework is assembled in information-theoretic terms: quantum mutual information between system and environment fragments, redundancy at an information deficit, the Holevo quantity separating accessible from total correlation, spectrum broadcast structure as the record-bearing state, and the quantum Chernoff information fixing the rate of accumulation. A proposition proving that spectrum broadcast structure forces the classical plateau fragment by fragment is given. Three media — spin registers, oscillator baths, and scattered photons — are examined, followed by account of a record’s life cycle: formation, degradation under intra-environmental scrambling and information backflow, and counterexamples where decoherence proceeds while no record is deposited. The relation between redundancy and consensus, consistent histories, laboratory evidence, and unresolved problems are treated in closing.
Quantum Darwinism explains classical objectivity through the redundant broadcast of a system's records into many environment fragments. A companion paper asks how hard it is to certify objectivity from a description of a state; this one asks the dual, dynamical question — how hard is it to produce objectivity in the first place? The paper defines the broadcast complexity — the minimum number of elementary two-qubit operations needed to turn a fresh environment into an objective record of a given redundancy and precision — and determines it. Four results make objectivity a resource measured in gates: with bounded-strength operations the cost is exactly the redundancy times the number of copies needed per witness, achieved by an explicit construction; for arbitrary operations a causal-cone argument forces at least one gate per independent record; complexity is a faithful proxy for objectivity, since redundancy grows in strict proportion to the gates invested; and the total cost equals the volume of an emergent "broadcast geometry" whose boundary is the records, a redundancy area law. Notably, while the total cost grows with the number of witnesses, the time to make a fact objective does not — through many parallel channels it is just the time to write a single sharp record. In short: making the world classical is cheap but not free — it takes only a handful of gates per copy, and no extra time however many witnesses come to share the fact.
Quantum Darwinism explains classical objectivity through the redundant spread of information across fragments of the environment. This note studies the opposite regime: an environment whose record is distributed through entangled correlations rather than independent fragments. The model considers a mediator carrying a which-branch displacement record while being purified by an unobserved environment. The central question is: how much of the original quantum information can an observer recover when only part of the environment is accessible? The paper proves that any fragment excluding the record-bearing mode contains exactly zero which-branch information. No individual fragment carries a readable record, and redundancy disappears completely. A closed-form partial-information curve is derived. The amount of accessible evidence depends only on the captured fraction of the environment purification and is governed by a single rational function. The curve connects two limits: the minimal information available to an external observer and the full information recovered when the complete environment is captured. A precise threshold is obtained: strong environmental entanglement requires capturing almost the entire environment before objectivity emerges. The results show that decoherence and objectivity are not equivalent. The same loss of coherence can either create a public, redundantly observable fact or leave information hidden entirely in nonlocal correlations. The structure of the environment determines whether quantum information becomes an objective classical fact or remains inaccessible. All results are derived in exactly solvable Gaussian models, verified analytically and numerically, and accompanied by a reproducibility package.
Quantum Darwinism explains effective classical objectivity through the selective proliferation of information about preferred quantum states into multiple environmental fragments. Its mature formulations operate within quantum theory, using a system-environment decomposition, decoherence or einselection, and information-theoretic or broadcast-structure criteria for redundant accessibility.This paper asks an upstream question from Registration Theory (RT): what makes a physical distinction a record before that record is described as information available to an observer? Canonical RT answers that retained registration is persistent organization constraining later evolution, that a distinction becomes a physical record only when it remains consequential, and that persistent physical organization is prior to information extracted from it.The paper assembles a theorem-backed finite-model chain now available in the RT corpus. Canonical ACRF dynamics supplies write-retain-read consequential history. The exact finite-amplitude ACRF-4 quotient identifies a nine-dimensional host-host-boundary state. The certified consequence-minimality theorem (DOI 10.6084/m9.figshare.34063464) shows that the retained relational boundary contributes irreducible future-consequence directions. The spectrum-preserving ACRF-8 redundant-record theorem (DOI 10.6084/m9.figshare.34063500) then proves an explicit local witness in which one quotient-valid source distinction is written into two disjoint retained fragments that are locally recoverable, counterfactually separable, and independently readable, giving RRT(q)≥2R_{RT}(q)\ge 2RRT​(q)≥2 under the stated criteria.The resulting distinction from Quantum Darwinism is precise but bounded. RT recordhood is consequence-based, while Darwinian objectivity is an accessibility-and-redundancy criterion formulated within quantum theory. The ACRF witness demonstrates redundant retained record formation without assuming Hilbert space, density operators, mutual information, decoherence, einselection, Born probabilities, or an observer-defined environment. Therefore, multiple persistent records of one distinction are not definitionally identical to Quantum-Darwinian redundancy.The paper does not claim that Quantum Darwinism is false or that RT has derived quantum mechanics, decoherence, Born probabilities, a mutual-information redundancy plateau, or macroscopic objectivity. Instead, it proposes a disciplined hierarchy: consequential distinction → retention → persistent record → branching → redundant retained records → redundant accessibility → environmental objectivity. The final two steps remain open in RT. A future bridge must dynamically select and scale multi-host redundancy and map consequence-bearing RT records into quantum environmental fragments that satisfy decoherence and strong objectivity conditions.
Quantum Darwinism explains classical objectivity through the redundant proliferation of records in an environment. This note studies that mechanism inside a gauge theory, where Gauss's law constrains which records can exist and where they can be stored. Using exact lattice Schwinger model calculations with both open and periodic boundaries, including the Wilson-line rotor, this work shows that record redundancy is determined by the relation between information and gauge constraints. Three fundamental record classes are identified. First, unscreenable charges produce perfectly redundant records: every accessible downstream region can read the same information because the constraint itself distributes the record without requiring copying dynamics. Second, screened charges produce bilocal records, with information divided between the screening cloud and the boundary-delivered image while intermediate regions remain blind. Third, symmetry-breaking order produces graded redundant records, while topological flux records represent the opposite extreme: globally distinguishable but invisible to local matter fragments. The study also reveals a constraint-induced discordant split. In the flux sector, mutual information can remain large while readable evidence is almost absent, showing that a gauge theory can contain correlations that no local fragment can convert into classical information. The central conclusion is that classical objectivity in gauge theories is not determined only by environmental interaction. It is controlled by the structure of the constraints: Gauss's law decides which records can be broadcast, localized, or hidden. All results are obtained through exact diagonalization with independent verification checks and documented numerical convergence tests.