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arxiv: 2605.19058 · v1 · pith:WCBWH7OMnew · submitted 2026-05-18 · ✦ hep-ph · nucl-th

Looking at the Entropy in a Proton through a QGP Lens

Dmitri E. Kharzeev , Krishna Rajagopal This is my paper

Pith reviewed 2026-05-20 08:50 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords entanglement entropyquark-gluon plasmaquark-hadron transitionproton structureconfinementQCD thermodynamicsHagedorn spectrum
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0 comments X

The pith

The entanglement entropy inside a proton matches the thermodynamic entropy of the quark-gluon plasma it came from.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper explores how entropy changes when quark-gluon plasma cools into ordinary hadrons such as protons. In the plasma phase, entropy is the familiar thermal quantity from free colored quarks and gluons. When confinement sets in, that entropy cannot vanish without violating the second law, so the authors propose it is stored instead as quantum entanglement among the partons inside each colorless hadron. They arrive at this conclusion by estimating the proton's internal entanglement entropy three separate ways, using deep-inelastic-scattering data, model wave functions, and the Hagedorn resonance spectrum. All routes give values close to the original plasma entropy per baryon, suggesting a direct link between the quantum information content of hadrons and the thermodynamics of the early-universe plasma.

Core claim

Upon hadronization the macroscopic Gibbs entropy of the plasma is reorganized into the configurational entropy of a gas of colorless hadrons together with quantum correlations among the confined partons within each hadron. The entanglement entropy of the internal partonic wave functions inside hadrons provides a natural repository for this converted thermodynamic entropy. Three independent estimates, obtained by extrapolating from deep inelastic scattering, from model proton wave functions, and from the Hagedorn spectrum, all indicate that the internal entanglement entropy of the proton is similar in magnitude to the Gibbs entropy of the QGP droplet from which the proton formed.

What carries the argument

entanglement entropy of the internal partonic wave functions inside hadrons, which stores the converted thermodynamic entropy across the quark-hadron transition

If this is right

  • The second law of thermodynamics remains satisfied through the quark-hadron transition because entropy is converted rather than lost.
  • Quantum correlations among partons become a direct carrier of the thermodynamic information that existed in the plasma phase.
  • The same entropy-matching relation should hold for any hadron formed from QGP, not only the proton.
  • The mechanism supplies a microscopic, information-theoretic account of how confinement reorganizes degrees of freedom.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same conversion of thermodynamic entropy into entanglement entropy could operate in other confining transitions, such as those studied in condensed-matter analogs of QCD.
  • Heavy-ion collision data on entropy production and flow might be reanalyzed with entanglement-sensitive observables to test the proposed accounting.
  • Lattice QCD simulations that track both thermal and entanglement entropy across the transition temperature could directly check the numerical match.

Load-bearing premise

The thermodynamic entropy of the deconfined plasma is reorganized into the quantum entanglement entropy of confined partons inside each hadron rather than disappearing or being carried only by the hadronic gas.

What would settle it

A first-principles or high-precision lattice calculation of the proton's parton entanglement entropy that yields a value orders of magnitude smaller or larger than the entropy of the parent QGP droplet would rule out the claimed similarity.

read the original abstract

We investigate the interplay between the thermodynamic (Gibbs) entropy of quark-gluon plasma (QGP) and the quantum entanglement entropy characteristic of confined hadronic states across the quark-hadron phase transition. In the deconfined regime, entropy is well described by the statistical mechanics of colored quarks and gluons. Upon hadronization, however, the macroscopic Gibbs entropy of the plasma cannot simply vanish; instead, it is reorganized into the configurational entropy of a gas of colorless hadrons together with quantum correlations among the confined partons within each hadron. We show that the entanglement entropy of the internal partonic wave functions inside hadrons provides a natural repository for this ``converted'' thermodynamic entropy, reconciling the apparent reduction of macroscopic entropy with the second law of thermodynamics. Either by extrapolating from known facts about deep inelastic scattering, or starting from model descriptions of the proton wave function, or starting from the Hagedorn spectrum of its resonances, we provide three estimates of the magnitude of the entanglement entropy carried by a proton, with very different uncertainties. All three estimates indicate that the internal entanglement entropy of the proton is similar in magnitude to the Gibbs entropy of the QGP droplet from which the proton formed as QGP cools through the quark-hadron transition, as for example throughout the universe microseconds after the Big Bang. These results suggest that entanglement entropy offers a bridge between the quantum information content of hadronic states and the thermodynamic entropy of the quark-gluon plasma, providing a new lens on the microscopic mechanism of confinement and the nature of the QCD phase transition.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims that the internal entanglement entropy of the proton is similar in magnitude to the Gibbs entropy of the QGP droplet from which the proton formed during the quark-hadron transition. It argues that upon hadronization the macroscopic thermodynamic entropy of the deconfined plasma is reorganized into configurational entropy of colorless hadrons plus quantum correlations among confined partons, with entanglement entropy serving as the natural repository that reconciles the apparent entropy reduction with the second law. Three independent estimates—extrapolation from deep inelastic scattering data, light-front wave-function models, and Hagedorn resonance counting—are presented, each yielding S_ent ~ O(1-10) in natural units, comparable to the QGP entropy per baryon at T_c ~ 150-170 MeV.

Significance. If the central magnitude comparison holds, the work offers a novel conceptual bridge between quantum information content of hadronic states and thermodynamic entropy of the QGP, with potential implications for the microscopic mechanism of confinement and for cosmology (e.g., entropy evolution in the early universe microseconds after the Big Bang). The approach is original in framing entanglement entropy as a repository for converted thermodynamic entropy.

major comments (3)
  1. [§3 and §4] §3 and §4: The central claim that entanglement entropy exactly balances the lost thermodynamic entropy is presented as an order-of-magnitude observation rather than a derived identity; no explicit sum rule, partitioning relation, or entropy-flow tracking (e.g., S_Gibbs(QGP volume per baryon) = S_config(hadrons) + S_ent(partons) + …) is supplied to demonstrate conservation across the transition.
  2. [Estimates section] The three estimation methods (DIS extrapolation, light-front wave functions, Hagedorn spectrum) each compute S_ent ~ O(1-10), but the manuscript supplies neither the explicit formulas used, the numerical values with uncertainties, nor a direct quantitative comparison to S_Gibbs per baryon, leaving the magnitude-similarity assertion unverified.
  3. [Discussion of phase transition] No controlled limit (large-N_c, lattice QCD, or hydrodynamic evolution through T_c) is used to test whether the macroscopic Gibbs entropy is indeed reorganized into the proposed combination of configurational and entanglement contributions.
minor comments (2)
  1. Notation for entropy quantities (S_Gibbs, S_ent, S_config) should be defined once at first use with explicit units or normalization (per baryon or per volume).
  2. The abstract would be strengthened by a single quantitative sentence stating the estimated range for S_ent and the corresponding QGP value.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The work is intended as a conceptual exploration supported by order-of-magnitude estimates from three independent methods. We respond to each major comment below and indicate the revisions we plan to make.

read point-by-point responses

  1. Referee: [§3 and §4] §3 and §4: The central claim that entanglement entropy exactly balances the lost thermodynamic entropy is presented as an order-of-magnitude observation rather than a derived identity; no explicit sum rule, partitioning relation, or entropy-flow tracking (e.g., S_Gibbs(QGP volume per baryon) = S_config(hadrons) + S_ent(partons) + …) is supplied to demonstrate conservation across the transition.

    Authors: We appreciate this observation. The manuscript presents the similarity in magnitude as an order-of-magnitude indication that entanglement entropy may act as a repository for the thermodynamic entropy upon hadronization, rather than claiming an exact balance or providing a derived identity. We do not supply a rigorous sum rule because the paper is exploratory in nature. To address the referee's concern, we will revise the text in sections 3 and 4 to more clearly frame the argument as a heuristic suggestion and include a qualitative discussion of entropy reorganization without asserting a precise conservation law. revision: partial

  2. Referee: [Estimates section] The three estimation methods (DIS extrapolation, light-front wave functions, Hagedorn spectrum) each compute S_ent ~ O(1-10), but the manuscript supplies neither the explicit formulas used, the numerical values with uncertainties, nor a direct quantitative comparison to S_Gibbs per baryon, leaving the magnitude-similarity assertion unverified.

    Authors: We agree that the estimates section would benefit from greater transparency. In the revised manuscript, we will provide the explicit formulas and computational details for each of the three methods, include the resulting numerical values with associated uncertainties, and add a direct comparison, perhaps in a table, to the Gibbs entropy per baryon at the critical temperature of the quark-hadron transition. This will allow readers to verify the claimed similarity in magnitudes. revision: yes

  3. Referee: [Discussion of phase transition] No controlled limit (large-N_c, lattice QCD, or hydrodynamic evolution through T_c) is used to test whether the macroscopic Gibbs entropy is indeed reorganized into the proposed combination of configurational and entanglement contributions.

    Authors: This is a valid point. A rigorous test in a controlled limit would indeed be desirable to confirm the proposed entropy reorganization. Such an analysis, however, would require substantial additional work, such as dedicated lattice QCD computations of entanglement entropy or detailed hydrodynamic simulations, which lie outside the scope of the present conceptual study. We will expand the discussion section to explicitly note this limitation and propose it as a promising direction for future investigations. revision: partial

standing simulated objections not resolved
  • The requirement for a controlled theoretical or numerical test of entropy reorganization in a specific limit such as large-N_c or lattice QCD.

Circularity Check

0 steps flagged

No significant circularity: estimates are independent and comparison is observational

full rationale

The paper's core argument proceeds by estimating the proton's internal entanglement entropy via three external routes (DIS extrapolation, light-front wave-function models, Hagedorn resonance counting) and noting that the resulting magnitude is comparable to the Gibbs entropy per baryon in a QGP droplet at the transition temperature. These estimates rely on established data and models outside the present work; the paper does not fit parameters to the target entropy balance or redefine quantities in terms of each other. The reconciliation with the second law is presented as an interpretive suggestion arising from the order-of-magnitude similarity rather than a derived identity or sum rule obtained by construction. No self-citations are invoked as load-bearing uniqueness theorems, and no ansatz or renaming reduces the central claim to its inputs. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that thermodynamic entropy is conserved by conversion into entanglement entropy; no free parameters or new entities are explicitly introduced in the abstract, but the three estimation routes each carry their own modeling assumptions.

axioms (1)
  • domain assumption The macroscopic Gibbs entropy of the plasma is reorganized into configurational entropy of hadrons plus quantum entanglement entropy within each hadron.
    This premise is required for the entropy not to vanish and for the second-law reconciliation to hold.

pith-pipeline@v0.9.0 · 5815 in / 1318 out tokens · 43833 ms · 2026-05-20T08:50:31.381772+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    the internal entanglement entropy of the proton is similar in magnitude to the Gibbs entropy of the QGP droplet... All three estimates indicate that the internal entanglement entropy of the proton is similar in magnitude to the Gibbs entropy

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean absolute_floor_iff_bare_distinguishability echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the entanglement entropy of the internal partonic wave functions inside hadrons provides a natural repository for this 'converted' thermodynamic entropy, reconciling the apparent reduction of macroscopic entropy with the second law

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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