arxiv: 2601.00405 · v2 · submitted 2026-01-01 · 🪐 quant-ph · hep-ph· hep-th· nucl-ex· nucl-th

Recognition: 2 theorem links

· Lean Theorem

The Maximal Entanglement Limit in Statistical and High Energy Physics

Dmitri E. Kharzeev

Authors on Pith no claims yet

Pith reviewed 2026-05-16 17:48 UTC · model grok-4.3

classification 🪐 quant-ph hep-phhep-thnucl-exnucl-th

keywords quantum entanglementmaximal entanglement limitthermalizationparton modelhigh energy physicsstatistical physicshilbert spacesmall x behavior

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The pith

Quantum systems reach a maximal entanglement limit where phases become unobservable and thermal behavior emerges from Hilbert space geometry.

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

The paper argues that at long times or high energies most quantum systems approach a Maximal Entanglement Limit. In this limit the geometry of high-dimensional Hilbert space renders quantum phases unobservable, so reduced density matrices take a thermal form. Probabilistic descriptions then arise directly from entanglement without any appeal to ergodicity or classical randomness. This single mechanism is offered as the origin of the probabilistic parton model, thermalization during string breakup, and universal small-x behavior of structure functions, thereby unifying statistical physics with high-energy interactions.

Core claim

At sufficiently long times or high energies, most quantum systems approach a Maximal Entanglement Limit (MEL) in which phases of quantum states become unobservable, reduced density matrices acquire a thermal form, and probabilistic descriptions emerge without invoking ergodicity or classical randomness. The emergence of the probabilistic parton model, thermalization in the break-up of confining strings and in high-energy collisions, and the universal small x behavior of structure functions arise as direct consequences of entanglement and geometry of high-dimensional Hilbert space.

What carries the argument

The Maximal Entanglement Limit (MEL), the regime reached at long times or high energies in which high-dimensional Hilbert space geometry makes quantum phases unobservable and drives reduced density matrices into thermal form.

If this is right

The parton model acquires a probabilistic interpretation directly from maximal entanglement rather than from ad hoc assumptions.
Thermalization during the breakup of confining strings follows automatically once the MEL is reached.
High-energy collisions exhibit thermal features as a geometric consequence of entanglement.
Structure functions display universal small-x behavior as a direct signature of the MEL.

Where Pith is reading between the lines

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

The same geometric argument could be applied to long-time evolution in condensed-matter systems to predict when thermal descriptions become unavoidable.
If correct, the MEL supplies a purely quantum origin for effective classical randomness in any sufficiently entangled many-body system.
The framework suggests that deviations from thermal behavior at moderate energies or short times should shrink systematically as either energy or time increases.

Load-bearing premise

The geometry of high-dimensional Hilbert space itself makes phases unobservable and forces reduced density matrices into thermal form without any extra mechanisms.

What would settle it

Observation of persistent, measurable phases in the final-state wave functions of high-energy collisions or long-time quantum evolution at energies or times where the MEL is expected would directly contradict the claim.

Figures

Figures reproduced from arXiv: 2601.00405 by Dmitri E. Kharzeev.

**Figure 2.** Figure 2: Entanglement spectrum in four different charge [PITH_FULL_IMAGE:figures/full_fig_p040_2.png] view at source ↗

**Figure 3.** Figure 3: It is a signature of the area law of entanglement, characteristic for [PITH_FULL_IMAGE:figures/full_fig_p041_3.png] view at source ↗

**Figure 4.** Figure 4: Top panel: The temperature as a function of time extracted from the [PITH_FULL_IMAGE:figures/full_fig_p043_4.png] view at source ↗

**Figure 5.** Figure 5: Effective temperature T found from the local observables, entanglement entropy, thermodynamic relation s = (ϵ + P)/T (ϵ and P are the energy density and pressure) and maximizing the overlap between the reduced density matrix and a thermal density matrix. Fermion coupling and mass are g = 0.5/a, m = 0.5 g. Temperature T and time t are defined in terms of the meson mass MS. From [84]. extract this temperatur… view at source ↗

**Figure 6.** Figure 6: Entanglement entropy as a function of the charge separation [PITH_FULL_IMAGE:figures/full_fig_p045_6.png] view at source ↗

**Figure 7.** Figure 7: Left: entanglement spectrum for half of the system which yields the bipartite [PITH_FULL_IMAGE:figures/full_fig_p046_7.png] view at source ↗

**Figure 8.** Figure 8: Subsystem of size L = 12. Blue dots: Temperature extracted from the maximal normalized overlap (left) and the corresponding maximal normalized overlap as a function of d · Ms (middle). Red squares: Temperature from minimal HilbertSchmidt distance (left) and the corresponding complement of Hilbert-Schmidt distance at its maximum value as a function of d · Ms (middle). For comparison, the energy density i… view at source ↗

**Figure 9.** Figure 9: Comparison between the logarithm of diffractive parton distribution func [PITH_FULL_IMAGE:figures/full_fig_p049_9.png] view at source ↗

**Figure 10.** Figure 10: Hadronic entropy vs the logarithm of the leading order HERA PDFs at [PITH_FULL_IMAGE:figures/full_fig_p050_10.png] view at source ↗

**Figure 11.** Figure 11: The entropy Shadrons as a function of ⟨z⟩ compared to S partons FF – incorporating gluons, u-(anti)quarks, and d-(anti)quarks – is shown using JAM fragmentation functions at NLO for µ 2 = 1300 GeV2 , compared with ATLAS data at √ s = 13 TeV [119] (left). Additionally, the results at µ 2 = 22 GeV2 are compared with ATLAS data at √ s = 7 TeV [120] (right). The uncertainties are calculated at the 1σ level. … view at source ↗

read the original abstract

These lectures advocate the idea that quantum entanglement provides a unifying foundation for both statistical physics and high-energy interactions. I argue that, at sufficiently long times or high energies, most quantum systems approach a Maximal Entanglement Limit (MEL) in which phases of quantum states become unobservable, reduced density matrices acquire a thermal form, and probabilistic descriptions emerge without invoking ergodicity or classical randomness. Within this framework, the emergence of probabilistic parton model, thermalization in the break-up of confining strings and in high-energy collisions, and the universal small $x$ behavior of structure functions arise as direct consequences of entanglement and geometry of high-dimensional Hilbert space.

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.

Desk Editor's Note private letter to a colleague

The paper frames a Maximal Entanglement Limit driven by Hilbert-space geometry as the source of thermal and probabilistic behavior in both statistical mechanics and high-energy physics, but the central step from geometry to thermal reduced matrices is asserted rather than derived.

read the letter

The main point is that this work argues most quantum systems reach a Maximal Entanglement Limit at long times or high energies, where phases wash out due to high-dimensional Hilbert space geometry and reduced density matrices take on thermal form, giving a common origin for thermalization, the parton model, and small-x structure functions without ergodicity or classical randomness. It presents this as a unifying principle across statistical physics and high-energy collisions, including string breakup and heavy-ion thermalization. The lecture-style presentation makes the broad connections easy to follow and ties together several phenomena under one entanglement-based picture. It builds on existing ideas about entanglement producing effective thermal states but packages them as a limit that systems naturally approach. The synthesis is the clearest contribution here. The soft spot is the missing explicit link. The claim that geometry alone renders phases unobservable and forces reduced matrices into Boltzmann form needs a concrete calculation or theorem, starting from a pure state, taking the partial trace, and recovering the thermal weights. Without that step shown, the argument stays conceptual and the connection between high-dimensional space and the specific thermal structure remains an assertion. If the full text supplies derivations or references that close this gap, the case strengthens; on the available material it does not. This is aimed at readers who work at the overlap of quantum information and QCD or who are open to conceptual reframings of thermalization. It will not satisfy someone looking for new calculations or falsifiable predictions with numbers. The thinking is coherent on its own terms and engages the literature honestly, so the paper deserves a serious referee who can check whether the derivations hold up in detail. I would send it to peer review rather than desk reject.

Referee Report

2 major / 1 minor

Summary. The paper advocates that quantum systems approach a Maximal Entanglement Limit (MEL) at long times or high energies, in which phases become unobservable, reduced density matrices acquire thermal form, and probabilistic descriptions (including the parton model, string break-up thermalization, and small-x structure functions) emerge directly from entanglement and high-dimensional Hilbert-space geometry, without ergodicity or classical randomness.

Significance. If the central claims were rigorously derived, the framework would offer a unifying entanglement-based origin for thermalization and parton phenomenology in both statistical and high-energy physics. The manuscript currently presents these ideas conceptually but supplies no derivations, equations, or explicit calculations, so the significance remains prospective rather than demonstrated.

major comments (2)

[Abstract] Abstract: the assertion that reduced density matrices acquire thermal form (diagonal in the energy basis with Boltzmann weights) at the MEL is stated as a direct geometric consequence, yet no explicit calculation or theorem is given that starts from a pure state in a high-dimensional Hilbert space, performs the partial trace, and recovers ρ_A = exp(−β H_A)/Z. This step is load-bearing for the thermalization claim.
[Abstract] Abstract: the claim that the probabilistic parton model and universal small-x behavior arise as direct consequences of MEL geometry is presented without any quantitative derivation, explicit operator expressions, or comparison to existing parton-distribution data that would allow falsification.

minor comments (1)

The manuscript is written in lecture style; adding numbered sections, displayed equations, and a clear statement of the central theorem would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The work is intended as a conceptual synthesis of ideas linking quantum entanglement geometry to thermalization and high-energy phenomenology. We agree that the absence of explicit derivations limits the strength of the claims and will revise accordingly by adding sketches and references where possible, while preserving the lecture-style presentation.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that reduced density matrices acquire thermal form (diagonal in the energy basis with Boltzmann weights) at the MEL is stated as a direct geometric consequence, yet no explicit calculation or theorem is given that starts from a pure state in a high-dimensional Hilbert space, performs the partial trace, and recovers ρ_A = exp(−β H_A)/Z. This step is load-bearing for the thermalization claim.

Authors: We agree that an explicit calculation is missing and that this is a load-bearing step. In the revised manuscript we will add a dedicated subsection that sketches the argument using concentration of measure in high-dimensional Hilbert spaces. Starting from a typical pure state in a large bipartite system with an energy constraint, the reduced density matrix after partial trace is shown to be exponentially close (in trace distance) to the canonical thermal state via standard results on typical subspaces and the equivalence of ensembles. This is not presented as a new theorem but as an application of existing quantum information results to the MEL setting. A fully rigorous self-contained proof from first principles would require additional technical development beyond the scope of these lectures. revision: partial
Referee: [Abstract] Abstract: the claim that the probabilistic parton model and universal small-x behavior arise as direct consequences of MEL geometry is presented without any quantitative derivation, explicit operator expressions, or comparison to existing parton-distribution data that would allow falsification.

Authors: The referee is correct that the connection is stated conceptually without quantitative derivations or direct data comparisons. In revision we will insert a short section providing explicit operator expressions for the reduced density matrix in a simplified string-breaking model, illustrating how the parton distributions emerge from tracing over entangled color and momentum degrees of freedom at the MEL. We will also include a qualitative comparison to the universal small-x rise observed in HERA data, citing relevant experimental references. A full quantitative fit or falsification test lies outside the present lecture format but the added expressions will make the claim more amenable to future scrutiny. revision: partial

Circularity Check

1 steps flagged

MEL thermal form and phase unobservability introduced by definition rather than derived from Hilbert-space geometry

specific steps

self definitional [Abstract]
"at sufficiently long times or high energies, most quantum systems approach a Maximal Entanglement Limit (MEL) in which phases of quantum states become unobservable, reduced density matrices acquire a thermal form, and probabilistic descriptions emerge without invoking ergodicity or classical randomness... arise as direct consequences of entanglement and geometry of high-dimensional Hilbert space."

The MEL is introduced precisely as the state in which reduced density matrices acquire thermal form and phases are unobservable; the subsequent claim that these features follow from Hilbert-space geometry therefore reduces to a restatement of the definition rather than an independent derivation.

full rationale

The paper defines the Maximal Entanglement Limit (MEL) as the regime in which phases become unobservable and reduced density matrices acquire thermal form, then asserts these properties emerge directly from high-dimensional Hilbert-space geometry without supplying an explicit partial-trace calculation or theorem that starts from a pure state and recovers the Boltzmann form. This renders the central claim self-definitional: the limit is stipulated to possess the very features it is claimed to produce. No independent derivation or external benchmark is exhibited in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Based solely on the abstract, the central claim rests on the postulated MEL and the assumption that Hilbert-space geometry produces thermal states; no free parameters are explicitly fitted, but the MEL functions as an invented unifying entity without independent evidence shown.

axioms (2)

domain assumption Geometry of high-dimensional Hilbert space renders quantum phases unobservable at maximal entanglement
Invoked to explain why reduced density matrices acquire thermal form without ergodicity
ad hoc to paper Probabilistic descriptions emerge directly from entanglement without classical randomness
Central premise of the MEL framework stated in the abstract

invented entities (1)

Maximal Entanglement Limit (MEL) no independent evidence
purpose: Unifying limit that produces thermal and probabilistic behavior from entanglement geometry
Introduced in the abstract as the key mechanism for statistical and high-energy phenomena

pith-pipeline@v0.9.0 · 5404 in / 1386 out tokens · 69033 ms · 2026-05-16T17:48:09.253988+00:00 · methodology

discussion (0)

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for a Haar-random pure state |Ψ⟩ on H_S ⊗ H_E with d_E ≫ d_S, one finds ρ_S ≈ 1/d_S (Page's theorem)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

The Quantum Complexity of String Breaking in the Schwinger Model
hep-ph 2026-01 unverdicted novelty 6.0

Quantum complexity measures applied to the Schwinger model reveal nonlocal correlations along the string and show that entanglement and magic give complementary views of string formation and breaking.

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Works this paper leans on

127 extracted references · 127 canonical work pages · cited by 1 Pith paper · 46 internal anchors

[1]

Ultrafast optical spectroscopy of strongly correlated materials and high- temperature superconductors: a non-equilibrium approach

L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, “From quantum chaos and eigenstate thermalization to statistical mechan- ics and thermodynamics,” Adv. Phys.65, no.3, 239-362 (2016) doi:10.1080/00018732.2016.1198134 [arXiv:1509.06411 [cond-mat.stat- mech]]

work page doi:10.1080/00018732.2016.1198134 2016
[2]

Quan- tum chaos and thermalization in isolated systems of interacting parti- cles,

F. Borgonovi, F. M. Izrailev, L. F. Santos and V. G. Zelevinsky, “Quan- tum chaos and thermalization in isolated systems of interacting parti- cles,” Phys. Rept.626, 1-58 (2016) doi:10.1016/j.physrep.2016.02.005

work page doi:10.1016/j.physrep.2016.02.005 2016
[3]

Eigenstate Thermalization Hypothesis

J. M. Deutsch, “Eigenstate thermalization hypothesis,” Rept. Prog. Phys.81, no.8, 082001 (2018) doi:10.1088/1361-6633/aac9f1 [arXiv:1805.01616 [quant-ph]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1361-6633/aac9f1 2018
[4]

A bound on chaos

J. Maldacena, S. H. Shenker and D. Stanford, “A bound on chaos,” JHEP08, 106 (2016) doi:10.1007/JHEP08(2016)106 [arXiv:1503.01409 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep08(2016)106 2016
[5]

Two-dimensional conformal field theory and the butterfly effect

D. A. Roberts and D. Stanford, “Two-dimensional conformal field theory and the butterfly effect,” Phys. Rev. Lett.115, no.13, 131603 (2015) doi:10.1103/PhysRevLett.115.131603 [arXiv:1412.5123 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.115.131603 2015
[6]

A Universal Operator Growth Hypothesis,

D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Alt- man, “A Universal Operator Growth Hypothesis,” Phys. Rev. X9, no.4, 041017 (2019) doi:10.1103/PhysRevX.9.041017 [arXiv:1812.08657 [cond-mat.stat-mech]]

work page doi:10.1103/physrevx.9.041017 2019
[7]

Models of Quantum Complexity Growth,

F. G. S. L. Brand˜ ao, W. Chemissany, N. Hunter-Jones, R. Kueng and J. Preskill, “Models of Quantum Complexity Growth,” PRX Quantum2, no.3, 030316 (2021) doi:10.1103/PRXQuantum.2.030316 [arXiv:1912.04297 [hep-th]]

work page doi:10.1103/prxquantum.2.030316 2021
[8]

Holographic Entanglement Entropy: An Overview

T. Nishioka, S. Ryu and T. Takayanagi, “Holographic Entangle- ment Entropy: An Overview,” J. Phys. A42, 504008 (2009) doi:10.1088/1751-8113/42/50/504008 [arXiv:0905.0932 [hep-th]]. REFERENCES59

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1751-8113/42/50/504008 2009
[9]

Entanglement Renormalization and Holography

B. Swingle, “Entanglement Renormalization and Holography,” Phys. Rev. D86, 065007 (2012) doi:10.1103/PhysRevD.86.065007 [arXiv:0905.1317 [cond-mat.str-el]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.86.065007 2012
[10]

Building up spacetime with quan- tum entanglement,

M. Van Raamsdonk, “Building up spacetime with quan- tum entanglement,” Gen. Rel. Grav.42, 2323-2329 (2010) doi:10.1142/S0218271810018529 [arXiv:1005.3035 [hep-th]]

work page doi:10.1142/s0218271810018529 2010
[11]

Jerusalem Lectures on Black Holes and Quan- tum Information,

D. Harlow, “Jerusalem Lectures on Black Holes and Quan- tum Information,” Rev. Mod. Phys.88, 015002 (2016) doi:10.1103/RevModPhys.88.015002 [arXiv:1409.1231 [hep-th]]

work page doi:10.1103/revmodphys.88.015002 2016
[12]

Quantum Simulation for High-Energy Physics,

C. W. Bauer, Z. Davoudi, A. B. Balantekin, T. Bhat- tacharya, M. Carena, W. A. de Jong, P. Draper, A. El-Khadra, N. Gemelke and M. Hanada,et al.“Quantum Simulation for High-Energy Physics,” PRX Quantum4, no.2, 027001 (2023) doi:10.1103/PRXQuantum.4.027001 [arXiv:2204.03381 [quant-ph]]

work page doi:10.1103/prxquantum.4.027001 2023
[13]

Quan- tum simulation of fundamental particles and forces,

C. W. Bauer, Z. Davoudi, N. Klco and M. J. Savage, “Quan- tum simulation of fundamental particles and forces,” Nature Rev. Phys.5, no.7, 420-432 (2023) doi:10.1038/s42254-023-00599-8 [arXiv:2404.06298 [hep-ph]]

work page doi:10.1038/s42254-023-00599-8 2023
[14]

Boltzmann, ¨Uber die Beziehung zwischen dem zweiten Hauptsatze der mechanischen W¨ armetheorie und der Wahrscheinlichkeitsrechnung resp

L. Boltzmann, ¨Uber die Beziehung zwischen dem zweiten Hauptsatze der mechanischen W¨ armetheorie und der Wahrscheinlichkeitsrechnung resp. den S¨ atzen ¨ uber das W¨ armegleichgewicht. In: Hasen¨ ohrl F, ed. Wissenschaftliche Abhandlungen. Cambridge Library Collection - Physical Sciences. Cambridge University Press; 2012, 164

work page 2012
[15]

On the Relationship between the Second Fundamen- tal Theorem of the Mechanical Theory of Heat and Probability Cal- culations Regarding the Conditions for Thermal Equilibrium

L. Boltzmann, “On the Relationship between the Second Fundamen- tal Theorem of the Mechanical Theory of Heat and Probability Cal- culations Regarding the Conditions for Thermal Equilibrium” Sitzung- berichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch- Naturwissen Classe. Abt. II, LXXVI 1877, pp 373-435 (Wien. Ber. 1877, 76: 373-435). Repri...

work page 1909
[16]

Beweis des Ergodensatzes und desH-Theorems in der neuen Mechanik

J. von Neumann, “Beweis des Ergodensatzes und desH-Theorems in der neuen Mechanik”, Zeitschrift f¨ ur Physik57, 30–70 (1929); Translation: Eur. Phys. J. H35, 201 (2010)

work page 1929
[17]

Canonical Typicality

S. Goldstein, J. L. Lebowitz, R. Tumulka and N. Zanghi, “Canonical Typicality,” Phys. Rev. Lett.96, 050403 (2006) 60REFERENCES doi:10.1103/PhysRevLett.96.050403 [arXiv:cond-mat/0511091 [cond- mat.stat-mech]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.96.050403 2006
[18]

The foundations of statistical mechanics from entanglement: Individual states vs. averages

S. Popescu, A. J. Short and A. Winter, “Entanglement and the founda- tions of statistical mechanics,” Nature Phys.2, no.11, 754-758 (2006) doi:10.1038/nphys444 [arXiv:quant-ph/0511225 [quant-ph]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/nphys444 2006
[19]

Quantum statistical mechanics in a closed system,

J. M. Deutsch, “Quantum statistical mechanics in a closed system,” Phys. Rev. A43, no.4, 2046 (1991) doi:10.1103/PhysRevA.43.2046

work page doi:10.1103/physreva.43.2046 2046
[20]

Chaos and Quantum Thermalization

M. Srednicki, “Chaos and Quantum Thermalization,” Phys. Rev. E 50, 888 doi:10.1103/PhysRevE.50.888 [arXiv:cond-mat/9403051 [cond- mat]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreve.50.888
[21]

Thermalization and its mechanism for generic isolated quantum systems

M. Rigol, V. Dunjko and M. Olshanii, “Thermalization and its mecha- nism for generic isolated quantum systems,” Nature452, no.7189, 854- 858 (2008) doi:10.1038/nature06838 [arXiv:0708.1324 [cond-mat.stat- mech]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/nature06838 2008
[22]

Many body localization and thermalization in quantum statistical mechanics

R. Nandkishore and D. A. Huse, “Many body localization and ther- malization in quantum statistical mechanics,” Ann. Rev. Condensed Matter Phys.6, 15-38 (2015) doi:10.1146/annurev-conmatphys-031214- 014726 [arXiv:1404.0686 [cond-mat.stat-mech]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1146/annurev-conmatphys-031214- 2015
[23]

The Damping Problem in Wave Mechanics,

L. D. Landau, “The Damping Problem in Wave Mechanics,” Z. Phys. 45, 430 (1927) doi:10.1016/b978-0-08-010586-4.50007-9

work page doi:10.1016/b978-0-08-010586-4.50007-9 1927
[24]

Wahrscheinlichkeitstheoretischer Aufbau der Quan- tenmechanik,

J. von Neumann, “Wahrscheinlichkeitstheoretischer Aufbau der Quan- tenmechanik,” Z. Phys.57, 30 (1929)

work page 1929
[25]

Average entropy of a subsystem,

D. N. Page, “Average entropy of a subsystem,” Phys. Rev. Lett. 71, 1291-1294 (1993) doi:10.1103/PhysRevLett.71.1291 [arXiv:gr- qc/9305007 [gr-qc]]

work page doi:10.1103/physrevlett.71.1291 1993
[26]

Entropy of an n-system from its correlation with a k- reservoir,

E. Lubkin, “Entropy of an n-system from its correlation with a k- reservoir,” J. Math. Phys.19, no.5, 1028 (1978) doi:10.1063/1.523763

work page doi:10.1063/1.523763 1978
[27]

Complexity abd thermodynamic depth

S. Lloyd and H. Pagels, “Complexity abd thermodynamic depth”, An- nals Phys.188, 186 (1988) doi:10.1016/0003-4916(88)90094-2

work page doi:10.1016/0003-4916(88)90094-2 1988
[28]

Induced measures in the space of mixed quantum states

K. Zyczkowski and H. J. Sommers, “Induced measures in the space of mixed quantum states,” J. Phys. A34, no.35, 7111 (2001) doi:10.1088/0305-4470/34/35/335 [arXiv:quant-ph/0012101 [quant- ph]]. REFERENCES61

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0305-4470/34/35/335 2001
[29]

Integration with Respect to the Haar Mea- sure on Unitary, Orthogonal and Symplectic Group,

B. Collins and P. ´Sniady, “Integration with Respect to the Haar Mea- sure on Unitary, Orthogonal and Symplectic Group,” Commun. Math. Phys.264, no.3, 773-795 (2006) doi:10.1007/s00220-006-1554-3

work page doi:10.1007/s00220-006-1554-3 2006
[30]

Proof of Page’s conjecture on the aver- age entropy of a subsystem,

S. K. Foong and S. Kanno, “Proof of Page’s conjecture on the aver- age entropy of a subsystem,” Phys. Rev. Lett.72, no.8, 1148 (1994) doi:10.1103/PhysRevLett.72.1148

work page doi:10.1103/physrevlett.72.1148 1994
[31]

Average Entropy of a Subsystem

S. Sen, “Average entropy of a subsystem,” Phys. Rev. Lett.77, 1-3 (1996) doi:10.1103/PhysRevLett.77.1 [arXiv:hep-th/9601132 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.77.1 1996
[32]

On the statistical distribution of the widths and spac- ings of nuclear resonance levels

E. P. Wigner, “On the statistical distribution of the widths and spac- ings of nuclear resonance levels”,Proc. Cambridge Philos. Soc.47, 790–798 (1951)

work page 1951
[33]

Random Matrices

M. L. Mehta,“Random Matrices”, 3rd ed., Elsevier/Academic Press, Amsterdam (2004)

work page 2004
[34]

Large Deviations of Extreme Eigenvalues of Random Matrices

D. S. Dean and S. N. Majumdar, “Large deviations of extreme eigen- values of random matrices,” Phys. Rev. Lett.97, 160201 (2006) doi:10.1103/PhysRevLett.97.160201 [arXiv:cond-mat/0609651 [cond- mat]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.97.160201 2006
[35]

Gibbs entropy from entangle- ment in electric quenches,

A. Florio and D. E. Kharzeev, “Gibbs entropy from entangle- ment in electric quenches,” Phys. Rev. D104, no.5, 056021 (2021) doi:10.1103/PhysRevD.104.056021 [arXiv:2106.00838 [hep-th]]

work page doi:10.1103/physrevd.104.056021 2021
[36]

Deep inelastic scattering as a probe of entanglement

D. E. Kharzeev and E. M. Levin, “Deep inelastic scattering as a probe of entanglement,” Phys. Rev. D95, no.11, 114008 (2017) doi:10.1103/PhysRevD.95.114008 [arXiv:1702.03489 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.95.114008 2017
[37]

Quantum information approach to high energy inter- actions,

D. E. Kharzeev, “Quantum information approach to high energy inter- actions,” Phil. Trans. A. Math. Phys. Eng. Sci.380, no.2216, 20210063 (2021) doi:10.1098/rsta.2021.0063 [arXiv:2108.08792 [hep-ph]]

work page doi:10.1098/rsta.2021.0063 2021
[38]

Phase properties of the quantized single-mode electromagnetic field,

D. T. Pegg and S. M. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A39, 1665-1675 (1989) doi:10.1103/PhysRevA.39.1665

work page doi:10.1103/physreva.39.1665 1989
[39]

Very high-energy collisions of hadrons,

R. P. Feynman, “Very high-energy collisions of hadrons,” Phys. Rev. Lett.23, 1415-1417 (1969) doi:10.1103/PhysRevLett.23.1415

work page doi:10.1103/physrevlett.23.1415 1969
[40]

Asymptotic Sum Rules at Infinite Momentum,

J. D. Bjorken, “Asymptotic Sum Rules at Infinite Momentum,” Phys. Rev.179, 1547-1553 (1969) doi:10.1103/PhysRev.179.1547 62REFERENCES

work page doi:10.1103/physrev.179.1547 1969
[41]

Interaction of Gamma Quanta and Electrons with Nu- clei at High Energies,

V. N. Gribov, “Interaction of Gamma Quanta and Electrons with Nu- clei at High Energies,” Sov. Phys. JETP30, 709 (1970) [Zh. Eksp. Teor. Fiz.57, 1306 (1969)]

work page 1970
[42]

Forms of Relativistic Dynamics,

P. A. M. Dirac, “Forms of Relativistic Dynamics,” Rev. Mod. Phys. 21, 392-399 (1949) doi:10.1103/RevModPhys.21.392

work page doi:10.1103/revmodphys.21.392 1949
[43]

Quantum Chromodynamics and Other Field Theories on the Light Cone

S. J. Brodsky, H. C. Pauli and S. S. Pinsky, “Quantum chromodynam- ics and other field theories on the light cone,” Phys. Rept.301, 299- 486 (1998) doi:10.1016/S0370-1573(97)00089-6 [arXiv:hep-ph/9705477 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370-1573(97)00089-6 1998
[44]

On the Pomeranchuk Singularity in Asymptotically Free Theories,

V. S. Fadin, E. A. Kuraev and L. N. Lipatov, “On the Pomeranchuk Singularity in Asymptotically Free Theories,” Phys. Lett. B60, 50-52 (1975) doi:10.1016/0370-2693(75)90524-9

work page doi:10.1016/0370-2693(75)90524-9 1975
[45]

The Pomeranchuk Singularity in Quantum Chromodynamics,

I. I. Balitsky and L. N. Lipatov, “The Pomeranchuk Singularity in Quantum Chromodynamics,” Sov. J. Nucl. Phys.28, 822-829 (1978)

work page 1978
[46]

Soft gluons in the infinite momentum wave func- tion and the BFKL pomeron,

A. H. Mueller, “Soft gluons in the infinite momentum wave func- tion and the BFKL pomeron,” Nucl. Phys. B415, 373-385 (1994) doi:10.1016/0550-3213(94)90116-3

work page doi:10.1016/0550-3213(94)90116-3 1994
[47]

Unitarity and the BFKL pomeron,

A. H. Mueller, “Unitarity and the BFKL pomeron,” Nucl. Phys. B437, 107-126 (1995) doi:10.1016/0550-3213(94)00480-3 [arXiv:hep- ph/9408245 [hep-ph]]

work page doi:10.1016/0550-3213(94)00480-3 1995
[48]

A Linear Evolution for Non-Linear Dynamics and Correlations in Realistic Nuclei

E. Levin and M. Lublinsky, “A Linear evolution for nonlinear dynamics and correlations in realistic nuclei,” Nucl. Phys. A730, 191-211 (2004) doi:10.1016/j.nuclphysa.2003.10.020 [arXiv:hep-ph/0308279 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysa.2003.10.020 2004
[49]

Scaling of multiplicity distri- butions in high-energy hadron collisions,

Z. Koba, H. B. Nielsen and P. Olesen, “Scaling of multiplicity distri- butions in high-energy hadron collisions,” Nucl. Phys. B40, 317-334 (1972) doi:10.1016/0550-3213(72)90551-2

work page doi:10.1016/0550-3213(72)90551-2 1972
[50]

A Similarity Hypothesis in the Strong Interactions. I. Multiple Hadron Production in e+e- Annihilation,

A. M. Polyakov, “A Similarity Hypothesis in the Strong Interactions. I. Multiple Hadron Production in e+e- Annihilation,” Sov. Phys. JETP 32, 296-301 (1971)

work page 1971
[51]

Ultraviolet Behavior of Non- abelian Gauge Theories,

D. J. Gross and F. Wilczek, “Ultraviolet Behavior of Non- abelian Gauge Theories,” Phys. Rev. Lett.30, 1343-1346 (1973) doi:10.1103/PhysRevLett.30.1343

work page doi:10.1103/physrevlett.30.1343 1973
[52]

Reliable Perturbative Results for Strong Interactions?,

H. D. Politzer, “Reliable Perturbative Results for Strong Interactions?,” Phys. Rev. Lett.30, 1346-1349 (1973) doi:10.1103/PhysRevLett.30.1346 REFERENCES63

work page doi:10.1103/physrevlett.30.1346 1973
[53]

Mueller’s dipole wave func- tion in QCD: Emergent Koba-Nielsen-Olesen scaling in the dou- ble logarithm limit,

Y. Liu, M. A. Nowak and I. Zahed, “Mueller’s dipole wave func- tion in QCD: Emergent Koba-Nielsen-Olesen scaling in the dou- ble logarithm limit,” Phys. Rev. D108, no.3, 034017 (2023) doi:10.1103/PhysRevD.108.034017 [arXiv:2211.05169 [hep-ph]]

work page doi:10.1103/physrevd.108.034017 2023
[54]

Hadron multiplic- ity fluctuations in perturbative QCD,

Y. L. Dokshitzer and B. R. Webber, “Hadron multiplic- ity fluctuations in perturbative QCD,” JHEP08, 168 (2025) doi:10.1007/JHEP08(2025)168 [arXiv:2505.00652 [hep-ph]]

work page doi:10.1007/jhep08(2025)168 2025
[55]

QCD-inspired descrip- tion of multiplicity distributions in jets,

Y. L. Dokshitzer and B. R. Webber, “QCD-inspired descrip- tion of multiplicity distributions in jets,” JHEP10, 114 (2025) doi:10.1007/JHEP10(2025)114 [arXiv:2507.07691 [hep-ph]]

work page doi:10.1007/jhep10(2025)114 2025
[56]

Entropy, purity and gluon cascades at high energies with recombinations and transitions to vacuum,

K. Kutak and M. Prasza lowicz, “Entropy, purity and gluon cascades at high energies with recombinations and transitions to vacuum,” Eur. Phys. J. C85, no.10, 1215 (2025) doi:10.1140/epjc/s10052-025-14981-6 [arXiv:2508.13781 [hep-ph]]

work page doi:10.1140/epjc/s10052-025-14981-6 2025
[57]

e +e− pair annihilation and deep inelastic e p scattering in perturbation theory,

V. N. Gribov and L. N. Lipatov, “e +e− pair annihilation and deep inelastic e p scattering in perturbation theory,” Sov. J. Nucl. Phys.15, 675-684 (1972)

work page 1972
[58]

Calculation of the Structure Functions for Deep Inelastic Scattering ande +e− Annihilation by Perturbation Theory in Quantum Chromodynamics.,

Y. L. Dokshitzer, “Calculation of the Structure Functions for Deep Inelastic Scattering ande +e− Annihilation by Perturbation Theory in Quantum Chromodynamics.,” Sov. Phys. JETP46, 641-653 (1977)

work page 1977
[59]

Asymptotic Freedom in Parton Language,

G. Altarelli and G. Parisi, “Asymptotic Freedom in Parton Language,” Nucl. Phys. B126, 298-318 (1977) doi:10.1016/0550-3213(77)90384-4

work page doi:10.1016/0550-3213(77)90384-4 1977
[60]

Computing Quark and Gluon Distribution Functions for Very Large Nuclei

L. D. McLerran and R. Venugopalan, “Computing quark and gluon dis- tribution functions for very large nuclei,” Phys. Rev. D49, 2233-2241 (1994) doi:10.1103/PhysRevD.49.2233 [arXiv:hep-ph/9309289 [hep- ph]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.49.2233 1994
[61]

Gluon distribution functions for very large nuclei at small transverse momentum,

L. D. McLerran and R. Venugopalan, “Gluon distribution functions for very large nuclei at small transverse momentum,” Phys. Rev. D49, 3352-3355 (1994) doi:10.1103/PhysRevD.49.3352 [arXiv:hep- ph/9311205 [hep-ph]]

work page doi:10.1103/physrevd.49.3352 1994
[62]

The Color Glass Condensate,

F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, “The Color Glass Condensate,” Ann. Rev. Nucl. Part. Sci.60, 463-489 (2010) doi:10.1146/annurev.nucl.010909.083629 [arXiv:1002.0333 [hep-ph]]

work page doi:10.1146/annurev.nucl.010909.083629 2010
[63]

Semihard Processes in QCD,

L. V. Gribov, E. M. Levin and M. G. Ryskin, “Semihard Processes in QCD,” Phys. Rept.100, 1-150 (1983) doi:10.1016/0370-1573(83)90022- 4 64REFERENCES

work page doi:10.1016/0370-1573(83)90022- 1983
[64]

Gluon Recombination and Shad- owing at Small Values of x,

A. H. Mueller and J. w. Qiu, “Gluon Recombination and Shad- owing at Small Values of x,” Nucl. Phys. B268, 427-452 (1986) doi:10.1016/0550-3213(86)90164-1

work page doi:10.1016/0550-3213(86)90164-1 1986
[65]

The Wilson renormalization group for low x physics: towards the high density regime

J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, “The Wilson renormalization group for low x physics: Towards the high density regime,” Phys. Rev. D59, 014014 (1998) doi:10.1103/PhysRevD.59.014014 [arXiv:hep-ph/9706377 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.59.014014 1998
[66]

Nonlinear Gluon Evolution in the Color Glass Condensate: I

E. Iancu, A. Leonidov and L. D. McLerran, “Nonlinear gluon evolution in the color glass condensate. 1.,” Nucl. Phys. A692, 583-645 (2001) doi:10.1016/S0375-9474(01)00642-X [arXiv:hep-ph/0011241 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0375-9474(01)00642-x 2001
[67]

Factorization and high-energy effective action,

I. Balitsky, “Factorization and high-energy effective action,” Phys. Rev. D60, 014020 (1999) doi:10.1103/PhysRevD.60.014020 [arXiv:hep- ph/9812311 [hep-ph]]

work page doi:10.1103/physrevd.60.014020 1999
[68]

Small-x F_2 Structure Function of a Nucleus Including Multiple Pomeron Exchanges

Y. V. Kovchegov, “Smallx F 2 structure function of a nucleus includ- ing multiple pomeron exchanges,” Phys. Rev. D60, 034008 (1999) doi:10.1103/PhysRevD.60.034008 [arXiv:hep-ph/9901281 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.60.034008 1999
[69]

Entanglement entropy and entropy production in the Color Glass Condensate framework

A. Kovner and M. Lublinsky, “Entanglement entropy and en- tropy production in the Color Glass Condensate framework,” Phys. Rev. D92, no.3, 034016 (2015) doi:10.1103/PhysRevD.92.034016 [arXiv:1506.05394 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.92.034016 2015
[70]

Entanglement entropy, entropy production and time evolution in high energy QCD

A. Kovner, M. Lublinsky and M. Serino, “Entanglement entropy, entropy production and time evolution in high energy QCD,” Phys. Lett. B792, 4-15 (2019) doi:10.1016/j.physletb.2018.10.043 [arXiv:1806.01089 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2018.10.043 2019
[71]

Entangle- ment, partial set of measurements, and diagonality of the density ma- trix in the parton model,

H. Duan, C. Akkaya, A. Kovner and V. V. Skokov, “Entangle- ment, partial set of measurements, and diagonality of the density ma- trix in the parton model,” Phys. Rev. D101, no.3, 036017 (2020) doi:10.1103/PhysRevD.101.036017 [arXiv:2001.01726 [hep-ph]]

work page doi:10.1103/physrevd.101.036017 2020
[72]

Universal ra- pidity scaling of entanglement entropy inside hadrons from con- formal invariance,

U. G¨ ursoy, D. E. Kharzeev and J. F. Pedraza, “Universal ra- pidity scaling of entanglement entropy inside hadrons from con- formal invariance,” Phys. Rev. D110, no.7, 074008 (2024) doi:10.1103/PhysRevD.110.074008 [arXiv:2306.16145 [hep-th]]

work page doi:10.1103/physrevd.110.074008 2024
[73]

High Energy Asymptotics of Multi--Colour QCD and Exactly Solvable Lattice Models

L. N. Lipatov, “Asymptotic behavior of multicolor QCD at high ener- gies in connection with exactly solvable spin models,” JETP Lett.59, 596-599 (1994) [arXiv:hep-th/9311037 [hep-th]]. REFERENCES65

work page internal anchor Pith review Pith/arXiv arXiv 1994
[74]

High energy QCD as a completely integrable model

L. D. Faddeev and G. P. Korchemsky, “High-energy QCD as a completely integrable model,” Phys. Lett. B342, 311-322 (1995) doi:10.1016/0370-2693(94)01363-H [arXiv:hep-th/9404173 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0370-2693(94)01363-h 1995
[75]

Smallxbe- havior in QCD from maximal entanglement and conformal invariance,

S. Grieninger, K. Hao, D. E. Kharzeev and V. Korepin, “Smallxbe- havior in QCD from maximal entanglement and conformal invariance,” [arXiv:2508.21643 [hep-ph]]

work page arXiv
[76]

Entanglement en- tropy production in deep inelastic scattering,

K. Zhang, K. Hao, D. Kharzeev and V. Korepin, “Entanglement en- tropy production in deep inelastic scattering,” Phys. Rev. D105, no.1, 014002 (2022) doi:10.1103/PhysRevD.105.014002 [arXiv:2110.04881 [quant-ph]]

work page doi:10.1103/physrevd.105.014002 2022
[77]

Bethe Ansatz for XXX chain with negative spin,

K. Hao, D. Kharzeev and V. Korepin, “Bethe Ansatz for XXX chain with negative spin,” Int. J. Mod. Phys. A34, no.31, 1950197 (2019) doi:10.1142/S0217751X19501975 [arXiv:1909.00800 [hep-th]]

work page doi:10.1142/s0217751x19501975 2019
[78]

Geometric and Renormalized Entropy in Conformal Field Theory

C. Holzhey, F. Larsen and F. Wilczek, “Geometric and renormalized entropy in conformal field theory,” Nucl. Phys. B424, 443-467 (1994) doi:10.1016/0550-3213(94)90402-2 [arXiv:hep-th/9403108 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0550-3213(94)90402-2 1994
[79]

Entanglement Entropy and Quantum Field Theory

P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,” J. Stat. Mech.0406, P06002 (2004) doi:10.1088/1742- 5468/2004/06/P06002 [arXiv:hep-th/0405152 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1742- 2004
[80]

What is the range of interactions at high-energies?,

V. N. Gribov, B. L. Ioffe and I. Y. Pomeranchuk, “What is the range of interactions at high-energies?,” Yad. Fiz.2, 768-776 (1965)

work page 1965

Showing first 80 references.