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The Maximal Entanglement Limit in Statistical and High Energy Physics

Dmitri E. Kharzeev

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.

The Maximal Entanglement Limit in Statistical and High Energy Physics

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 behavior of structure functions arise as direct consequences of entanglement and geometry of high-dimensional Hilbert space.
Paper Structure (37 sections, 126 equations, 11 figures)

This paper contains 37 sections, 126 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Classical ensembles and quantum mechanics
  3. Classical Statistical Physics and Ergodicity
  4. Microcanonical ensemble
  5. Canonical ensemble from entropy maximization
  6. Ergodic hypothesis
  7. Maximally Entangled States from Geometry of Hilbert Space
  8. Reduced density matrices and entanglement entropy
  9. Typical states and maximal entanglement
  10. Typical states and random matrices
  11. Maximal entanglement and $2D$ Coulomb gas
  12. Maximal entanglement from typicality
  13. Maximal entanglement and nuclear level statistics
  14. Maximal Entanglement and Thermodynamic Entropy
  15. Gibbs state from a maximally entangled state
  16. ...and 22 more sections

Figures (11)

  • Figure 1: The bipartition of the system used to study the entanglement of the central region of length $L$ (subsystem $A$) with the complement (subsystem $B$). Subsystem $A$ is centered around the center of the system of size $N$. From Florio:2025hoc.
  • Figure 2: Entanglement spectrum in four different charge $Q$ sectors for a subsystem of size $L=12$ as a function of time (logarithms of eigenvalues are shown for legibility). Total size of the system is $N=100$, fermion coupling and mass are $g = 0.5/a, m = 0.5 g$. From Florio:2025hoc.
  • Figure 3: Top panel: time dependence of entanglement entropy $S(L)$ (times are shifted so that the jets cross the subsystem boundary at $t-t_\times=0$). Matching of the initial shape of the curves is the signature of an area law scaling of the entanglement entropy, expected for the ground state of a gapped theory. Bottom panel: vacuum--subtracted entanglement entropy per unit length, with the time shifted same as above. Different subsystem size curves reach a plateau at late times at the same value of entropy density, signifying the volume law of entanglement that signals thermalization. System parameters are $m=0.5 g, g=0.5/a$; values of $L$ are separated on the right and on the left panels due to staggering effects. From Florio:2025hoc.
  • Figure 4: Top panel: The temperature as a function of time extracted from the maximal normalized overlap (right) and minimal HS distance (left) for subintervals $L=4,8$ centered in the middle. Bottom panel: the value of the metric at the extremum shown in the top panel. For the thermal states we used $N_{\rm thermal}=12$ and traced down to $L$. From Florio:2025hoc.
  • Figure 5: Effective temperature $T$ found from the local observables, entanglement entropy, thermodynamic relation $s=(\epsilon+P)/T$ ($\epsilon$ 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 $M_S$. From Florio:2025hoc.
  • ...and 6 more figures