⚛️ nuclear theory

Quantum tomography at the Electron-Ion Collider

This paper proposes a method for performing quantum state tomography of quark-antiquark pairs at the Electron-Ion Collider by utilizing azimuthal angular correlations from fragmentation functions to experimentally reconstruct spin density matrices and investigate quantum properties like entanglement and Bell-nonlocality.

Original authors: Michael Fucilla, Yuxun Guo, Yoshitaka Hatta

Published 2026-09-29

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CC BY 4.0

Original authors: Michael Fucilla, Yuxun Guo, Yoshitaka Hatta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, particles do not merely exist; they carry a hidden internal orientation known as spin. While this property is often compared to a tiny bar magnet, it is more accurately a fundamental quantum characteristic that dictates how a particle interacts with others. When two particles are created together, such as a quark and its antimatter partner, their spins can become inextricably linked, a phenomenon called entanglement. In this state, measuring the spin of one particle instantly reveals the state of the other, regardless of the distance between them. For decades, physicists have studied these connections in high-energy collisions, but a new frontier has opened: using these collisions not just to discover new particles, but to map the complete quantum state of the particles themselves. This process, known as quantum state tomography, is akin to taking a full three-dimensional scan of a quantum object, revealing not just its average behavior, but the precise, complex web of correlations that define its existence.

Researchers are now proposing to bring this powerful diagnostic tool to the future Electron-Ion Collider, a massive machine currently under construction in the United States. The goal is to reconstruct the complete quantum state of quark-antiquark pairs produced when high-energy electrons smash into protons or heavy atomic nuclei. While previous experiments at the Large Hadron Collider have successfully mapped the spin states of top quarks, those measurements relied on the particles decaying through the weak nuclear force, a process that is rare and difficult to control. The new approach outlined in recent research suggests a different, more direct path. By analyzing the patterns in which the resulting quarks break apart into streams of ordinary particles called hadrons, scientists can infer the full spin relationship between the original pair without needing them to decay. This method relies on the strong nuclear force, which governs how quarks bind together, offering a much more abundant and accessible way to study these quantum connections.

The core of this proposal involves looking at the angles at which the resulting particles fly apart. When a quark and an antiquark are created, they do not simply vanish; they fragment into clouds of new particles. The direction these new particles take is not random; it carries a subtle imprint of the original spins of the quarks. Specifically, the researchers show that by measuring the relative angles between pairs of particles, or even pairs of particle pairs, they can extract every component of the spin relationship. This is a significant leap forward because it allows for the measurement of the entire "spin density matrix," a mathematical object that fully describes the quantum state. Previously, experiments could only glimpse parts of this picture. Now, by combining two specific types of particle fragmentation patterns, the team demonstrates that every piece of the puzzle becomes accessible.

One of the most striking predictions of this work concerns heavy quarks, specifically the charm quark. In certain conditions, the researchers predict that these heavy quark pairs will reach a state of "maximal entanglement," where their quantum connection is as strong as nature allows. In a standard scenario, the angular distribution of the resulting particles follows a familiar pattern. However, the team calculates that when this maximal entanglement occurs, the pattern changes dramatically. The usual correlation between the angles of the two particles disappears, replaced by a completely different relationship. Instead of the particles moving in a way that suggests a simple sum of angles, their behavior shifts to a form where their individual angles become independent in a specific, counterintuitive way. This shift serves as a clear signal, a bright flag in the data that tells physicists, "Here, the quantum state is maximally entangled."

The researchers also highlight that this technique offers a unique window into the internal structure of the targets being hit. When an electron strikes a proton or a heavy nucleus, the resulting spin correlations depend heavily on the arrangement of gluons—the particles that hold quarks together—inside that target. By measuring these correlations, scientists can effectively probe the three-dimensional structure of the proton and the nucleus in a new way. This complements the main scientific mission of the Electron-Ion Collider, which is to create a detailed map of the proton's interior. The spin tomography approach adds a new layer to this map, revealing not just where the parts are, but how they are quantum mechanically linked.

To test these ideas, the team performed numerical simulations using realistic models of how quarks fragment into particles. They focused on the production of charm quarks, which are heavy enough to exhibit these strong quantum effects but light enough to be produced in large numbers. Their calculations suggest that with the high intensity of the future collider, it will be possible to collect enough data to see these effects clearly. They estimate that in a specific range of collision energies, the signal for maximal entanglement should be strong enough to be distinguished from background noise. While the actual measurement will require the full capabilities of the collider, the theoretical groundwork is now laid. The study confirms that the necessary tools exist within the standard framework of particle physics to perform this quantum reconstruction.

This work represents a bridge between abstract quantum information theory and the concrete reality of high-energy physics. It suggests that the Electron-Ion Collider will function not only as a microscope for the smallest building blocks of matter but also as a laboratory for testing the deepest principles of quantum mechanics. By turning the collider into a machine capable of reading the full quantum state of produced particles, physicists can investigate properties like entanglement and non-locality directly in the debris of high-speed collisions. The findings indicate that the transition from observing simple particle properties to mapping complex quantum states is within reach, promising a new era where the quantum nature of the universe is measured with the same precision as its spatial structure.

Technical Summary: Quantum Tomography at the Electron-Ion Collider

Problem Statement
Recent developments at the LHC have established the application of quantum state tomography to top-antitop ( $t\bar{t}$ ) production, allowing for the experimental reconstruction of the complete spin density matrix and the investigation of quantum information properties such as entanglement and Bell nonlocality. However, extending this framework to the Electron-Ion Collider (EIC) presents specific challenges. While the spin density matrix elements ( $B, \bar{B}, C$ ) for quark-antiquark ( $q\bar{q}$ ) pairs produced in electron-proton ($ep$) and electron-nucleus ( $eA$ ) collisions have been theoretically computed, concrete observables for experimentally probing these matrix elements have not been clearly identified. Existing strategies often rely on parity-violating weak decays, which are difficult to utilize for light quarks and suffer from low branching ratios for heavy baryons. This paper addresses the gap in identifying robust, strong-interaction-based observables to perform quantum state tomography for both massless and massive $q\bar{q}$ pairs at the EIC.

Methodology
The authors develop a formalism connecting the spin density matrix of a produced $q\bar{q}$ pair to azimuthal angular correlations in the final state hadrons. The methodology proceeds through several key steps:

Spin Density Matrix Representation: The authors derive explicit formulas for the spin density matrix elements ( $B_a, \bar{B}_b, C_{ab}$ ) in terms of four-dimensional Dirac traces involving the production amplitude $\Gamma$ . This representation is valid for both massive and massless quarks and is expressed directly in the center-of-mass frame of the pair.
Fragmentation Function Integration: To connect the theoretical density matrix to experimental observables, the authors utilize Transverse Momentum Dependent (TMD) fragmentation functions (FFs).
- Collins FF ( $H_1^\perp$ ): Used to access transverse spin correlations ( $C_{ij}$ , where $i,j=x,y$ ) via single-hadron production.
- Dihadron FFs: Introduced to access longitudinal components ( $C_{zz}, C_{iz}, C_{zi}$ ) and polarization vectors ( $B_z, \bar{B}_z$ ). Specifically, the interference FF ( $H_1^\triangleright$ ) and the new longitudinal dihadron FF ( $G_1^\perp$ ) are employed.
- Mass Effects: The authors incorporate quark mass effects by modifying the standard collinear FF definitions to include twist-3 and twist-4 terms, ensuring consistency with the equation of motion relations for massive quarks.
Process Analysis: The framework is applied to two distinct processes:
- $e^+e^-$ Annihilation: Used as a baseline to reformulate known angular asymmetries (Collins and Artru-Collins) in terms of the spin density matrix, highlighting mass-dependent modifications to azimuthal modulations.
- Deep Inelastic Scattering (DIS): The primary focus is on diffractive $q\bar{q}$ production in $ep$ and $eA$ collisions. Here, the spin density matrix depends non-trivially on the target's nonperturbative structure (TMDs, GPDs, or GTMDs) and the kinematics of the virtual photon.

Key Contributions and Results

Complete Tomography via Strong Interactions: The paper demonstrates that all components of the spin density matrix ( $B, \bar{B}, C$ ) for $q\bar{q}$ pairs can be accessed purely through strong interaction processes using Collins and dihadron fragmentation functions. This avoids the experimental penalties associated with weak decays.
Mass-Dependent Angular Modulations: For massive quarks, the authors derive that the standard Collins angular dependence $\cos(\phi_P + \phi_{P'})$ is modified. A new term, $\cos(\phi_P - \phi_{P'})$ , appears, proportional to the mass squared ( $m_q^2/s$ ). This allows for the extraction of polarization terms ( $B_i, \bar{B}_i$ ) and full spin correlations even in the presence of mass.
Maximal Entanglement in Heavy Quarks: A central prediction is the existence of kinematic regions where the $q\bar{q}$ $q \overset{q}{ˉ}$ pair reaches a maximally entangled state.
- In the heavy-quark limit (specifically charm at the EIC), the authors predict a specific kinematic point (related to $\xi \approx 1/2$ and specific transverse momentum scales) where the spin correlation matrix $C_T$ takes a form corresponding to maximal entanglement ( $C_{yy} = -1$ ).
- Signature: At this point, the usual Collins asymmetry $\cos(\phi_P + \phi_{P'})$ is replaced by a distinct angular dependence: $\cos \phi_P \cos \phi_{P'} - \cos \psi \sin \phi_P \sin \phi_{P'}$ . For certain target models, this simplifies to $\cos \phi_P \cos \phi_{P'}$ , effectively "disentangling" the azimuthal angles in the Collins sector while maximizing other correlations.
Target Structure Sensitivity: Numerical results using the Golec-Biernat-Wusthoff (GBW) and McLerran-Venugopalan (MV) models show that the spin density matrix elements are highly sensitive to the target structure (proton vs. heavy nucleus) and the saturation scale. The MV model predicts a more general form of maximal entanglement compared to the GBW model, highlighting the interplay between gluon saturation and quantum information properties.
Feasibility Estimates: The authors provide a preliminary estimate for the cross-section of diffractive charm production in the kinematic window of maximal entanglement. For a representative EIC setup, they estimate a yield of approximately $4 \times 10^6$ $c\bar{c}$ pairs with an integrated luminosity of $100 \text{ fb}^{-1}$ , suggesting that experimental measurement is feasible despite the small cross-sections typical of diffractive processes.

Significance and Claims
The paper claims to initiate the study of quantum state tomography at the EIC, positioning the collider not just as a tool for mapping the 3D partonic structure of nuclei (conventional tomography) but also as a laboratory for characterizing the quantum mechanical nature of the final state.

The authors emphasize that this approach enriches the EIC's tomographic program by linking the spin density matrix elements directly to TMDs and GPDs. They argue that their method is the most practical path forward for heavy quark tomography, avoiding the severe suppression factors ( $10^{-4} \sim 10^{-6}$ ) associated with weak decays of heavy baryons.

The work is presented as a first step connecting formal QCD calculations to concrete experimental observables. The authors modestly note that while they have established the theoretical framework and identified promising kinematic windows, further work is required to address higher-twist effects, decoherence in inclusive production, and the quantitative extraction of the currently unknown longitudinal dihadron FF ( $G_1^\perp$ ). The paper concludes that the observation of the predicted angular modifications would serve as a direct signal of maximal entanglement and Bell nonlocality in the heavy-quark sector, simultaneously probing the nonperturbative structure of the target nucleus.

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