# Deep inelastic scattering as a probe of entanglement

## Abstract

Using nonlinear evolution equations of QCD, we compute the von Neumann entropy of the system of partons resolved by deep inelastic scattering at a given Bjorken x and momentum transfer q ^{2} = - Q ^{2} . We interpret the result as the entropy of entanglement between the spatial region probed by deep inelastic scattering and the rest of the proton. At small x the relation between the entanglement entropy S ( x ) and the parton distribution x G ( x ) becomes very simple: S ( x ) = ln [ x G ( x ) ] . In this small x , large rapidity Y regime, all partonic microstates have equal probabilities—the proton is composed by an exponentially large number exp ( Δ Y ) of microstates that occur with equal and exponentially small probabilities exp ( - Δ Y ) , where Δ is defined by x G ( x ) ~ 1 / x Δ . For this equipartitioned state, the entanglement entropy is maximal—so at small x , deep inelastic scattering probes a maximally entangled state. Here, we propose the entanglement entropy as an observable that can be studied in deep inelastic scattering. Thismore »

- Authors:

- Stony Brook Univ., NY (United States). Dept. of Physics and Astronomy; Brookhaven National Lab. (BNL), Upton, NY (United States). RIKEN Research Center, Dept. of Physics
- Tel Aviv Univ., Ramat Aviv (Israel). Dept. of Particle Physics; Federico Santa MarÃÂa Technical Univ. and Science and Technology Center, Valparaiso (Chile). Dept. of Physics

- Publication Date:

- Research Org.:
- Brookhaven National Laboratory (BNL), Upton, NY (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1392245

- Alternate Identifier(s):
- OSTI ID: 1363840

- Report Number(s):
- BNL-114269-2017-JA

Journal ID: ISSN 2470-0010; PRVDAQ; R&D Project: KB0301020

- Grant/Contract Number:
- SC0012704; FG-88ER40388; AC02-98CH10886

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 95; Journal Issue: 11; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS

### Citation Formats

```
Kharzeev, Dmitri E., and Levin, Eugene M..
```*Deep inelastic scattering as a probe of entanglement*. United States: N. p., 2017.
Web. doi:10.1103/PhysRevD.95.114008.

```
Kharzeev, Dmitri E., & Levin, Eugene M..
```*Deep inelastic scattering as a probe of entanglement*. United States. doi:10.1103/PhysRevD.95.114008.

```
Kharzeev, Dmitri E., and Levin, Eugene M.. Sat .
"Deep inelastic scattering as a probe of entanglement". United States.
doi:10.1103/PhysRevD.95.114008. https://www.osti.gov/servlets/purl/1392245.
```

```
@article{osti_1392245,
```

title = {Deep inelastic scattering as a probe of entanglement},

author = {Kharzeev, Dmitri E. and Levin, Eugene M.},

abstractNote = {Using nonlinear evolution equations of QCD, we compute the von Neumann entropy of the system of partons resolved by deep inelastic scattering at a given Bjorken x and momentum transfer q 2 = - Q 2 . We interpret the result as the entropy of entanglement between the spatial region probed by deep inelastic scattering and the rest of the proton. At small x the relation between the entanglement entropy S ( x ) and the parton distribution x G ( x ) becomes very simple: S ( x ) = ln [ x G ( x ) ] . In this small x , large rapidity Y regime, all partonic microstates have equal probabilities—the proton is composed by an exponentially large number exp ( Δ Y ) of microstates that occur with equal and exponentially small probabilities exp ( - Δ Y ) , where Δ is defined by x G ( x ) ~ 1 / x Δ . For this equipartitioned state, the entanglement entropy is maximal—so at small x , deep inelastic scattering probes a maximally entangled state. Here, we propose the entanglement entropy as an observable that can be studied in deep inelastic scattering. This will then require event-by-event measurements of hadronic final states, and would allow to study the transformation of entanglement entropy into the Boltzmann one. We estimate that the proton is represented by the maximally entangled state at x ≤ 10 -3 ; this kinematic region will be amenable to studies at the Electron Ion Collider.},

doi = {10.1103/PhysRevD.95.114008},

journal = {Physical Review D},

number = 11,

volume = 95,

place = {United States},

year = {Sat Jun 03 00:00:00 EDT 2017},

month = {Sat Jun 03 00:00:00 EDT 2017}

}