Quark Entanglement in the Proton on the Light Front*

We consider entanglement of quarks in the proton on the light

front. For $N_c$ colors the anti-symmetric valence quark color space

singlet state $\sim \epsilon_{i_1\cdots i_{N_c}} |i_1,\cdots,

i_{N_c}\rangle$ of the proton corresponds to the reduced density

matrix $\rho_{ij}=(1/N_c) \delta_{ij}$ for a single color degree of

freedom. Its degenerate spectrum of eigenvalues $\lambda_i=1/N_c$, the

purity $\text{tr}\, \rho^2 = 1/N_c$, and the von~Neumann entropy

$S_\mathrm{vN}=\log(N_c)$ all indicate maximal entanglement of color.

On the other hand, for $N_c\to\infty$ the spatial wave function of the proton

factorizes into valence quark wave functions determined by a mean

field (E.~Witten, Nucl.\ Phys.\ B 160 (1979) p.\ 57), and there is no

entanglement of spatial degrees of freedom.

A model calculation at $N_c=3$, using a well known valence quark

light-front wave function by Brodsky and Schlumpf, predicts percent

level entanglement of spatial degrees of freedom. We also illustrate the

violation of a Bell-CHSH inequality by color correlations in the state

$\sim \epsilon_{i_1 i_2 i_3} |i_1, i_2, i_3 \rangle$.