Emergent Bell-Triplet State in Proton-Proton Scattering

Abstract

Entanglement is a fundamental resource in quantum information science, with profound implications for computing, communication, and metrology. Nuclear scattering processes, dominated by rich spin-dependent interactions, offer a natural platform for generating complex spin entanglement. Here, using proton-proton scattering as a quantum laboratory, we report the emergence of a near-pure Bell-triplet state at a laboratory energy of 151 MeV and a center-of-mass scattering angle of 90 degrees, with the spin amplitude a transition operator connecting two different Bell states. In contrast to the low-energy singlet state governed by the Pauli principle and the S-wave dominance, this second maximally entangled state is directly shaped by tensor forces beyond leading-order chiral effective field theory, providing a distinct quantum-information signature for realistic nuclear forces. These findings, invisible to traditional scattering observables, establish proton-proton scattering as a robust source of triplet entanglement and pave the way for next-generation nuclear Bell tests.

Figures (4)

• Figure 1: (a) Entanglement power $\mathcal{E}$ and (b) concurrence $\mathcal{C}$ for $pp$ scattering, shown as functions of the laboratory kinetic energy $E$ and the center–of–mass angle $\theta$. Both quantities are calculated from the Nijmegen PWA93 database, with the concurrence evaluated for a completely unpolarized initial state. The red pentagrams mark the second local maxima in each distribution at $(E,\theta)=(151\ \text{MeV}, 90^\circ)$.
• Figure 2: (a) Concurrence $\mathcal{C}$ calculated using chiral EFT at different orders. (b) Partial-wave decomposition of the Bell-triplet weight $W_{10}$ at LO in chiral EFT. The three panels, from top to bottom, respectively show the results excluding the contributions from $^1S_0$ and $^3P_0$ components.
• Figure 3: (a) Concurrence $\mathcal{C}$ calculated using the N$^3$LO chiral EFT interaction with the tensor strength $c_T$ varied. (b) Concurrence $\mathcal{C}$ and (c) Bell-triplet weight $W_{10}$ at $(E,\theta)=(151\ \text{MeV}, 90^\circ)$, calculated using both LO and N$^3$LO chiral EFT interactions, plotted as a function of $c_T$.
• Figure 4: (a) Horodecki function $H(\rho_f)$ derived from the Nijmegen PWA93 database. (b) Spin correlation function $F(\phi)=\text{Tr}[ (\bm{\sigma}_1\cdot\bm{n}_1) \otimes (\bm{\sigma}_2\cdot\bm{n}_2)\rho_f]$ computed from the same database. The measurement geometry is defined by a fixed analyzer along $\bm{n}_1=(1,0,0)$ and a second analyzer rotating in the $x$--$z$ plane as $\bm{n}_2=(\sin\phi,0,\cos\phi)$.