Local Minimum of Spin-Sector Magic at the CP-Conserving Point in Low-Energy Neutron-Proton Scattering

Abstract

We study Magic generation in elastic neutron-proton scattering within a leading low-energy spin-sector ansatz that retains the one-pion-exchange spin structures and treats each scattering direction as a conditional two-qubit spin map. We show that the direction-averaged Magic is locally minimized at the CP-conserving (CPC) point $\bar{\theta }=0$ at the Clifford point $f_{\mathrm{CPC}}=\pi /4$, and for the representative non-Clifford CPC backgrounds analyzed here. At $f_{\mathrm{CPC}}=\pi /4$, the CPC spin map reduces to SWAP up to a phase and therefore generates zero Magic from stabilizer inputs. We further evaluate the complete spin-sector Magic functional by averaging over all 60 two-qubit stabilizer inputs and over scattering directions, and find that the curvature at $\bar{\theta }=0$ is positive only within specific windows of the effective CPC phase $f_{\mathrm{CPC}}$. These results identify the CPC point as a local Magic minimum within the restricted low-energy spin sector considered here.