Can Mirror Symmetry Challenge Local Realism? Probing Photon Entanglement from Positronium via Compton Scattering

This study investigates photon entanglement generated from para-positronium decay by analyzing azimuthal correlations after the double Compton scattering with stationary electrons. We introduce a normalized correlation observable $\mathcal{O}_1 = \cos(2\phi_1 - 2\phi_2)/C_1$ to witness entanglement. In the absence of decoherence, $\langle\mathcal{O}_1\rangle = -1$, corresponding to a maximally entangled Bell state. With decoherence parameterized by $\rho$, the expectation becomes $-(1-\rho)$, allowing direct experimental quantification of coherence loss. A prior symmetry analysis of the Compton scattering process within the quantum field theory (QFT) is provided, which establishes the mirror-symmetric nature of the single-photon angular distribution. We further examine a local hidden-variable theory (LHVT) under the angular-momentum conservation. Imposing the mirror symmetry with respect to the plane defined by the photon spin and momentum leads to a non-negative LHVT prediction for $\langle \sin^2\theta_1 \sin^2\theta_2 \cos(2\phi_1-2\phi_2)\rangle$, contradicting the negative QFT prediction value for any $\rho < 1$. Thus, mirror symmetry serves as a novel criterion to exclude LHVT descriptions of the entangled state, whereas without preserving this symmetry, LHVTs can reproduce the correlations.