Linear Polarization-Based Entanglement of a Single Photon, 2-Qubit Spatial Mode System

Quantum communications offers a revolutionary solution to future data security challenges. Photons, the building blocks of quantum communications, encode quantum information, such as polarization and momentum. We investigate the enhancement of communications protocols by treating the spatial modes of a single polarization-encoded photon as two separate qubits. In effect, we can recreate traditional 2-photon operations in an individual photon, improving the scability of quantum devices. To demonstrate the robustness of our approach, we created path entanglement between the spatial mode qubits in the polarization basis. By introducing a linear approach, we avoid low conversion efficiencies and spectral limitations associated with nonlinear processes like spontaneous parametric down conversion (SPDC). We explore extending the concept of two-photon Hong-Ou-Mandel (HOM) interference to demonstrate instinguishability between two spatial modes. Using the Python-based photonic quantum computing library Perceval, we leveraged a single photon source as the input, combined with beam splitters (BS), waveplates (WP), and polarizing beam splitters (PBS). The HOM effect is observed when both spatial modes bunch together at the port to form a photon coincidence. We used Perceval's Strong Linear Optical (SLOS) simulator to run 10,000 shots and measured equal probabilities at all four detectors, confirming consistent indistinguishability of the modes when the HWP is rotated $\pi/4$ radians. Finally by parameterizing the equivalent circuit in Qiskit with angles from $[0,2\pi]$, we showed a clear violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, more definitively indicating successful entanglement.