Measuring Quantum State Purity from Configurable Sources of Polarization Entangled Photon Pairs

High-purity, non-classical light, such as two-mode entanglement, is essential for quantum sensing, quantum communications and networking, and quantum computing. Two-mode entanglement is typically generated via spontaneous parametric down-conversion (SPDC) processes in nonlinear, periodically poled electro-refractive crystals, producing bi-photon pairs entangled in polarization, frequency, time-bin, or optical quadrature. Although these practical sources of two-mode entanglement aim to achieve high photon pair rates and optical wavelength flexibility, they often face quantum state loss due to sub-unit efficiencies in waveguides, fiber coupling, wavelength conversion, non-ideal phase matching in the crystal and residual noise from pump light. To address this, we derive a complete parametric quantum mechanical description of these processes using density matrices to model the mixed quantum states produced by the entanglement source. The result enables the calculation of the quantum state purity of practical sources of two-mode entanglement and helps identify key parameters to retain high purity. Additionally, we propose diagnostic measurements to quantify purity and identify sources of deterioration.