Nonlocal realism tests and quantum state tomography in Sagnac-based type-II polarization-entanglement SPDC-source

We have experimentally created a robust, ultrabright and phase-stable polarization-entangled state close to maximally entangled Bell-state with $ \% 98 $-fidelity using the type-II spontaneous parametric down-conversion (SPDC) process in periodically-poled KTiOPO$ _4 $ (PPKTP) collinear crystal inside a Sagnac interferometer (SI). Bell inequality measurement, Freedman's test, as the different versions of CHSH inequality, and also visibility test which all can be seen as the nonlocal realism tests, imply that our created entangled state shows a strong violation from the classical physics or any hidden-variable theory. We have obtained very reliable and very strong Bell violation as $ S=2.78 \pm 0.01 $ with high brightness $ \mathcal{V}_{\rm HV}= \% (99.969 \pm 0.003) $ and $\mathcal{V}_{\rm DA}= \% (96.751 \pm 0.002) $ and very strong violation due to Freedman test as $ \delta_{\rm F} = 0.01715 \pm 0.00001 $. Furthermore, using the tomographic reconstruction of quantum states together a maximum-likelihood-technique (MLT) as the numerical optimization, we obtain the physical non-negative definite density operator which shows the nonseparability and entanglement of our prepared state. By having the maximum likelihood density operator, we calculate some important entanglement-measures and entanglement entropies. The Sagnac configuration provides bidirectional crystal pumping yields to high-rate entanglement source which is very applicable in quantum communication, sensing and metrology as well as quantum information protocols, and has potential to be used in quantum illumination-based LIDAR and free-space quantum key distribution (QKD).