Quantum entanglement is a phenomenon in which a group of particles that are either generated, interacting with each other, or close in proximity to each other, have a property that the quantum states of each particle cannot be described independently of the states of the other particles. This phenomenon was initially investigated by Albert Einstein, Boris Poldosky, and Nathan Rosen in their landmark paper known as the EPR paradox, in which Einstein described this behavior as "spooky action at a distance''. This thesis presents a mathematical and theoretical approach in defining quantum entanglement by detecting photons in their entangled state through a set of photon-number-resolving (PNR) detectors and threshold detectors. This theoretical approach is made rigorous by including the notion of a dark count, a phenomenon in which a detector incidentally detects a photon when it should not have been detected. With this dark count model, we define the probabilities of finding a coincidence of such entangled photons through a combination and configuration of PNR detectors and threshold detectors. Then, we find the coincidence probabilities of detecting a single coincidence of photons within the detector system, the total coincidence probabilities of detecting this coincidence with respect to ground truth, and the effective density matrix that characterizes how well each combination and configuration of detectors can detect photon coincidences. By making mathematical and probabilistic assumptions on the distribution of photon types and counts with respect to ground truth, we are able to compute these quantities and analyze their expressions based on a mathematical and conceptual context.

Quantum entanglement, a phenomenon first introduced in the realm of quantum mechanics by the famous Einstein-Podolsky-Rosen (EPR) paradox, has intrigued physicists and philosophers alike for nearly a century. Its implications for the nature of reality, particularly its apparent violation of local realism, have sparked intense debate and spurred numerous experimental investigations. This thesis presents a comprehensive examination of quantum entanglement with a focus on probing its non-local aspects. Central to this thesis is the development of a detailed project document outlining a proposed experimental approach to investigate the non-local nature of quantum entanglement. Drawing upon recent advancements in quantum technology, including the manipulation and control of entangled particles, the proposed experiment aims to rigorously test the predictions of quantum mechanics against the framework of local realism. The experimental setup involves the generation of entangled particle pairs, such as photons or ions, followed by the precise manipulation of their quantum states. By implementing a series of carefully designed measurements on spatially separated entangled particles, the experiment seeks to discern correlations that defy explanation within a local realistic framework.