Extracting Entanglement from a Four-Level Quantum System

The four-level system is the simplest system from which entanglement can be extracted and analyzed; as such, I will derive the appropriate observable operators that can extract entanglement from a four-level mixed state. The entanglement of a general four-level quantum system is an open problem in quantum information theory. Quantum entanglement is a counterintuitive phenomenon in quantum mechanics with no analog in classical mechanics; it allows for systems of multiple objects in which measurements on these objects are more strongly correlated than any possible classical system. These correlations hold even if the objects are separated by considerable distances. These properties make quantum entanglement a crucial resource for developing quantum encryption and quantum computing technologies. Using the mathematics of representation theory and linear algebra, I will separate a general foul-level state into two entangled two-level states, from which I can analyze the extent of the entanglement between the two systems.