Graph state properties and applications in the stabilizer formalism

Graph states form a class of entangled quantum states that have multiple useful applications within quantum computing and quantum communication protocols. The stabilizer formalism offers an efficient mathematical description of graph states and the effect of operations acting on such states. Here, we consider three scenarios where the stabilizer formalism can be utilised to investigate the structure, manipulation and generation of graph states. First, we propose a method to calculate the purity of reduced states of graph states entirely within the stabilizer formalism, using only the stabilizer generators for a given state and apply this method to find the Concentratable Entanglement of graph states. Next, we reduce the number of qubits required within a graph state used as a resource for the measurement based implementation of a general two-qubit unitary, using the stabilizer formalism to track supplementary operations that are required. Finally, we examine spin-photon interactions followed by single-qubit measurements as a process to probabilistically generate states described by the stabilizer formalism, particularly when the phase shift induced is small.