Ground State Entanglement in a Combination of Star And Ring Geometries Of Interacting Spins

We compare a star and a ring network of interacting spins in terms of the entanglement they can provide between the nearest and the next to nearest neighbor spins in the ground state. We then investigate whether this entanglement can be optimized by allowing the system to interact through a weighted combination of the star and the ring geometries. We find that such a weighted combination is indeed optimal in certain circumstances for providing the highest entanglement between two chosen spins. The entanglement shows jumps and counterintuitive behavior as the relative weighting of the star and the ring interactions is varied. We give an exact mathematical explanation of the behavior for a five qubit system (four spins in a ring and a central spin) and an intuitive explanation for larger systems. For the case of four spins in a ring plus a central spin, we demonstrate how a four qubit GHZ state can be generated as a simple derivative of the ground state. Our calculations also demonstrate that some of the multi-particle entangled states derivable from the ground state of a star network are sufficiently robust to the presence of nearest neighbor ring interactions.