Combinatorial Structures of Quantum Entangled States

We investigate the combinatorial structures of multipartite quantum systems based on algebraic varieties. In particular, we study the relations between resolution of conifold, toric varieties, separable states, and quantum entangled states. We show that the resolved or deformed conifold is equivalent with the space of a pure entangled two-qubit state. We also generalize this result into multi-qubit states. The results give new insight about multipartite systems and also a new way of representing quantum entangled multipartite systems and quantum operations with potential applications in quantum computing.