Entangled graphs: Bipartite entanglement in multiqubit systems

Quantum entanglement in multipartite systems cannot be shared freely. In order to illuminate basic rules of entanglement sharing between qubits, we introduce a concept of an entangled structure (graph) such that each qubit of a multipartite system is associated with a point (vertex), while a bipartite entanglement between two specific qubits is represented by a connection (edge) between these points. We prove that any such entangled structure can be associated with a pure state of a multiqubit system. Moreover, we show that a pure state corresponding to a given entangled structure is a superposition of vectors from a subspace of the $2^N$-dimensional Hilbert space, whose dimension grows linearly with the number of entangled pairs.