A quantum state on a Hilbert space is represented by a positive operator of trace one, called a density matrix in finite dimension. A pure state is a rank-one projection onto a unit vector; vectors differing by a phase factor represent the same pure state. General states include convex combinations of pure states.
The space for a bipartite system is a tensor product of the two component Hilbert spaces. Quantum entanglement distinguishes states expressible as convex combinations of product states from states that cannot be expressed in that form.
