 Entanglement entropy

For a bipartite pure state, entanglement entropy is the Von Neumann entropy of either reduced density matrix. A Schmidt decomposition gives squared coefficients $p_j$ as the nonzero eigenvalues of both reductions, soA product state has one coefficient equal to one and entropy zero. A Bell state has two equal coefficients and entropy one bit when logarithms have base two.

This entropy quantifies how much the subsystems are entangled even though the joint pure state itself has zero Von Neumann entropy. The purity assumption is essential: a classically correlated mixed state can have mixed reductions without entanglement. Reduced-state entropy alone is therefore not a general entanglement measure for arbitrary mixed bipartite states.