A bipartite state is separable if it can be written as $\rho=\sum_k p_k\,\rho_k^A\otimes\rho_k^B$. Nonseparable states exhibit correlations that cannot be explained by shared classical randomness alone.

The four Bell states form an orthonormal maximally entangled basis for two qubits. They are central to teleportation, superdense coding, entanglement swapping, and device-independent tests of quantum theory.

The CHSH inequality bounds classical correlations by $|S|\le 2$, while quantum mechanics allows up to $2\sqrt2$. Experiments violating CHSH provide strong evidence for entanglement and against local hidden-variable theories.

For a pure bipartite state $|\psi\rangle_{AB}$, the entanglement entropy is $S(\rho_A)=-\mathrm{Tr}(\rho_A\log\rho_A)$, where $\rho_A=\mathrm{Tr}_B(|\psi\rangle\langle\psi|)$. It quantifies bipartite entanglement.

Entanglement is constrained by monogamy relations: strong entanglement with one system limits entanglement with others. This is essential in quantum networks, many-body physics, and quantum cryptography.