# Entanglement

**Entanglement** is a quantum correlation between two or more qubits such that the combined state cannot be written as a product of independent qubit states. An entangled state exhibits non-local correlations: measuring one qubit instantly affects the measured outcome probabilities of distant qubits, even if no signal travels between them.

A two-qubit state $|\psi\rangle$ is separable if $|\psi\rangle = |\psi_A\rangle \otimes |\psi_B\rangle$—it factors into independent qubit states. If no such factorization exists, the state is entangled.

## Bell States

The four [[quantum-state-bell|Bell states]] are the maximally entangled two-qubit states: $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ and three others. Measuring either qubit in the computational basis gives 0 or 1 with equal probability, but the results are perfectly correlated: outcomes always match.

## Multipartite Entanglement

Three or more qubits can be entangled in different ways. The [[quantum-state-ghz|GHZ state]] $\frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$ is fully entangled but fragile—loss of one qubit breaks the correlations. The [[quantum-state-w|W state]] $\frac{1}{\sqrt{3}}(|001\rangle + |010\rangle + |100\rangle)$ distributes entanglement more robustly.

## Entanglement and Quantum Computing

Entanglement is the defining resource that separates quantum from classical computation. A system of $n$ unentangled qubits can be simulated classically; entanglement enables exponential speedup for certain problems. Entangling gates like CNOT and CZ create entanglement from separable states.

## Measurement and Disentanglement

Measuring one qubit of an entangled state projects the other qubits onto definite subspaces. This is the basis of [[quantum-state-entanglement#teleportation|quantum teleportation]] and other quantum protocols.