The Bell States

The four Bell states form a maximally entangled orthonormal basis for two qubits and anchor most entanglement protocols.

The maximally entangled basis

The Bell states are four two-qubit states, each maximally entangled and mutually orthogonal, so they span the whole two-qubit space. They are Phi+ = (|00>+|11>)/sqrt(2), Phi- = (|00>-|11>)/sqrt(2), Psi+ = (|01>+|10>)/sqrt(2), and Psi- = (|01>-|10>)/sqrt(2). Each has both reduced density matrices equal to I/2, the signature of maximal entanglement.

Preparation

Starting from |00>, a Hadamard on the first qubit followed by a CNOT produces Phi+. The other three arise by applying local Pauli operations to one qubit: X gives Psi+, Z gives Phi-, and XZ (up to phase, Y) gives Psi-. This shows the four states are related by local operations on a single qubit, a fact exploited by teleportation and dense coding.

import numpy as np
H = np.array([[1,1],[1,-1]])/np.sqrt(2)
CNOT = np.array([[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]])
psi = np.kron(H,np.eye(2)) @ np.array([1,0,0,0])
psi = CNOT @ psi
print(np.round(psi,3))   # (|00>+|11>)/sqrt2

Bell measurement

Measuring in the Bell basis, a Bell measurement, projects two qubits onto one of the four states and is the joint measurement at the heart of teleportation and entanglement swapping. It is implemented by reversing the preparation circuit (CNOT then Hadamard) and measuring both qubits in the computational basis, mapping each Bell state to a distinct two-bit outcome.

Why they matter

Bell states are the standard unit of entanglement, the ebit. They saturate the quantum bound in the CHSH inequality, provide the shared resource for superdense coding, and serve as the target of distillation. Nearly every protocol that consumes or certifies entanglement is stated in terms of how close a state is to a Bell state.