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The Bell basis is the two-qubit case of the generalized Bell basis. Its four vectors are the Bell states, and a Bell-basis measurement projects onto them.

Apply a Hadamard gate to the first qubit, then a controlled-NOT gate from the first to the second. This maps a computational-basis state $|ij⟩$ to the Bell state with phase bit $i$ and parity bit $j$. Its inverse uses the same gates in reverse order because each gate is self-inverse but they do not commute. Applying the forward sequence twice is not generally a decoding operation.

A Bell-basis measurement is the four-outcome projective measurement whose projectors are onto the four Bell states. It jointly measures two qubits and is the sender's measurement in quantum teleportation.

A Bell pair consists of two qubits prepared jointly in a Bell state. It is a maximally entangled two-qubit resource with Schmidt rank two. Two independent Bell pairs have resource Schmidt rank four across the same party partition.

The Bell states are joint eigenstates of commuting $Z_A{Z}_B$ and $X_A{X}_B$. Two shared Bell state meter pairs realize their two entanglement-assisted nondemolition parity measurements using spacelike local circuits. Their projector products give the four rank-one Bell state outcomes and preserve each input Bell state. The globally identified outcome requires later local operations and classical communication.

Share a Bell state meter pair, apply local system-to-meter CNOT gates, and measure each meter in the computational basis. The parity of the local meter records measures system $Z_A{Z}_B$ while preserving all coherence within either parity sector. The complete tuple of local records has Kraus operator $P_{u\oplus v}/\sqrt{2}$. Each local record is uniform, and the nonlocal result requires local operations and classical communication to compare records. A local Hadamard gate conjugation supplies the corresponding $X_A{X}_B$ measurement.

A generalized Bell state is one member of a generalized Bell basis. Its reduced density matrices are both $I/n$, so it is a maximally entangled state. For $n=2$ these are the four Bell states.

For $|{\alpha }_{\theta }⟩=\cos \theta |0⟩+\sin \theta |1⟩$, Alice measures her half of a Bell state in the real basis $\{|{\alpha }_{\theta }⟩,|{\beta }_{\theta }⟩\}$, with $|{\beta }_{\theta }⟩=-\sin \theta |0⟩+\cos \theta |1⟩$. Bob receives one outcome bit and applies $I$ or $iY$. Since $iY|{\beta }_{\theta }⟩=|{\alpha }_{\theta }⟩$ for every $\theta$, the preparation is exact and the correction is independent of the unknown parameter.