Generalized Bell basis ($|{\psi }_{rs}⟩=n^{-1/2}{\sum }_j{\omega }^{rj}|j,j+s⟩$)

For a primitive root of unity $\omega =e^{2\pi i/n}$ and labels $r,s$ modulo $n$, the displayed $n^2$ maximally entangled states form an orthonormal basis. Inner products equal ${\delta }_{s{s}^{\prime }}{n}^{-1}{\sum }_j{\omega }^{j(r^{\prime }-r)}={\delta }_{s{s}^{\prime }}{\delta }_{r{r}^{\prime }}$, by the geometric series. A nonprimitive root repeats phase labels and does not give a complete orthonormal basis.

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.

The Bell basis diagonalizes the commuting Pauli operators $Z\otimes Z$ and $X\otimes X$. Two local anticommutations make the joint operators commute. Their signs encode parity and relative phase. The displayed projectors turn the signed eigenvalues into literal zero-or-one bit eigenvalues: even parity and plus phase have bit zero.

Two identical copies of an unknown Bell state suffice for perfect identification by LOCC. Alice and Bob each measure $Z$ on the first copy and $X$ on the second. Comparing their local signs supplies the eigenvalues of $Z\otimes Z$ and $X\otimes X$. These are respectively $(+,+)$ for ${\Phi }^{+}$, $(+,-)$ for ${\Phi }^{-}$, $(-,+)$ for ${\Psi }^{+}$, and $(-,-)$ for ${\Psi }^{-}$. Each parity is certain even though individual local outcomes are random. Different copies provide the two complementary measurements, so measuring the first copy does not erase the phase information needed on the second.

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.