Generalized Bell basis
= Generalized Bell basis
{title2=$|\psi_{rs}\rangle=n^{-1/2}\sum_j\omega^{rj}|j,j+s\rangle$}
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 $\delta_{ss'}n^{-1}\sum_j\omega^{j(r'-r)}=\delta_{ss'}\delta_{rr'}$, by the <geometric series>. A nonprimitive root repeats phase labels and does not give a complete orthonormal basis.