A shift-invariant basis is a common <eigenbasis> of all <cyclic shift operator>[cyclic shift operators] $U(\gamma)$. For the displayed Fourier states, <orthonormal set>[orthonormality] follows from the <root-of-unity filter>:
$$
\langle\xi_\alpha|\xi_{\alpha'}\rangle
=\frac1N\sum_{\beta\in\mathbb Z_N}
\omega^{(\alpha-\alpha')\beta}
=\delta_{\alpha,\alpha'}.
$$
There are $N$ vectors in this <orthonormal set> in the $N$-dimensional <Hilbert space>, so they form a <basis>. Reindexing the finite sum gives
$$
\begin{aligned}
U(\gamma)|\xi_\alpha\rangle
&=\frac1{\sqrt N}\sum_\beta
\omega^{-\alpha\beta}|\beta+\gamma\rangle\\
&=\omega^{\alpha\gamma}|\xi_\alpha\rangle.
\end{aligned}
$$
Thus every $|\xi_\alpha\rangle$ is simultaneously an <eigenvector> of every shift, with <eigenvalue> $\omega^{\alpha\gamma}$ for $U(\gamma)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 324 / 1 / b / Solution

On a basis indexed by $\mathbb Z_N$, a cyclic shift operator has the form $U(a)|b\rangle=|a+b\bmod N\rangle$. The <quantum Fourier transform> simultaneously diagonalizes all such shifts.

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 Incoming links: Cyclic shift operator