I'm curious about for which $n$ there exists a $2\times 2$ unitary matrix $U(n)$ such that for $1 \le k \le n$ $$|U^k_{1,0}(n)|^2 = \frac kn,$$ where $U^k_{1,0}(n)$ is the lower left element of the $k$th power of $U(n)$.

The idea is that if such a matrix exists, then applying it $k$ times to the quantum state $|0\rangle$ will produce the quantum state $\sqrt{1-k/n}|0\rangle + \sqrt{k/n}|1\rangle$ modulo some phases, a "quantum coin" with probability $k/n$ of giving outcome 1.

It's easy to see that it exists for $n=2$, with an example being $$U(2) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix},$$ but I couldn't find an example for any other $n$. A bit of brute force shows that no such matrices exist for $n=3$, and suggests that indeed no other $n$ works. Perhaps there's an elegant way to prove this? Or maybe there is a construction that I missed?


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