Let $U$ and $S$ be operators in the projective unitary group $PU(d)$. Additionally, let $\Omega$ be a denumerable subset of the positive integers. If $|| U^{\otimes n}-S^{\otimes n}||_{\mathrm{op}} \leq \epsilon$ for all $n \in \Omega$ and $\epsilon > 0$ sufficiently small, is it the case that $U = S$? Or is it the case that no matter how small an $\epsilon$ I choose, $U$ need not equal $S$?
I'm not exactly sure how to proceed. Of course I know $||U^{\otimes n} - S^{\otimes n} ||_{\mathrm{op}} \leq n ||U - S||_{\mathrm{op}}$, but this doesn't seem to help. One way might be to provide a nontrivial lower bound on $|| U^{\otimes n}-S^{\otimes n}||_{\mathrm{op}}$ in terms of $|| U-S||_{\mathrm{op}}$, but I also don't see how this can be done.