Show that no $d×d$ unitary matrix $U$ satisfies the condition $|\langle \psi | U | \psi \rangle|^2 = 0$ for every $|\psi_i\rangle \in C^d$.
How would I show this? So far, I've observed that it's essentially saying that no unitary matrix $U$ exists such that when we measure phi when the transformation $U$ on $\phi$ is applied, the probability of getting $\phi$ is $0$, or in other words the transformation can't result in getting back the same result.
Here is a simple approach.
Note that every operator over $\Bbb C^d$ has an eigenvector, so there exist some $\lambda \in \Bbb C$ and some non-zero $|\psi\rangle \in \Bbb C^d$ such that $U|\psi \rangle = \lambda |\psi \rangle$. On the other hand, from the fact that $U$ is unitary we deduce that we must have $|\lambda| = 1$. It follows that $$ |\langle \psi | U |\psi \rangle| = |\lambda\langle \psi | \psi\rangle| = |\lambda| \cdot \langle \psi | \psi\rangle = \langle \psi | \psi\rangle \neq 0. $$