I'm trying to show/convince myself the following statement is correct (I haven't been able to find any similar posts):
"There is no reversible quantum operation that transforms any input state to a state orthogonal to it."
I can see how this could be true based on the operation being unitary and that you could likely find some input state that isn't transformed into an orthogonal state. Is there a simple way to show/prove this? My unsuccesful approach has been:
Let $U$ be some unitary transformation/operation, and $|x\rangle$ some input state decomposed as $$|x\rangle=\lambda_1|0\rangle + \lambda_2|1\rangle + \lambda_1|2\rangle + \cdots + \lambda_n|n\rangle$$ where $$\lambda_1 \lambda_1+\lambda_2 \lambda_2+\cdots+\lambda_n\lambda_n=1,$$ and
$$U|x\rangle = \langle x|U^\dagger$$
I'm trying to show that there exists an input state $x$ such that $\langle x|U^\dagger|x\rangle = 0$ isn't true.
I've tried to make use of the properties of unitary matrices but haven't had much luck.
Any assistance or suggestions on alternative approaches on how to show this would be greatly appreciated.