I have a unitary matrix $U$ and a quantum state $\vert \Psi \rangle$ such that $$ U \vert \Psi \rangle = e^{i \theta} \vert \Psi \rangle.$$ I also know that my unitary matrix and my quantum state can be written as $U = U_1\otimes U_2$ and $\vert \Psi \rangle = \vert \Psi_1\rangle\otimes\vert\Psi_2\rangle$ with matching dimensions (i.e. $U_1$ acts on the Hilbert state $\vert\Psi_1\rangle$ belongs to, same for $U_2$ and $\vert\Psi_2\rangle$).
My first equation then becomes $$U_1\vert\Psi_1\rangle \otimes U_2 \vert\Psi_2 \rangle = e^{i\theta} \vert \Psi_1 \rangle \otimes \vert \Psi_2 \rangle.$$
Can I deduce the 3 following assertions from my formula above?
- $U_1 \vert\Psi_1\rangle = e^{i\theta_1}\vert\Psi_1\rangle$
- $U_2 \vert\Psi_2\rangle = e^{i\theta_2}\vert\Psi_2\rangle$
- with $\theta = \theta_1 + \theta_2 + 2k\pi$.
The implication seems "logical" as we just separate $2$ non-related quantum states from each other, but I do not have enough confidence in my reasoning to accept this result yet. I searched for a mathematical proof, but the closer I found is the following theorem from here:
Theorem: Let $A$ and $B$ be two complex square matrices. If $\lambda$ is an eigenvalue of $A$ with corresponding eigenvector $x$ and $\mu$ is an eigenvector of $B$ with corresponding eigenvector $y$, then $\lambda\mu$ is an eigenvalue of $A\otimes B$ with corresponding eigenvector $x\otimes y$. Moreover, every eigenvalue of $A\otimes B$ arises as such a product.
that does not fit with my problem (not the good implication and more oriented toward eigenvectors).