Let $|\psi\rangle = \frac{1}{\sqrt{2}}\sum_{k=0}^{1}(-1)^{ka}|k, k \oplus b\rangle $ so that $|\psi'\rangle = \frac{1}{\sqrt{2}}[\sum_{k=2}^{1}(-1)^{ka}(U_{A}|k\rangle \otimes U_{B}|k \oplus b\rangle)]$ where $U_{A}, U_{B}$ are unitary operators.
The action of $U_{A}|k\rangle$ gives $U_{A}|k\rangle = |k_{A}\rangle$ which is equivalent to rotating $|k\rangle$; likewise, $U_{B}|k\rangle = |k_{B} \oplus b_{B}\rangle$.
But correctly: $U_{B}|k\rangle = |k_{B} \oplus b\rangle$.
Why is this so?
