Difference between unitary transformations and measurement

Say I have some qubit $$|q\rangle = \alpha |0\rangle + \beta |1\rangle$$. If I apply some unitary $$U$$, I get $$U |q\rangle$$; great. I can also think of $$U$$ as a change-of-basis matrix that maps the $$|0\rangle, |1\rangle$$ basis to the $$U|0\rangle, U|1\rangle$$ basis. If I just decide to measure the original $$|q\rangle$$ in that new basis, then all of my very limited knowledge about all of this would lead me to believe I get $$U|q\rangle$$. So is there any actual difference between the two situations? This seems like a well-known or obvious thing but I really don't know.

• The question in its current form does not make sense to me: there is little or no in common between unitary transformations and measurement. Maybe OP wanted to ask about the difference between unitary operators and unitary basis transformation, but this is just my guess. Nov 18, 2019 at 6:57

Application of a unitary transformation $$U$$ on a state $$|q\rangle$$ really leads to a new state $$U|q\rangle$$. What you get after a measurement is a one particular outcome of $$U|q\rangle$$ state because the measurement leads to colapse of the state wave function.