I am a little confused by an elementary point. Something must be wrong with the following: On one hand, a Hermitian operator (which is not necessarily unitary) takes one state to another state. Hence a state need not be represented as a unit vector; its norm can be greater (or less than) one. On the other hand, the probability amplitude for one state to collapse upon measurement into another (or the same) state is the inner product of the two states; therefore the norm of the state's representative vector must be one. These contradict each other. What is wrong? Does it have anything to do with a distinction between pure and mixed states?