Suppose that the spin-state happens to be |α〉, but we do not know this; that is, we do not know the direction α in which the electron is supposed to be spinning. Can we determine this direction by measurement? No, we cannot. The best that we can do is extract ‘one bit’ of information – that is, the answer to a single yes/no question. We may select some direction β in space and measure the electron’s spin in that direction. We get either the answer YES or NO, but thereafter, we have lost the information about the original direction of spin. With a YES answer we know that the state is now proportional to |β〉, and with a NO answer we know that the state is now in the direction opposite to β. In neither case does this tell us the direction α of the state before measurement, but merely gives some probability information about α.
On the other hand, there would seem to be something completely objective about the direction α itself, in which the electron ‘happened to be spinning’ before the measurement was made.* For we might have chosen to measure the electron’s spin in the direction α – and the electron has to be prepared to give the answer YES, with certainty, if we happened to have guessed right in this way! Somehow, the ‘information’ that the electron must actually give this answer is stored in the electron’s state of spin.
It seems to me that we must make a distinction between what is ‘objective’ and what is ‘measurable’ in discussing the question of physical reality, according to quantum mechanics. [...] (However, we shall be seeing later that this ‘objective’ picture is much stranger with more complicated systems – even for a system which consists merely of a pair of spin-one-half particles.)
— emperors-new-mindch. 6