To be precise, if your prior is 50% left and 50% right, then you generally believe that the world you are in is either a left world or a right world, and you don’t know which. A left or right world itself factorises into a tensor product of (rest of the world) × (superposition of up particle and down particle). Measuring the particle along the up/down axis causes the rest of the world to be become entangled with the particle along that axis, splitting it into two worlds, of which you observe yourself to be in the ‘up’ one.
Of course, observing the particle along the up/down axis tells you nothing about whether its original spin was left or right, and leaves you incapable of finding out, since the two new worlds are very far apart, and it’s the phase difference between those two worlds that stores that information.
To be precise, if your prior is 50% left and 50% right, then you generally believe that the world you are in is either a left world or a right world, and you don’t know which. A left or right world itself factorises into a tensor product of (rest of the world) × (superposition of up particle and down particle). Measuring the particle along the up/down axis causes the rest of the world to be become entangled with the particle along that axis, splitting it into two worlds, of which you observe yourself to be in the ‘up’ one.
Of course, observing the particle along the up/down axis tells you nothing about whether its original spin was left or right, and leaves you incapable of finding out, since the two new worlds are very far apart, and it’s the phase difference between those two worlds that stores that information.