As mentioned elsewhere, bits are probably the wrong way to think about it. Does it really double the size of your Hilbert space? Yes. Perhaps this will help cl

As mentioned elsewhere, bits are probably the wrong way to think about it. Does it really double the size of your Hilbert space? Yes. Perhaps this will help clarify why it is different. In the classical case, adding one more spin is doubling the size of your set. In the quantum mechanical case, adding one more spin instead doubles the dimensionality of the space of possible states.