A Quoran asks Is distance in Hilbert space a measure of similarity between quantum states?
To which I answer:
Yes, but only to a limited extent. Quantum states are not represented by general vectors but just by unit vectors (or equivalently by “rays” – each of which consists of all real multiples of some unit vector). As such, the distance between them can never be greater than $#\sqrt{2}#$ (which happens when they are mutually orthogonal – ie perpendicular to one another).
Any two states which can be experimentally distinguished with certainty are eigenstates with distinct eigenvalues of some observable (namely the one that responds with a 1 or 0 depending on which of the two states is present) and so their unit vectors are mutually orthogonal and therefore maximally separated. On the other hand, if the distance is zero they can differ only by a phase factor and so are almost the same.
For intermediate distances between $#0#$ and $#\sqrt{2}#$ , the angle is between $#0#$ and $#\pi/2#$, so each state’s unit vector projects partially onto the other and the absolute value of their inner product (which measures the probability of one being identified as the other) is between $#0#$ and $#1#$.
But distance is not a very direct way of getting at the quantity of interest which, as noted above, is captured much more directly by just looking at the inner product.