A quantum state has to have a global phase, but this can never be measured, so it isn't information (?!)

So what's left? The global phase is included (it has to be) in measurement operator notation; measurement in QM is the same operation as converting the momentum into a single bit of . . . information.

So, unpick the standard double-slit experiment in terms of a set of measurements of individual particle states; the global phase vanishes for each particle that hits the screen (and was never going to be part of the measurement).

To get the probability of where each dot will appear you need an expression that accounts for the global phase which is "gauged out" by the operation of measurement; in the case of the double slit experiment, the measurement operators are what create the dots on a screen, the position of the screen is another kind of gauge factor (!).

Or Something.