What does the lost information in the wavefunction mean?

[Runner 1]

The wavefunction is defined on the domain of complex numbers. To find the probability of discovering a particle in a certain region, the amplitude of the wavefunction is integrated over that region. The problem is that you have an infinite set of complex numbers mapping to a single amplitude. What does the information that is lost when the wavefunction is "squared" represent?

 

[Demystifier]

A complex number z can be represented by two real numbers: the absolute value |z| and the phase of z. The probability density is equivalent to the absolute value of complex wave function and does not depend on the phase. If I understood your question correctly, you ask what the phase of complex wave function corresponds to, what is the physical meaning of it?

The answer is that the phase plays a role in calculating probability densities of other observables (not positions), such as momentum or various complicated functions of both position and momentum.

An alternative answer is that, according to the Bohmian interpretation, the phase determines a deterministic (not merely probabilistic) physical property - the velocity of the particle at a given position.