quantum mechanics - Where is the info of the wavefunction stored at?

I have read these questions:

Is the wavefunction a real physical wave or only a mathematical abstraction?

The wavefunction gives you the probability description of a particle's position (and other characteristics).

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi, respectively).

In Born's statistical interpretation in non-relativistic quantum mechanics,[8][9][10] the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle's being detected at a given place – or having a given momentum – at a given time, and possibly having definite values for discrete degrees of freedom.

But the wavefunction is just information about the particle. But where is this information stored?

Is it stored in the fabric of spacetime?

Or is it stored in the particle somehow that we do not see?

Or is it stored in an extra dimension that we do not understand?

Question:

1. Where is the information of the wavefunction stored?

After the comments, here is another question, assuming that the question where it is stored, can be understood as what the possible range of values for probabilities are:

2. Is it OK to say that the state of a quantum particle is stored in a projectivized Hilbert space?

Answer

Concepts of wave function and measurement are quite interrelated. You apply external macroscopic measuring device towards a microscopic object, then you observe how the state of this macrospic device changed. From this change you infer what was a state of microscopic quantum object. Please see Basic Concepts of Quantum Mechanics, Chapter 1, by L&L.

The state of microscopic object can be described, as well as its future behaviour can predicted with a high degree of precision by introduction of the abstraction called wave function. So the wave function, namely its parameters, itself defines microscopic state. Hence information about its parameters infered through macroscopic measurement is stored in the observer's dataset.

Hence:

1. 
   

   Q: Where is the information of the wavefunction stored?

   
   
   
   
   • A: In the observer's dataset.
   
   
   
   


2. 
   

   Is it OK to say that the state of a quantum particle is stored in a projectivized Hilbert space?

   
   
   
   
   
   • A: No, see the item above.