How can something be an axiom and at the same time has a mathematical derivation well understood from Quantum Mechanics [1] and are supported by experimental observations ?
The idea of axioms in mathematics is not quite the same as it is with physics.
Physicists don't need the type of mathematical rigor such as the set theory axioms to build up all of mathematics. In fact, by keeping in mind the multitudes of paths from which one part of physics could be related or derived from another part, you are exploring the possibly undiscovered laws in nature.
It will only be relevant to axiomatically define physics (from a set of basic axioms to reach all laws) after we confirmed to have discovered all of physics. That day hasn't come yet - there's no grand unified theory so far, and there's still stuff that we dont know surely.
> How can something be an axiom and at the same time has a mathematical derivation well understood from Quantum Mechanics
See my notes, it's not an axiom, just take it for granted for the explanation to work. You can derive it from experiments like squeezing light, or you can derive it from other things as you mentioned.
[1] https://courses.physics.illinois.edu/phys580/fa2013/uncertai...