“Uncertainty is not a statement about the limits of measurement, it’s a statement about the limits of reality. Asking for the precise position and momentum of a particle doesn’t even make sense, because those quantities do not exist. This fundamental uncertainty is a consequence of the dual nature of quantum particles.”
In “How to Teach Quantum Physics to Your Dog” by Chad Orzel
“CENTRAL PRINCIPLES OF QUANTUM MECHANICS:
1 – Wavefunctions: Every object in the universe is described by a quantum wavefunction;
2 – Allowed states: A quantum object can only be observed in one of a limited number of allowed states;
3 – Probability: The wavefunction of an object determines the probability of being found in each of the allowed states;
4 – Measurement: Measuring the state of an object absolutely determines the state of that object.
In “How to Teach Quantum Physics to Your Dog” by Chad Orzel
Fairly basic take on Quantum Mechanics. It had to be to make it intelligible to a make-believe-Orzel-disguised-as-a-dog…
It doesn't fully address the core ambiguity in the 4 postulates Orzel uses (see quote above, namely the 4th postulate): What exactly is measurement? That is, the new postulates that they propose simply assume measurement to be a primitive notion of the theory, not reducible to anything more fundamental. Orzel can’t answer the troublesome question of why measurement outcomes are unique; rather, it makes that uniqueness axiomatic, turning it into part of the very definition of a measurement. And since it does not address what the measurement process is actually doing, it also does not address the issue "Did you even succeed in measuring the thing you thought you were measuring?" It is easy to show that "quantum correlations" and entanglement have nothing to do with either spooky action at a distance or hidden variable. It is caused by the purely classical phenomenon of inter-symbol interference together with noise, associated with Shannon's definition of a single bit of information. You cannot measure two independent parameters from an entity manifesting only one such bit. This is the ultimate origin of Heisenberg's Uncertainty Principle. Any attempt to even try to perform a second measurement, is guaranteed to be corrupted by the intrinsic properties (noise and inter-symbol interference) inherent to one of Shannon's "bits".
The question of "what exactly is measurement?" is squarely addressed and answered in The Transactional Interpretation, which yields a physical (as opposed to decision-theoretic) derivation of the Born Rule (Orzel only mentions it en passant, preferring to dedicate a whole chapter to the MWI). For specifics, including calculations, see which provides an explicit derivation of the Born Rule for radiative processes) and see this too... I’m not sure the dog would be able to understand it though…loved the way Orzel explains the Uncertainty Principle by adding wave-functions and using this approach to also explain Schrödinger’s Cat.
On the other hand, there is no mystery to the Born rule. The entire process of computing a wave-function and then computing the sum of the squares of its real and imaginary parts, amounts to nothing more than computing the power spectrum of a Fourier transform. The power spectrum (as the name implies) simply measures the energy accumulated/detected within each "channel' of a filter bank. When the energy happens to arrive in discrete, equal quanta, the ratio of (total energy)/(energy per quanta) yields the number of quanta accumulated within each channel. In other words, the entire mathematical procedure amounts to nothing more than the description of a histogram which is why it yields probability estimates. Every photon, gauge boson or quantum object that appears travelling at c to us is at the same time an "observer" of that part of the universe that involves its emission-flight-detection path (or better, "process"). It feels that space-time "chunk" of the universe as a single point of existence, with no distances and no intervals involved. That's why "paths" have no sense for them, or why wave/particle duality is not resolved until detection: because time intervals (or space distances, for that matter) have no sense for quantum objects that go at c. Their single bit of existence (from their point of view) is a probability function for us, until it collapses when we detect them. But for them, that collapse happens at the same "time" they are emitted, because there's no time involved in their experience of the universe, the whole "emission-flight-detection" process is experienced at once from their POV. So the universe would "exist" the same way without conscious beings, only it will not be "perceived" the same way you are used to (the space-time ratios conscious beings create in our brains). My point is specifically that the multiple interpretations of quantum mechanics means the philosophical question of whether the world is deterministic or not is still unsolved. By neglecting pilot wave theory due to its impracticabilities is reasonable concluding that the world is fundamentally random is not. All reasonable alternatives world need to be discarded not just a few fringe models.
Of course, The Uncertainty Principle is more fundamental than the Born rule. The former arise from the logical contradiction of trying to locate a particle with non-zero size in a point in space. A particle is not located in any one point in space but in a region of space that mathematically contains infinite number of points. There is contradiction between the math we're using and our physical intuition. We remedy this logical contradiction by imagining that the location of a point particle in a region of space is governed by probabilities. The Born rule uses the square of the amplitude and it works because it represents the area perpendicular to the velocity vector of an imaginary point particle.
One last piece of advice: next time lose the dog...