“The quantum world is context dependent“
Before we even attempt to explain quantum theory it is absolutely essential to know what needs explaining. There are certainly many weird properties of the quantum world – but which are fundamental? What can be derived? Most telling of all is not so much the fundamental properties of the theory – after all a better theory could come along. What do we know about the quantum world that distinguishes it from the classical world?
The uncertainty principle, the complex functions, and entanglement are not fundamentally quantum.
The starting point is that the wavefunction is a type of probability function.
“Fact – The wave function is a probability function“
It allows us to calculate the probabilistic results of experiments. That is a fact. Now some scientists attribute additional meaning to the wavefunction, but that is an interpretation, not a fact. Being a probability function is in itself not remarkable -think of a theory of dice throws or playing cards. The most useful theories are probabilistic, they are good theories and they are not trivial.
The probability function for a dice throw is (1/6,1/6,1/6,1/6,1/6, 1/6) for the outcomes (1,2,3,4,5,6). For a fair dice it cannot be anything else. That is quite a deep result. We might think that a six is less likely than 3 or 4, but symmetry says it cannot be. And note that the distribution does not depend on previous dice throws. The idea that after several rolls without a six, then a six becomes more likely is not true. (if anything a long run without sixes might lead to us questioning if it is a fair dice and judging a six less likely in the future).
It is highly significant that the probability function does not depend on what we look for. We might bet on a six, or might only be interested in even vs odd; or high vs low, but that makes no difference.
“Classical probabilities can always be represented by a Venn diagram”
Classical probability functions are the subject of rigorous mathematical probability theory. Essentially the probabilities can be represented by, and even defined by, a Venn diagram where the areas are proportional to the probability. In technical terms they are a weighted volume measure.
Quantum theory represents probabilities differently. And we now know that probabilities in the quantum world cannot be defined or represented in a classical way. This is a powerful statement, well two powerful statements actually. it is a mathematical result that quantum theory is incompatible with classical probability assignments (read about Bell’s inequalities derived by John Bell in 1960). And it is an experimental result that the quantum world cannot be described by any classical theory (read about Alain Aspect’s 2022 Nobel prize winning experiments)
We can restore classicality to a limited extent because for any given experiment, quantum theory can be described in a classical way, there are good reasons why this has to be so. But we can’t draw the Venn diagram without knowing what will be measured – even if the measurement won’t be chosen until a future time. Technically, the Yes/No questions that can be asked in quantum theory form an orthomodular lattice of propositions, while the propositions of classical physics satisfies a Boolean logic.
“You can’t describe a quantum state with a Venn diagram“
Everything about quantum theory follows from the orthomodular logical structure plus a few standard assumptions about symmetry and continuity. That, very briefly, is the essence of quantum theory. In simple terms quantum theory is context dependent. It is only possible to describe a quantum state when you know what measurements will be made.
The entire history and development of quantum theory has been a successful attempt to disguise the context dependence. States are described independently of the measurement, but that requires the complex state vector and a quadratic rule to pull probabilities out for different measurements. Probability is redefined by physicists with surprisingly little concern.
The links above to wikipedia articles are clear and concise descriptions. My own language on this page is quite lax, so if you want a more precise explanation then the Wiki articles are a good place to go next before perhaps reading my original publications.