How do we show that no hidden variable theories can replace QM?
Jherico, I see that you are keen in finding answers to your questions, or putting your views across for a debate, and this is really good. This is what science is all about. I think your questions deserve attention and proper debate.
Here is an effort from my side to help dilute some of the misunderstanding through the comments section of this forum.
(1) Your opening statement “Quantum mechanics is deterministic …”
The deterministic nature of QM is only in relation to the inherent structure of Schrodinger equation, which gives us the ability to predict the *probability for an event to take place. We can only predetermine probabilities, and this does not make QM deterministic in any way. Even if we had the most accurate Hamiltonian or Lagrangian (whatever we want to call it) we would still be calculating probabilities.
The probabilistic nature of QM is related to the fact that nature always has numerous options available to choose from, when she does something and, amazingly she picks just the right amount from each option and does a perfect job out of it!! ADMIRABLE + FASCINATING!!
(2) “We use computers …”
This has been answered by @Lubos Motl very well. I will only add the following: The use of computers to solve the complex problems in the quantum world is not a compromise over the richness of subtleties of QM. The algorithms used are of purely mathematical nature and only help us with the finding solutions to the complex mathematics involved in our problems. The hidden variables you are referring to probably relate to errors propagating and accumulating, and therefore might obscure the accuracy of the answers we obtain. But that is a purely computational problem and has nothing to do with the hidden variables physicists are referring to when they talk about hidden variables.
I hope I have cleared up some of the misunderstanding, and please do keep in touch with physics. You will find it is one of the most fulfilling enterprises undertaken by mankind.
Some discussion on Bell’s inequalities can be found in several places in this forum, but if you wish more detail you could try this book:
Speakable and Unspeakable in Quantum Mechanics (Any new edition)
Cambridge University Press
John S Bell;
I answer my own question only because the original answer provided doesn't directly answer it... although it lead me to the correct interpretations.
So to address my points...
We can absolutely replace QM with a deterministic theory and get the same predictions, we don't actually have to stop before we turn our wave functions into probabilities either as the original question suggests...
Local hidden variable theory does not simply mean 'an underlying deterministic theory' - it is meant to imply a concept referred to as 'local realism'. The computer models we use do not include 'local realism' as a constraint - importantly we can show that if they did they would be inaccurate.
What Bell's theorem shows us is that a fully deterministic theory with these properties of 'local realism' does not agree with experiment. This looks like a very special and weird case to consider without context - but previously the idea of 'local realism' was held in high regard.
(Please do correct me further if you can - particularly I can not yet agree with the comment 2) from Lubos Motl - I do not see how merely being a simulation will break Lorentz invariance - more importantly I can construct simulations where I can rotate the universe or make time go backwards without altering the algorithm - making it evolve forwards in time is a combination of choosing a simple integration strategy and providing visual feedback which is intuitive to grasp - without changing the underlying laws I can make it step 'diagonally' along some arbitrary 4-vector to produce some arbitrary foliation and the results are the same up-to approximation errors (rather than fundamental errors))
To this date there is no valid argument against the existence of deterministic, local, hidden-variable theories. Bell's theorem and its modifications only deal with non-deterministic theories because they require non-determinism (often encountered as "free-will") as their fundamental assumption.
The possibility of such theories is accepted by John Bell and also by the authors of the so-called "free-will theorem".
Until someone proposes such a theory, or a valid no-go theorem is demonstrated, there is no way to know where the truth is. QM might be fundamental or just a statistical approximation.