The equivalent formulations aren't the alternate theories that information theorists study when they say, "QM without complex numbers." They study very much non-equivalent theories, although they feel unnatural enough that it's hard to see what they're all about unless you closely study the technical details. To tell you the truth I don't have any picture of what restricting the matrices to real numbers means physically.
These alternate theories that use quaternions and whatnot are just mathematical marvels that have nothing to do with physics, QM can be well explained with conventional probability theory already (even without generalized probability theories that people also study). Seems like negative conditional probabilities governing quantum processes can be understood as intrinsic Bayesian inference or Particle filter estimators. I wish there were more research in this direction ...
That's not what I'm saying. Bell or any CHSH-like experiment can be equivalently described using random variables and quasi-stochastic processes instead of quantum states, unitaries, and measurements. It would still involve non-local correlations and inequality violations, but without mentioning the Born rule, phases, and interference with imaginary numbers. It is just an equivalent mathematical framework.
This paper [1] doesn't stoop to providing an example, but isn't that just the thing where you can write 1 and i as 2x2 matrices? I don't think that's what Scott is talking about. Requiring that the elements of the density matrix be real (or allowing them to be quaternion) creates a non-equivalent theory.
Right, these are different questions indeed. Scott wonders what happens to amplitudes as they already appear in the theory but with numbers being no longer complex. But those lifted representations effectively change to a specific basis in higher dimensions (think qubit's 2x2 density matrix becoming a 4-dimensional distribution vector, with the same 3 real degrees of freedom) where everything is real and interpreted as probabilities.
Well, yes, but real matrices are also a subspace of complex matrices, you don't have to switch to a real valued representation of GL(n) to arrive at that.
The measurement operators are matrices that come about as a result of assigning real eigenvalues (these are your possible measurement outcomes) to orthonormal vectors (your arbitrary coordinate system). The results are hermitian, complex-valued matrices, because that's just what comes out if you try to engineer a matrix to have those eigenvalues and vectors. The rest follows from that.
Trying to fit a real number constraint somewhere, other than the one that's already there (real measurement outcomes), to me seems like the step you would have to justify, not the absence of one.
The complex numbers in the matrices appear as a consequence of trying to do something else. I don't think they have much physical meaning on their own, which is why I am surprised that people ask what it would mean if they had to all be real numbers.
