[whatshisface]

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.

[1] https://arxiv.org/pdf/1704.08525.pdf

[sw1sh]

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.

[whatshisface]

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.