in 1905, some guy whose name I forget derived the otherwise strange-looking Lorentz transformations purely from the assumption that the laws of physics (including a fixed, finite value for the speed of light) take the same form in every inertial frame

You might be thinking of Vladimir Ignatowski, who attempted to derive the Lorentz transform without the speed of light assumption. Some modern papers along the same line include:

– Palash B. Pal, Nothing But Relativity

– Joel W. Gannett, Nothing But Relativity, Redux

– Jean-Marc Lévy-Leblond, One More Derivation of the Lorentz Transformation

]]>Said another way, unitary operators that act on a qubit include as a subset all the unit quaternions.

]]>1) There’s a point of view according to which, while “quanternionic quantum mechanics” isn’t a thing (because there’s no sensible tensor product), there *does* exist a sensible hybrid of real and quaternionic QM, in which the tensor product of two quaternionic Hilbert spaces is a real Hilbert space, and the tensor product of a real and a quaternionic Hilbert space is quaternionic. This is explained well, and I think compellingly, by Baez in “Division algebras and quantum theory”, https://arxiv.org/pdf/1101.5690.pdf. Thus, the various pathologies of the quaternionic tensor product aren’t in themselves a conclusive argument against the reasonableness of individual quaternionic quantum systems.

2) While I don’t find local tomography completely compelling as an axiom, neither do I think it’s just an ad hoc technical device for ruling out non-complex quantum systems. In fact, I think it’s a sufficiently principle that anyone starting to think abstractly about what a composite system (one describing joint measurements and joint probabilities) should look like, would be very likely to assume it, at least on a first pass.

]]>In QCSD you say (slightly reordering phrases for clarity) “There are exactly N^2 independent real parameters in an N-dimensional mixed state – provided we assume, for convenience, that the state doesn’t have to be normalized. … Intuitively, it seems like the number of parameters needed to describe [a composite system] AB … should equal the product of the number of parameters needed to describe A and the number of parameters needed to describe B.”

Indeed that would be a natural supposition, but it doesn’t hold in complex QM! I never understood your argument, because your neglect of normalization “for convenience” always seemed to completely undermine the argument. A physical mixed state *doesn’t* uniquely correspond to a (positive-semidefinite) Hermitian operator, but to a *trace-one* (positive-semidefinite) Hermitian operator. When your entire argument hinges on counting degrees of freedom, it seems awfully suspicious to insert an extra dummy d.o.f. “for convenience”.

But I now see (I think) that what you were really getting at was the axiom of local tomography, not just “d.o.f.’s of a composite system is product of d.o.f.’s of individual systems.”

]]>The key is that, because quaternions don’t commute, multiplying the wavefunction by a global phase can actually matter. And multiplying by a global phase is something that Bob can notice even if Alice does it.

Does that also apply to the *global* phase for a subset of qubits, or is there only one global phase for the universe?

What constitutes a system in this model? That’s the part i find confusing.

I guess there’s only one system (the universe) since apparently there’s no locality.

But then the superluminal channel would operate on the shared global phase for the whole universe, which makes it really noisy.

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