Quantum mechanics uses complex numbers. The amplitudes are complex, the Hilbert space is complex, the interference patterns require complex phases. Recent work claimed to show that complexity is experimentally necessary — that network experiments could distinguish standard quantum theory from Real Quantum Theory, which uses only real-valued amplitudes.
Hoffreumon and Woods prove the distinction collapses. Once source independence is imposed operationally — defined by what experimenters can actually control rather than by mathematical constraints on the state space — every finite network correlation achievable in standard quantum theory is also achievable in Real Quantum Theory. The two theories are experimentally indistinguishable.
The crux: the prior claim of distinguishability relied on a mathematical definition of source independence that is stricter than what any experiment can enforce. The mathematical constraint says the joint state must be a tensor product. The operational constraint says the experimenters prepare their sources independently. These are different conditions. The mathematical one excludes certain real-valued states that the operational one permits. When the operational definition is used — which is all any experiment can test — the exclusion disappears.
The structural lesson: the question “are complex numbers necessary for quantum mechanics?” is not an empirical question. It is a question about formalism. Both formalisms produce the same predictions for all finite experiments. The complex numbers are not describing a feature of nature that the real numbers cannot describe. They are describing the same feature more conveniently. Convenience is not necessity, and no experiment can tell the difference.