The Unnecessary Complex

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.