The Real Indistinguishability

Standard quantum mechanics uses complex numbers. Amplitudes are complex; the Hilbert space is over the complex field; interference patterns arise from complex phases. Recent work claimed to experimentally distinguish complex quantum theory from a real-number variant by showing that certain network correlations require complex amplitudes.

Hoffreumon and Woods show that claim relied on an untestable assumption.

The previous experiments imposed source independence as a mathematical constraint: the joint state of particles from different sources must factor as a tensor product. This is a natural assumption in complex quantum theory, where independence and tensor products are synonymous. But in real quantum theory, the relationship between independence and tensor structure is different. The mathematical constraint was doing work that the physics didn’t require.

When source independence is defined operationally — in terms of observable correlations rather than mathematical structure of the state space — every finite network correlation achievable in complex quantum theory is also achievable in real quantum theory with the same locality structure. The two theories predict identical experimental outcomes.

The theories are structurally different. They represent different mathematical frameworks with different correlation structures. But the differences are invisible to any experiment. No observation can tell them apart.

This doesn’t mean complex numbers are unnecessary — they may be mathematically simpler, more natural, or more explanatory. But the argument that nature requires complex quantum mechanics, that experiments have proven real quantum theory wrong, doesn’t hold. The experimental evidence is equally consistent with both.

The complexity of quantum mechanics may be a choice, not a discovery.