Locality Implies Complex Numbers in Quantum Mechanics

We show that a real-number quantum theory, compatible with the independent source assumption, requires the inclusion of a nonlocal map. This means that if the independent source assumption holds, complex-number quantum theory is equivalent to a real-number quantum theory with hidden nonlocal degrees of freedom. This result suggests that complex numbers are indispensable for describing the process involving entanglement between two independent systems. That is, quantum theory fundamentally requires complex numbers; otherwise, one may have to accept a nonlocal real-number quantum theory.