> So I can't tell what any of this is supposed to mean at all? Quantum mechanics is defined by three axioms: 1. States are unit vectors. Vectors whose 2-norm

> So I can't tell what any of this is supposed to mean at all? Quantum mechanics is defined by three axioms: 1. States are unit vectors. Vectors whose 2-norm is 1. 2. Operations preserve the 2-norm. They are described by unitary matrices. 3. Systems are combined using the tensor product. If system A has state u and system B has state v, then the combined system (A, B) has state u⊗v. If you apply operation x to system A and operation y to system B, the operation on the combined system is x⊗y. What this paper proved is that, if you use these axioms but limit yourself to unit vectors and unitary matrices with real entries, you cannot explain some experiments. This is surprising because a complex number can be thought of as just a pair of real numbers. For example, in numpy, you can turn any complex ndarray into a real ndarray by making an ndarray with one additional index of length 2, like this: def complex_to_real(complex_ndarray): real_ndarray = np.zeroes(shape=(*complex_ndarray.shape, 2), dtype=np.float64) real_ndarray[..., 0] = np.real(complex_ndarray) real_ndarray[..., 1] = np.imag(complex_ndarray) return real_ndarray These real ndarrays can represent all states and operations that the complex ndarrays could. So how could they possibly fail to explain any experiment? The problem is axiom (3), where systems are combined using the tensor product. The issue is that complex_to_real(A ⊗ B) has one more index than A⊗B, but complex_to_real(A)⊗complex_to_real(B) has two additional indices because you gained one real-vs-imaginary index from A and also one from B. The tensor product doesn't understand that these indices should be merged, instead of concatenated. Using complex numbers tweaks the definition of the tensor product so that it does merge these indices. The experiment is basically a way of checking that you contracted those extra indices instead of keeping them both. So really this result is not about complex numbers vs real numbers. It's about how the states of quantum systems are combined. If you want to use real numbers, you can't use the normal tensor product; you have to use a modified one that understands every system has a special real-vs-imaginary index and that when combining systems you must contract these indices together. Complex numbers just happen to have a tensor product that packages this contract-one-index functionality nicely.