Quantum Mechanics without Complex Numbers in a multipartite setting

Question

I was fairly convinced that usual QM formalism didn't necessitate the use of complex numbers and that ultimately they're just a matter of convenience and utility rather than anything fundamental. This paper claims that isn't the case. But their demonstration doesn't satisfy me because it's muddled up in this Bell-like experiment. Can anyone explain what's going on in a simplified manner? Their argument does not make it clear to me why splitting the wave function into its real and imaginary portions and treating them as two quantities which are coupled via Schrodinger evolution isn't always possible---even in the case of multipartite states.

Answer 1

It is indeed true that generally, quantum mechanics can be phrased with real amplitudes by adding an extra two-level system (let's call it RI), whose state encodes whether the amplitude is the real part or the imaginary part of the "original" amplitude. This is discussed at great length in Quantum mechanics without complex numbers.

However, in order to implement an operation which itself involves complex numbers, one has to act on both the original system and the extra RI system. For instance, to implement a rotation $$ U_C = \begin{pmatrix}\sin\phi & i \cos\phi \\ i\cos\phi & \sin\phi\end{pmatrix} $$ using a real two-qubit encoding $$ |\psi_R\rangle = a|0\rangle_A|0\rangle_{RI} + b|0\rangle_A|1\rangle_{RI} + c|1\rangle_A|0\rangle_{RI} +d|1\rangle_A|1\rangle_{RI} $$ of a complex state $$ |\psi_C\rangle = (a+ib)|0\rangle_A + (c+id)|1\rangle_A\ , $$ one has to apply a rotation $$ U_R = \begin{pmatrix}\sin\phi & 0 & 0 & -\cos\phi\\ 0 & \sin\phi & \cos\phi & 0 \\ 0 & -\cos\phi & \sin\phi & 0 \\ \cos\phi & 0 & 0 & \sin\phi \end{pmatrix}\ . $$

This works fine, and allows to replace quantum mechanics with complex amplitudes by quantum mechanics with real amplitudes.

The situation chances, however, if more than one party is involved. Then, a complex-valued state in the basis $|i\rangle_A|j\rangle_B$ is replaced by a real-valued one in the basis $|i\rangle_A|j\rangle_B|c\rangle_{RI}$. Now imagine that Alice wants to implement the operation $U_C$ on her qubit. In the real-valued basis, this requires her to implement $U_R$ jointly on the registers $A$ and $RI$ -- thus, she must be holding both qubits $A$ and $RI$.

But now let's imagine Bob wants to implement the same operation: He is in deep trouble now, because he is not holding the qubit $RI$, and thus, he is not able to carry out this operation, which he would be able to carry out in the setting with complex amplitudes!

This is the basic idea why a situation with more than one party is different.

Now, this is of course not a proof -- it is simply an illustration that the situation is different with several parties. It does indeed seem plausible that there is a way to overcome this, by using several qubits to store the real/imaginary part information.

The paper you quote proves that this is in fact not possible in a setting with three parties, and that there is a Bell-type inequality which is obeyed by quantum mechanics with real numbers, but can be violated by quantum mechanics with complex numbers (just as standard Bell inequalities, CHSH, etc., are obeyed by local hidden variable models, but can be violated in quantum theory.) Note that this is a highly non-trivial result and settled a long-standing open question, as illustrated by the attention it received.