How does this experiment "rule out real-valued standard formalism of quantum theory"? I came cross this paper: https://arxiv.org/abs/2103.08123
To be frank I don't understand most of it, but the summary seems a bit shocking.
I found it rather strange. There have been multiple discussions on this topics on the site ( see here, here or here ). The consensus seems to be like complex number is just a mathematical trick.
If they two formations are different, wouldn't we be able to mathematically prove that? why do we need an experiment?
I also don't understand how an "Bell-like game experiment" can prove that we can only use complex numbers in QM. This sounds like a theoretical problem?
 A: The paper you linked (Chen et al.) is an experimental realisation of a game proposed in another paper (Renou et al.), and this game is only able to be accurately explained with a complex quantum theory (or something more complicated, see below).
You're right that are complex-quantum-theory and real-quantum-theory mathematically different? is an entirely theoretical problem, and that problem is answered in the positive by Renou et al. (As you can see from the previous discussions of you linked, this answer was suprising!) They proposed a spesific Bell-like gate that will have one behaviour in complex-quantum, and a different predicted behaviour in real-quantum. They also include much greater theoretical discussion of their definitions, assumptions and implications, so I'd recomend this as the paper to read if you're interested in that.
The experimental qustion then answered by Chen et al is is the real world then complex-quantum or real-quantum?. They run such a game and find that complex-quantum-theory is necessary to explain the experimental results.
What is meant by complex vs real quantum theory
Renou et al begins with a clear definition of '(complex) quantum theory', containing the usual elements:

*

*A physical system corresponds to a complex valued hilbert space $H$

*Measurements correspond to projection operators on $H$

*The liklihood of a given measurement outcome is given by the inner product of the eigenstate representing that outcome with the current state of the system; the Born rule.

*Combining two systems is just a tensor product: $H_a \otimes H_b$
'Real quantum theory' is defined analagously, but with a real-valued Hilbert space in (1).
Renou et al's game
They give a spesific game involding 'entanglement teleportaiton' which complex quantum theory allows but real quantum theory does not. Basically, it works by having A and B share a Bell pair, B and C share a different Bell pair, and B performing a Bell measurement on the two halves of the pairs they have access to, resulting (in complex quantum theory) in A and C now sharing a Bell pair. The entanglement between AB and BC has been 'teleported' to be between AC, despite A and C never interacting. Reproducing this in a real values quantum theory is not posible. Interestingly, they need a 'multi-party' Bell-like game to separate real and complex theories; a regular 'single-party' Bell game keeps the same predictions in both. Still, multiparty games of this type (if not exactly this game) have been performed before.
Chen et al impliments this game precisely on superconducting quanutm hardware. You can see in Fig2 they make 2 Bell pairs (labeled EPR), and then perform a bell measurements (labeled BSM). They show what we should expect is true for complex quantum theory. (If you're especially interested, I'll note that this game is quite easy to impliment on current superconducting hardware, and it would not be a stretch to reproduce it on IBM's quantum experience.)
Locality
Assumption (4) above is perhaps non-descript but important; it corresponds to locality of information in the theory, sometimes called 'Local Tomography'. Renou et al notes that while their game distinguishes real and complex quantum theory with assumption (4), it is easy to construct a real valued theory that prefectly matches quantum theory by also violating (4) and introducing some non-locality. Examples they give include the path-integral formalism and Bohmian mechanics. To return to the game above, reproducing the result in a real quantum theory is only possible if you add some non-locality that lets you reproduce the necessary correlations between A and C.
We should then perhaps rephrase the overall claim that we can distinguish that quanutm mechanics in our world is either complex-valued or inherently non-local (in which case the using complex or real returns to being only a methamatical convenience). The second case is 'considered nasty by physicists', which explains the choice of definitions they use: 'real-valued quantum theory' is taken to mean 'real and local quantum theory', which has now been effectively falsified.
A: *

*Complex numbers in QM (and anywhere else) can be replaced by a tuple of two real numbers, with a non-trivial action on them.


*From a quantum information point of view, this means that by attaching an additional qubit to any system, we can replace complex quantum mechanics by real quantum mechanics. Complex operations acting on part of the system can then be replaced by real actions on the same part of the system + the extra qubit.


*In a situation where we care about locality - e.g., we want to describe two (complex) qubits held e.g. by Alice and Bob - this raises a new question: What do we do with this extra qubit used to make the system real?


*This is a non-trivial constraint: If we give this qubit to Alice, she can do all "complex" operations in this "real" basis. But Bob cannot do, since then he would have to also act on this extra qubit, which is not in his possession. (At least, it is not obvious how Bob should do that. Adding another extra qubit for Bob does not obviously work, since then $i\otimes 1\ne 1\otimes i$.)


*Now I haven't looked at either of the papers, but as far as I understand this is precisely what they formalize - that no quantum theory with real numbers and locality will be able to fully capture complex local quantum theory, and they devise a specific task which only works in a local complex theory.  (This clearly goes beyond what my argument above does, since I just say that the "standard" way to make quantum mechanics real will fail.)
