Marc-Olivier Renou, David Trillo, Mirjam Weilenmann
Thinh P. Le, Armin Tavakoli, Nicolas Gisin, Antonio Acin, Miguel Navascues
2021 · arXiv
While complex numbers are essential in mathematics, they are not needed to describe physical experiments, expressed in terms of probabilities, hence real numbers. Physics however aims to explain, rather than describe, experiments through theories. While most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural. In fact, previous works showed that such "real quantum theory" can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states. Thus, are complex numbers really needed in the quantum formalism? Here, we show this to be case by proving that real and complex quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment whose successful realization would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.
Metadata: arXiv. Abstracts retain their original authors’ rights.
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Perhaps of your interest might be this work https://arxiv.org/abs/2101.10873v1 on why quantum physics needs complex numbers to work. Interesting noting though that as for solving polynomials, quantum physics might be also considered a “convenience” within the Copenhagen interpretation
>These all rely on real numbers, so there's no computational complexity to talk about. There's a pretty decent argument real numbers are not enough: https://www.nature.com/articles/s41586-021-04160-4/ https://physics.aps.org/articles/v15/7
> Imaginary numbers are just 2D vectors. You meant complex numbers. I wrote "imaginary numbers" on purpose. > The imaginary part isnt any more mysterious than the real part. What's surprising is that imaginary numbers are required for QM. Not the 2d vector in one variable trick, actual imaginary numbers. See for example: https://www.nature.com/articles/s41586-021-04160-4
Sometimes math can be pretty surprising, such as when complex numbers might be shown to be necessary to explain quantum mechanics: https://www.nature.com/articles/s41586-021-04160-4 Or when group theory predicts subatomic particles through symmetries: https://www.britannica.com/science/subatomic-particle/Hidden... But other things like continuity and limits, as well as various topological spaces, seem to be purely mathematical constructs.
This is your regularly scheduled reminder that the author of these papers (https://arxiv.org/abs/2101.10873, https://arxiv.org/abs/2111.15128), which this article is based on know very well that complex numbers have real matrix representations. What they add which your comment discounts is the locality structure of quantum mechanics, i.e. what happens when you combine multiple quantum systems. Specifically if the state of one system lives in A, and the state of another system lives in B then the state of a combined system lives in the tensor product A ⊗ B. If you do the trick where you replace complex numbers by real matrices what you end up is having the state of the…
first system living in A = X ⊗ A', and the second is in B = X ⊗ B', where A' and B' are real and the X subsystem is the degree of freedom you're using to "fake" the complex numbers. Then if you try to combine the two systems you end up with a state that lives in A ⊗ B = (X ⊗ A') ⊗ (X ⊗ B') but what you need in order to get the behavior you want is actually for the combined system to live in X ⊗ (A'⊗ B'). What they do is a bit more complicated because they show that all ways to fake complex numbers using real numbers breaks this way of combining systems, but this is the gist.
I didn't have time to read all of it, but it seems to focus a bit much on the imaginary numbers aspect of complex numbers, and how they're all spooky and weird. Reading this abstract[1] and this[2] StackOverflow discussion about the topic, it seems the point is rather more along the lines that complex numbers aren't just plain two-dimensional vectors of real numbers. There's an extra constraint involved by requiring that i^2 = -1, which could be written other ways, and quantum physics requires this constraint to work. Then again, I'm just a programmer. [1]: https://www.nature.com/articles/s41586-021-04160-4 [2]: https://physics.stackexchange.com/questions/691623/how-does-...
Whole numbers can be defined and proven to be necessary to describe the world pretty easily. From there, rational numbers are trivial to define. Negative numbers are somewhat more abstract, but they have very intuitive definitions in many domains, such as accounting. It may be possible to avoid them in a theory of physics, though. The complex numbers (well, at least those with a rational imaginary part and a rational real part) have been recently proven to be necessary to describe the universe[0] (assuming quantum theory is correct). The irrational numbers are then are the only numbers that are harder to pin down, and I'm not sure that there is a way to prove that any physical quantity has an irrational value, vs a rational value that is arbitrarily close to that irrational value. [0] https://arxiv.org/abs/2101.10873
The paper is describing a test of a particular alternative to quantum mechanics called "real quantum mechanics", its rather more specific than you are suggesting https://arxiv.org/abs/2101.10873 I know a few of the people on that preprint, and I'm pretty certain they all know you can embed the complex numbers in the real 2x2 matrices ;) As an aside I'm pretty sure the matrix you need is [[0,-1],[1,0]], which squares to -I, unless you don't want to use matrix multiplication in your construction.
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