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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?

CuriousMind
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  • From the abstract. "between a high-fidelity multiqubit quantum experiment and players using only real-number quantum theory". Since it is a real experiment, since it is run on real superconducting matter, compared to a "calculation" using real numbers and real players it sounds definitive. It has been accepted by peer review ( Physical Review Letters, Volume 128, Issue 4, article id.040403) one has to wait for similar experiments for confirmation as with all experimental results. in case of unknown (to the peer reviewers) systematic errors – anna v Jan 30 '22 at 05:01
  • @MahirL If they two formations are different, wouldn't we be able to mathematically prove that? why do we need an experiment? – CuriousMind Jan 31 '22 at 03:42
  • @annav If they two formations are different, wouldn't we be able to mathematically prove that? why do we need an experiment? – CuriousMind Jan 31 '22 at 03:43
  • @CuriousMind the "experiment " one measures nature, the calculations are the mathematical method to model the measurements, and they disagree. – anna v Jan 31 '22 at 05:27
  • @annav aren't all complex numbers representable in real-valued matrices? see https://math.stackexchange.com/questions/180849/why-is-the-complex-number-z-abi-equivalent-to-the-matrix-form-left-begins for example. So shouldn't all complex forms have equivalent real-value-only forms? – Matt Frank Jan 31 '22 at 13:48
  • @MattFrank in the link you give the answer makes clear it is a more complicated way of modeling the square root of -1? . – anna v Jan 31 '22 at 14:48
  • @annav Yes, obviously it's more clumsy, but nevertheless it's mathematically equivalent. That's why I don't understand why the paper says it rules out the real-value formation of QM. It might rule out one formation (which must be non-standard anyways, since standard QM is based on complex numbers), but there must be an equivalent albeit clumsy real-value based QM formation, by replacing all complex numbers with real-value matrices? – Matt Frank Jan 31 '22 at 15:22
  • @MattFrank I do not think so , I think it would be the same mathematically, but it needs a theorist to reply. maybe you could ask a question on these lines. – anna v Jan 31 '22 at 16:22
  • @CuriousMind: We actually can prove that the two (or more of the) formalisms are inequivalent. For example, one of them gives an equality A, the other gives an inequality B whilst both with the same set of assumptions (e.g., as in a quantum game). An experiment is a different thing. You set it up to emulate the consequences of the assumptions of these two (which are the same), then see what happens (A, or B, or something else). Then it can be used to falsify one or the other (or both). – Mahir Lokvancic Jan 31 '22 at 16:23
  • @MattFrank: I think what they mean by the real-valued formation of QM is whereby the field of complex numbers is substituted with the field of reals -- with all else (the standard Axioms of the non-relativistic QM) the same. The two (formal systems) are not mathematically equivalent, and, luckily, the ones with reals is NOT how nature works. (Obviously, "luckily" is a subjective word; I am just speaking for myself, that life is so much for fun with quantum as it is.) – Mahir Lokvancic Jan 31 '22 at 16:32
  • @MahirL thanks. I found it hard to imagine this was not proven until this experiment? I mean there must be a reason Schrodinger and Heisenberg introduced complex numbers into QM. They definitely didn't do it for fun. It must be because complex numbers match the reality better even in their time? – Matt Frank Jan 31 '22 at 16:37
  • @MattFrank: Oh, I think this is a long and very complicated story, with many characters. From what I remember, even Schrödinger did not just declare it miraculously, but gradually arrived to it from higher order differential equations, in a series of papers. (Now of course we know many ways to derive it from other, more fundamental principles.) So, using complex numbers appears the simplest and connects beautifully with some mathematical structures -- while still works per experiments. – Mahir Lokvancic Jan 31 '22 at 16:49
  • @MattFrank: An important point is that the complex numbers are not just pairs of reals -- there is so much more in their algebra structure that seems to provide an excellent representation of reality. – Mahir Lokvancic Jan 31 '22 at 16:55
  • As J. Finkelstein pointed out in his paper, this experiment does not rule out any real quantum theory (nor do they claim to); instead, they rule out real quantum theories which respect the formalism of the standard complex theory. – mma Dec 19 '23 at 05:49

2 Answers2

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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:

  1. A physical system corresponds to a complex valued hilbert space $H$
  2. Measurements correspond to projection operators on $H$
  3. 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.
  4. 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.

MattMcEwen
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  • Thanks for a lot for the thoughtful answer. Not OP, but I found it hard to imagine this was not proven until this experiment? I mean there must be a reason Schrodinger and Heisenberg introduced complex numbers into QM. They definitely didn't do it for fun. It must be because complex numbers match the reality better even in their time? Is this experiment reinventing the wheels that Schrodinger and Heisenberg have seen 100+ years ago? – Matt Frank Jan 31 '22 at 18:24
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    I broadly think the same thing: multi-party Bell games are not unusual, and entanglement teleportation experiments have definitely been done before. My feeling is that Chan et al is published as a spesific instance directly pointing out the experimental case for Renou et al, but it not ground breaking experimentally, and I wouldn't have mentioned it without that OP asking about it spesifically. Renou et al I'd think of it as the major piece of progress here. – MattMcEwen Jan 31 '22 at 19:38
  • All that said, I think Schrodinger, Heisenburt etc introduced complex numbers because they're elegant mathematically, but they and later others didn't believe they were necessary until much more recently. Renou was significant in that it gave a very simple proof against the mostly accepted idea that complex was elegant but unecessary. – MattMcEwen Jan 31 '22 at 19:40
  • Thanks. Just to confirm, the "real-value QM" here means simply replacing all complex numbers with real numbers, thus we don't consider the mathematical tricks of replace complex numbers with a matrix like $a + ib \mapsto \left[\matrix{a&-b\cr b &a}\right]$ as "real-value QM"? – Matt Frank Jan 31 '22 at 19:58
  • @MattFrank Yes, the underlying Hilbert space is changed directly from 'complex-valued' to 'real-valued', not to say '2x2-matrix-with-constraints-valued'. Of course the latter could be equiv directly to the complex valued version. – MattMcEwen Jan 31 '22 at 22:04
  • @MattMcEwen: Re "...the mostly accepted idea that complex was elegant but unnecessary." Are you sure about this? I am wondering if there is some kind of poll that was done, or so, even if non-scientific, to testify to this? I actually think quite the opposite, that many physicists have long ago internalized that the complex numbers are not an elegant way to solve the problem, but indispensable (up to isomorphic embedding, of course, because you can always realize them in some other structure). – Mahir Lokvancic Jan 31 '22 at 22:13
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    @MattFrank Renou starts the paper with a quote that I find telling: "What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. Ψ is surely fundamentally a real function." Letter from Schrodinger to Lorentz. June 6th, 1926. – MattMcEwen Feb 01 '22 at 20:03
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  1. Complex numbers in QM (and anywhere else) can be replaced by a tuple of two real numbers, with a non-trivial action on them.

  2. 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.

  3. 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?

  4. 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 $i\otimes 1\ne 1\otimes i$.)

  5. 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.)

Norbert Schuch
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  • Re (1), what would be the non-trivial action? You must have meant something other than the structure of pairs of reals which, together with '+', the action, etc, is isomorphic to the field of complex numbers? (Because then we would just call them complex numbers.) – Mahir Lokvancic Feb 01 '22 at 21:14
  • Yes, I do mean that - it is isomorphic. But the isomorphism doesn't care about locality, and quantum physics does. (If you favor, the fact that the tensor product is $\otimes_{\mathbb C}$ does not behave nicely under the isomorphism, that is, you have redefine your tensor product to something other than $\otimes_{\mathbb R}$ to behave nicely under the isomorphism. And -apparently - there is no way to reconcile this which obeys locality.) – Norbert Schuch Feb 01 '22 at 21:31
  • Can you elaborate the part of extra complex qubits to make real-value QM? I'd have thought it'd be the other way around? You need two reals to make a complex number, but why do you need two complex qubits to make real QM? – CuriousMind Feb 02 '22 at 01:29
  • @CuriousMind I don't see where I say that. I am saying that by adding one extra qubit, everything which before could be done with complex amplitudes/operations can now be done with real ones. – Norbert Schuch Feb 02 '22 at 18:06