About Ruling Out Real-Valued Standard Formalism of Quantum Theory

[Haorong Wu]

TL;DR Summary

A recent experiment rules out the real-valued standard formalism of QM.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.128.040403

In QM, I was taught that the imaginary unit $i$ in wave functions is merely a mathematical tool. It has no physical meaning. We can always take the real part of the complex wave functions. Therefore, there should be some real-valued formalism of QM. However, a recent paper rules out this conclusion. It seems that the imaginary unit $i$ does have some physical meaning. What would you think about this?

 

[Demystifier]

I think that real-valued standard formalism of QM is an oxymoron. Standard QM formalism is not real valued. QM can be formulated equivalently with real numbers only, but that's not standard.

 

[PeterDonis]

We can always take the real part of the complex wave functions.

You could, but that would lose information. You would need to treat both the real and the imaginary parts as separate functions.

Therefore, there should be some real-valued formalism of QM.

Yes, but not by just taking the real part of complex functions.

a recent paper rules out this conclusion.

That would not seem possible, since any valid formalism for QM must make the same experimental predictions as the usual complex Hilbert space formalism, so it would not be possible to rule out any valid formalism by experiment, as the paper claims to be doing.

 

[anuttarasammyak]

For information commentary on this paper and another Chinese paper on this issue using photonic quantum processor is here https://physics.aps.org/articles/v15/7 Quantum Mechanics Must Be Complex Alessio Avella National Institute of Metrological Research (INRIM), Turin, Italy January 24, 2022.

 

[Haorong Wu]

@PeterDonis. But I think the information of a complex number is contained in its magnitude and its phase. By taking the real part, both parts are kept, since for a complex number $A=A_0 e^{i\phi}$, its real part is $A_0 \cos \phi$. What do I misunderstand?

 

[vanhees71]

If you have only $\text{Re} A$ you can't calculate both the magnitude and phase. You through away information. I must admit that I don't understand the hype the press makes out of this result. Standard QT is formulated over a complex Hilbert space, and I think that should be so, because we need the relation between self-adjoint operators, there eigenvalues, and the completeness and orthogonality of their (generalized) eigenvectors.

 

[Demystifier]

@PeterDonis. But I think the information of a complex number is contained in its magnitude and its phase. By taking the real part, both parts are kept, since for a complex number $A=A_0 e^{i\phi}$, its real part is $A_0 \cos \phi$. What do I misunderstand?

If you know $A_0 \cos \phi$, you don't know separately $A_0$ and $\cos \phi$. For instance, if I tell you that $A_0 \cos \phi=7$, can you tell me what's $A_0$ and what's $\cos \phi$?

 

[PeterDonis]

I think the information of a complex number is contained in its magnitude and its phase.

Yes, and both of those are real numbers, so the information in a complex number is two real numbers.

By taking the real part, both parts are kept

No, they aren't. They can't be, because the real part is just one real number, which can't possibly contain the same information as two.

 

[Haorong Wu]

Oh. Thanks! I mistakenly think it as in optics.

 

[PeroK]

In QM, I was taught that the imaginary unit $i$ in wave functions is merely a mathematical tool. It has no physical meaning. We can always take the real part of the complex wave functions. Therefore, there should be some real-valued formalism of QM. However, a recent paper rules out this conclusion. It seems that the imaginary unit $i$ does have some physical meaning. What would you think about this?

The unit $i = \sqrt{-1}$ is the standard mathematical way to obtain an algebraically closed field of scalars. Algebraic closure implies that the equation $x^2 + 1 = 0$ has a solution. You can equally achieve this with the equivalent formulation of complex numbers as 2x2 matrices of the form:
$$a + bi \leftrightarrow
\begin{bmatrix}
a& b\\
-b&a
\end{bmatrix}
$$In that formalism, we have:
$$i \leftrightarrow
\begin{bmatrix}
0& 1\\
-1&0
\end{bmatrix}
$$Isn't that a rotation matrix? That would rotate a 2D vector through an angle of $\frac \pi 2$ clockwise. Doesn't that have a physical meaning?

 

[vanhees71]

Oh. Thanks! I mistakenly think it as in optics.

In optics you deal with real fields only (that's possible for relativistic wave equations and usually means you deal with strictly neutral "particles" in the sense of QFT, and the photon is a strictly neutral "massless particles").

Since the Maxwell equations are linear ODEs with real coefficients for real fields you can take the real part of a complex solution as a solution. That's sometimes advantegeous when you deal with fields of harmonic time dependence, because then you can work with an exponential function $\exp(-\mathrm{i} \omega t)$ instead of $\cos(\omega t)$ and/or $\sin(\omega t)$, which is often more convenient.

 

[Demystifier]

Oh. Thanks! I mistakenly think it as in optics.

For one particle, there is no big difference between QM and optics. The true difference arises when you deal with two or more particles. For instance, consider a wave
$$\psi(x,t)=A(x,t)+iB(x,t)$$
where $A$ and $B$ are real. Such a wave can represent either a quantum particle or a wave in classical optics.

But now consider two independent (i.e. not entangled) quantum particles, each with wave function
$$\psi_1(x_1,t)=A_1(x_1,t)+iB_1(x_1,t)$$
$$\psi_2(x_2,t)=A_2(x_2,t)+iB_2(x_2,t)$$
Then the total quantum wave function is
$$\Psi(x_1,x_2,t)=\psi_1(x_1,t)\psi_2(x_2,t) {\;\;\;\;} (1)$$
In terms of real and imaginary parts, it is
$$\Psi=(A_1A_2-B_1B_2)+i(A_1B_2+B_1A_2)$$
In particular, note that
$$\Psi=\psi_1\psi_2 \neq A_1A_2 + i B_1B_2$$
so combination of two particles is simple in terms of complex waves $\psi_a$, but complicated in terms of real waves $A_a$, $B_a$. That's the true reason why QM is naturally formulated in terms of complex waves, rather than real ones.

On the other hand, the quantity (1) has no meaning in classical optics. Instead, a combination of two classical waves can only be a superposition
$$\psi(x,t)=\psi_1(x,t)+\psi_2(x,t)$$
which is simple even in terms of real waves
$$\psi=\psi_1+\psi_2=(A_1+A_2)+i(B_1+B_2)$$

Of course, a simple superposition as above makes sense in QM too, but such a quantum superposition describes one particle, not two particles. As I said, the true difference between QM and classical optics arises when QM deals with two or more particles.

 

[Cthugha]

That would not seem possible, since any valid formalism for QM must make the same experimental predictions as the usual complex Hilbert space formalism, so it would not be possible to rule out any valid formalism by experiment, as the paper claims to be doing.

I do not really see the point of this argument. The main point of the PRL in question is that real-valued QM indeed provides different predictions compared to complex-valued QM in entanglement swapping scenarios. That has been predicted already last year and the PRL is just an experimental verification. The original prediction can be found here:
Renou et al., Nature 600, 625 (2021)
The ArXiv version is available here:
Arxiv version
The proof can be found in the supplementary information. Loosely speaking, you run into problems when trying to get the tensor product structure of composite quantum systems using real-valued approaches. The definition of "real valued approach" can be seen, e.g., from Stückelberg's paper:
Quantum Theory in Real Hilbert Space, hosted by ETH Zurich

 

[PeterDonis]

The main point of the PRL in question is that real-valued QM indeed provides different predictions compared to complex-valued QM

In other words, "real-valued QM" is not QM. It's a different theory that makes different predictions from QM. So the term "real-valued QM" is a misnomer.

 

[vanhees71]

So it's a big hype in the popular media about nothing. It's just an alternative theory, which has been disproven by experiment. Given all the amazing successes of standard QT that's not so surprising after all...

 

[Cthugha]

In other words, "real-valued QM" is not QM. It's a different theory that makes different predictions from QM. So the term "real-valued QM" is a misnomer.

Well, yes. Of course I fully agree. I just would like to point out that "a different theory that makes different predictions" is a pretty dynamical characterization. The paper that demonstrated that this kind of theory actually yields different perdictions dates from December 2021. Up to this date it was (at least to the best of my knowledge) not obvious that these two kinds of theories actually yield different predictions. Up to Bell's seminal work, it was also not clear that local realistic theories and QM yield different predictions. Up to that point it would still have been reasonable to talk about local realistic QM and in a similar vein I think it would have been appropriate to talk about real-valued QM until about two months ago.

On a different note: Nature nowadays posts the full peer review file for any article and this also applies to the theory article predicting this difference. I find it quite interesting that at least to me the discussion between the authors and the referees explains much better what the real-valued theories the authors write about actually are as compared to the published manuscript.
Nature manuscript peer review file

 

[PeterDonis]

"a different theory that makes different predictions" is a pretty dynamical characterization. The paper that demonstrated that this kind of theory actually yields different perdictions dates from December 2021. Up to this date it was (at least to the best of my knowledge) not obvious that these two kinds of theories actually yield different predictions.

The peer review file you referenced says that the original real-valued approach in the Stueckelberg paper you linked to is equivalent to standard QM. So at the very least there seems to be some ambiguity about what theory the term "real-valued QM" actually refers to.

 

[Cthugha]

The peer review file you referenced says that the original real-valued approach in the Stueckelberg paper you linked to is equivalent to standard QM.

I am not sure where you get that from. The authors state that point quite clearly in the response to all 3 referees. Maybe most prominently in response to referee 2:
"Our result implies that no generalization of Stueckelberg’s theory will, at the same time, satisfy postulates (i’),(ii)-(iv) and lead to the same predictions as standard quantum theory. " or equivalently "no theory satisfying postulates (i’), (ii)-(iv), constrained or not, can explain the statistics observed in the proposed quantum tripartite experiment with two independent sources of entangled states. "

If you want to explain the results observed in experiments, you need to get rid of at least one of the postulates (i')-(iv). Postulate (i') is the postulate that the Hilbert space is real. If you want to stick to it, you need to get rid of one of the other postulates.
This is fully parallel to what Bell does. He assumes locality, reality and freedom of choice as postulates and says that you need to get rid of one of them to be in accordance with experimental results.

Here you have real Hilbert spaces, Born's rule, the construction of composite Hilbert spaces via the tensor product and the connection between measurements and projection operators as the four basic postulates.
Condensed to a single sentence, the result of the manuscript is: you need to get rid of one of these postulates as well. Real Hilbert spaces are the obvious choice, but of course not the only one.

This renders the title some kind of a misnomer: The authors then state that they consider the other three postulates as the essential definition of quantum theory. Loosely speaking they state that a theory that satisfies postulates (ii) to (iv) qualifies as a quantum theory. They then go on to say that real Hilbert spaces work, but the resulting theory should not carry the name quantum theory or quantum mechanics as you need to get rid of one of the three postulates they just used to define quantum theory. Although the authors' reasoning is in line with the history of QM and Von Neumann's definitions, indeed I consider this reasoning as some kind of a play with language.

Edit for clarity: If your statement "the original real-valued approach in the Stueckelberg paper you linked to is equivalent to standard QM" was intended to mean "the original real-valued approach in the Stueckelberg paper you linked to yields equivalent predictions for experimental results compared to standard QM", I fully agree.

 

[PeterDonis]

I am not sure where you get that from.

The very first page of the file, where referee #1 says of the Stueckelberg paper: "That paper shows how to construct a theory equivalent to quantum theory but based on a real Hilbert space."

If your statement "the original real-valued approach in the Stueckelberg paper you linked to is equivalent to standard QM" was intended to mean "the original real-valued approach in the Stueckelberg paper you linked to yields equivalent predictions for experimental results compared to standard QM", I fully agree.

It's not a matter of what I intended to mean, it's a matter of what referee #1 intended to mean in the statement I quoted above.

 

[PeterDonis]

the result of the manuscript is: you need to get rid of one of these postulates as well. Real Hilbert spaces are the obvious choice, but of course not the only one.

If I understand the comments of referee #1 correctly, the formulation in the Stueckelberg paper does not satisfy the tensor product postulate, correct?

 

[Cthugha]

The very first page of the file, where referee #1 says of the Stueckelberg paper: "That paper shows how to construct a theory equivalent to quantum theory but based on a real Hilbert space."It's not a matter of what I intended to mean, it's a matter of what referee #1 intended to mean in the statement I quoted above.

This becomes clearer when you scroll down to the second referee report of referee 1 starting on page 25 (where he or she is still not fully satisfied with the authors' statements). The referee indeed seems to consider the real Hilbert space as equivalent to quantum theory. This seems reasonable as the predictions are the same. The authors essentially reply that they do not see these theories as equivalent as they define a theory that satisfies either postulates (i) and (ii) to (iv) or (i') and (ii) to (iv) as a "quantum theory".

The authors then state that their paper demonstrates that it is possible to satisfy all of the postulates (i) to (iv) simultaneously, but that one cannot satisfy (i') to (iv) simultaneously. As the difference between (i) and (i') is complex Hilbert space versus real Hilbert space, this leads them to conclude that there is no quantum theory in real Hilbert space. Referee 1 then replies that this result depends on the very specific definition of "quantum theory" used by the authors and even on their formulation of quantum mechanics. As soon as you move on to the path integral formalism or Bohm's theory, the authors' argument does not really work anymore (there may be similar arguments in other formulations, though). In the final reply by the authors, they also agree to this point.

In my opinion, this full discussion between referee 1 and the authors represents the claim made by the authors and what it actually means better than the paper itself.

If I understand the comments of referee #1 correctly, the formulation in the Stueckelberg paper does not satisfy the tensor product postulate, correct?

Yes, absolutely correct.