## Comments about the article in Nature: Quantum theory and the square root of minus one

To study the full text, select this link: https://www.nature.com/articles/d41586-021-03678-x?
• The text in italics is copied from the article
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### Introduction

The article starts with this text:
Standard quantum theory contains square roots of negative numbers. But how essential are these ‘imaginary’ numbers? A way of disproving analogous theories that omit them has been proposed — and confirmed experimentally.
What this means that in order to understand Quantum Theory and or Quantum mechanics imaginary numbers have to be used. In a more global sense in order to understand all physical processes mathematics has to be used i.e. imaginary numbers. This raises a more detailed question: which are the specific processes that require imaginary numbers? Are these always processes which fall in the quantum mechanical realm like Qubits?
The square root of a negative number could easily be regarded as a curious mathematical construction that has little bearing on real life.
All negative numbers have no meaning in physics or every day life.
When there are 10 people in a room and during a week 2 people leave the room and 2 die then at the end of the week there are still 6 left. A description of what happened does not require the concept of negative numbers.
However, since the development of a quantum theory nearly a century ago, it has seemed likely that complex numbers - which combine imaginary and real numbers - are deeply embedded in the structure of the physical world.
Indeed, the imaginary unit that represents the square root of minus one occurs prominently in the basic equations of quantum theory.
The problem is when you do that you step out of the real physical world and you step into the (complex) mathematical world where no physical measurements can be made. To perform the predictions calculated in the mathematical world you have to return back into the physical world.
Renou et al show on page 625 how experiments could strengthen the evidence that complex numbers are key to describing the quantum world, by eliminating a class of theory that contains only real numbers.
See for a review of this article this link: Quantum theory based on real numbers can be experimental falsified
To understand the difference between standard quantum theory and analogous real-number theories, consider an electron confined to two separate sites.
Unfortunate no clear physical description is given what this means. What does it means that one electron exists is two separate locations?
In quantum theory, the electron can be in a superposition of the two locations - neither at one place nor the other.
Also this sentence is not clear.
A molecule H2 has 2 electrons and 2 protons.
A molecule O2 has 16 electrons and 16 protons.
A molecule H20 has 10 electrons and 10 protons.
For water this means that in the 1s layer there are 2 electrons, in the 2s layer there are 2 electrons and in the 2p layer there are 6 electrons. All layers are full.
I have doubt if at any instant there are superposition's involved implying that a certain electron is neither at one place nor the other. . This raises the question: what has to be done that we can speak of a superposition.
Such a superposition is represented as a point in abstract space compromising two dimensions.
Renou et al. sought to determine whether a real-space analogue of quantum theory could reproduce all the predictions of standard quantum theory - not just those for a single particle such as the electron in two locations but also those for a system of two or more particles.
In my opinion it is much easier in general to start with one experiment using two particles, observe first how the experiments evolves and secondly try to explain this behaviour. This explanation should come in two flavours: First using (quantum theory which involves) only real numbers and secondly using ( quantum theory) using complex numbers. The emphasis on a detailed description of what is physical involved.
Consider, for example, two particles prepared at a common source, possibly exhibiting the kind of correlation known as 'quantum entanglement'
First a question: why using the word possibly? When there is no quantum entanglement involved the whole article does not make sense. Quantum entangle is a must.
One important question is what exactly is quantum entanglement? How is it explained? Requires this explanation the concept of complex number? How do we verify that quantum entanglement is involved?
To answer these questions experiments are required.
In standard quantum theory, the abstract space of the two particles is related to the abstract spaces of the individual particles through a mathematical construction known as the (complex) tensor product.
That may be true, but first a detailed description of the experiment, that was the basis of the standard quantum theory, has to be given.
The real-space framework that Renou at al. analysed is based on an analogous construction, which can be called the real tensor product
The complex tensor product and real tensor products both allow the possibility of entanglement but they impose different constraints on the physics of the system
That is strange because only one experiment, which involves entanglement, should be used as the basis for a complete explanation.
In light of these results, it might be reasonable to expect that for every conceivable experiment, the real-number framework would always be able to imitate standard quantum theory.
This sentence should be rewritten as: 'In light of these results, it might be reasonable to expect that for every conceivable experiment, thereal-number frame work would be able to explain.'
If that is not the case then: 'the complex frame work should be able to explain.'
But Renou et al identified an experiment for which this is not the case.
See description below.
Particles B1 and B2, each from a different entangled pair, are jointly measured at the central location, while their partner particles, A and C, are measured separately.
The details of these measurements are not supplied, nor if these measurements influence each other.. This is a pity. See next sentence.
This particular combination of preparations and measurements turns out to be too much for the real tensor product.
I'm not amazed, because in its detail, this is a very complex process. To describe this process by means of mathematics IMO is impossible. See also: Reflection 3 - Detailed Discussion about Measurement outcome b
No theory in which the particles are linked in this way can reproduce the predictions of standard quantum theory no matter how many dimensions are assigned to the real spaces.
This is the wrong way to do science. Science starts with performing experiments. The second step is to explain the experiments in an orderly way, using logical physical terminology
 ``` Detector Source 1 Detector Source 2 Detector ----- ----- ----- ----- ----- | | A | | B1 | | B2 | | C | | | A | <--O | | O--> | B | <--O | | O--> | C | | | | | | | | | | | ----- ----- ----- ----- ----- | Entangled | Entangled | | Pair | Pair | V V V Measurement Measurement Measurement Outcome a Outcome b Outcome c ```

### Figure 1: An Experimental set-up to test real-number analogues of quantum theory

Particles A and B1 are generated at a single source at the same time that particles B2 and C are generated at a second source.
Particles A and B1 are always generated at the same time at Source 1 because they are supposed to be entangled, as demonstrated by individual experiments.
For the particles B2 and C created at Source 2, this is also the case.
To create both particle pairs (A,B1) and (B2,C) at the same time is extremely difficult. The whole issue is to what extend the experimenter can be sure that this is actual happening.
One important question to answer is to what extend the particle-pairs created at Source 1 are considered random events or can be controlled. For Source 2 the same question. When they are controlled than in principle it is possible that both events can be simultaneous.
Particles B and B1 are then jointly measured at a central location, while A and C are measured separately.
My expectation is that the chance that B1 and B2 are jointly measured is zero. See Reflection 2
Renou at al proposed this experiment to show that a quantum-like framework that imaginary numbers (square roots of negative numbers) cannot reproduce all the statistical predictions of standard quantum theory for the combination of measurement outcomes a,b and c.
(*) The emphasis on proposed. Nothing is shown about the (statistical) results of the experiment.
As already mentioned there are two pairs of entangled electrons involved. That means that the physical orientation of the axis of electrons created at source 1 and the physical orientation of the axis of electrons created at source 2 each are correlated, but that says nothing about the axis of the electrons B1 and B2 'measured' at Detector B.
Page 608
Renou et al put forward a measurable quantity whose value can distinguish between the real and complex frameworks - and this quantity has now been measured. See: https://arxiv.org/abs/2103.08123
In order to understand, for more detail, study next sentence.
In two experimental papers, the value of this key quantity (or a closely related quantity ^3) was found in conflict strongly with the real-number frame work.
In order to understand, for more detail, study next sentence.
Both experimental groups used binary quantum objects 'qubits' as prescribed by Renou et al. but the actual physical systems were quite different : super conducting qubits in one case and photons in the other ^3
For some quantum physicists, Renou and colleagues work will raise a long-standing question anew: why is the world structured such that complex numbers have a fundamental role?
To answer that question, first a much more important question has to be answered: What is this fundamental role of numbers? To understand the cause of a Tsunami, do we need complex numbers? No. In any physical process are numbers involved? No. My 'mathematical' teacher always told me that numbers don't exist. Only cows exist.
You can ask the same question for complex numbers. They have no fundamental role to understand the physical world i.e. the physical processes that take place around us. The most important 'tool' is chemistry, alas almost no mathematics. That does mean that it is not important to quantify the power of the Tsunami in 2022.
The current findings provide a concrete counterexample to the most natural real-number analogue of quantum theory, thereby strengthening the evidence that the complex structure is essential.
Let us wait and see.

### Reflection 1 - entanglement.

 ``` Detector Source 1 Detector ----- ----- ----- | | A | | B | | | A | <--O | | O--> | B | | | | | | | ----- ----- ----- | Entangled | | Pair | V V Measurement Measurement Outcome a Outcome b ```

### Experiment to demonstrate entanglement

The drawing at the left shows one source and two detectors
At the source a reaction takes place, which creates two electrons: A and B.
Electron A is transmitted towards the left and electron B is transmitted towards the right, where both are detected i.e. measured.
Measured in this case means that the direction spin of the electron is measured, which can be either UP or DOWN. In some cases this can be zero (0), which means that nothing is detected. For a standard reaction this can be the outcome:
 ```Detector A UP UP DOWN UP DOWN UP DOwn Detector B DOWN UP DOWN DOWN UP UP UP ```
If those are the results, then they are not correlated. However also the following is possible:
 ```Detector A UP UP DOWN UP DOWN DOWN UP Detector B DOWN Down UP DOWN UP UP DOWN ```
In this case they are correlated. We call the electrons A and B an entangled pair.
Now we have a more difficult question: how is this explained? The spin of an electron defines an axis of rotation. The direction can be any direction in (physical) space. When the direction of rotation of one electron, has the direction +x,+y,+z, than the entangled electron has the direction -x,-y,-z.
• Entanglement starts at a source. At the source a reaction takes place, which as a by-product creates two electrons. For example: A + B --> C + D + e1 + e2. What is not shown, but what is assumed, is that the total number of electrons (including e1 and e2), before the reaction and after the reaction are the same.
• What is special, when there is entanglement, that the spin of the two electrons involved, are correlated. There is no direct explanation for this behaviour, but in order to understand a more detailed physical investigation of this reaction is required. Specific the results have to be compared with reactions which show and which don't show entanglement. By preference the difference between the two should be as small as possible.
• What is important that this entanglement has nothing to do with the measurement as such, which is performed at the two detectors a and b. Neither any action at detector a influences detector b. Nor detector b influences a. What that means if the two electrons are measured 'simultaneous' and there is correlation then the cause is the reaction.
• When you study: https://en.wikipedia.org/wiki/Quantum_superposition#Examples you can read that one electron has the direction UP and the other one is called DOWN. This is completely arbitrary selection and it would be more accurate to call one direction -UP and the opposite direction -UP.
• This brings me to a much more touchier subject and that is because more or less a similar distinction is called a real direction and an imaginary direction. Or slightly different a real axis and an imaginary axis. Both concepts are borrowed from the field of mathematics by using the idea that a wave has a real and an imaginary part. The problem is that using these concepts, don't explain anything. Don't explain why certain parameters of electrons are correlated
• To be even more rigid: Nothing in physics can be explained by using imaginary number If you want to understand anything in physics, the cause can never be found by using mathematics, specific by using imaginary numbers.

### Reflection 2 - Figure 1 implementations

There are two proposals and or actual implementations of the experiment 'To test real-number analogues of quantum theory' as depicted in Figure 1 : superconducting qubits and photons. That means electrons were not involved.
• An experiment using of photons, is physical the easiest to discus. I assume that means the pairs (A,B1) and (B2,C) are each two entangled photons created respectively at source 1 and source 2. I assume also that reactions are the same.
The most important point is when exactly does this reaction takes place, what is the moment that those photons are released? This is not important for the detectors A and C, because their operation is only dependent when a photon arrives (that means photon A for Detector A) and not about each other.
For detector B this is different because what happens depends about both photons B1 and B2.

### Reflection 3 - Figure 1 Detailed Discussion about Measurement outcome b

Quantum entanglement is described in: https://en.wikipedia.org/wiki/Quantum_entanglement"> https://en.wikipedia.org/wiki/Quantum_entanglement as:
Quantum entanglement is a physical phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance.
The problem with that definition is that people forget that in order to establish a certain correlation you have to perform a certain number of experiments. In such experiments you have to measure the same parameter, on two particles. For example: the axis of rotation. When the result is always, that when one axis is measured and defined as UP, that then the axis of the second particle is measured as DOWN. Only in that case, both particles are defined as correlated.
The explanation that this correlation induced in the second particle, immediate after the first measurement, does not sound very scientific. The explanation that this is induced (caused) immediate after the two particles are created seems more physical correct.
The experiment of Figure 1 consists of two sources (1 and 2) were entangled pairs are created. Both pairs are created simultaneous.
The distance from "Source 1" to Detector A is the same as the distance to detector B and these distances are the same as the distance from "Source 2" to Detector B and the distance to detector C. That means if the pairs are created simultaneous all the 4 particles should also reach the detectors simultaneous.
It is my understanding that there is correlation between (the axis of rotation) the two individual particles created at source 1 and between the two individual particles created at source 2, but not between the individual particles of source 1 and source 2
The most important part is, that in such a setup the two particles B1 (from source 1) and B2 (from source 2) will arrive simultaneous at detector B.
This creates two symmetric questions:
• Will the 'Collision' or 'interference' of the two particles B1,B2 at detector B have any influence of the behaviour or measurement of particle A at detector A.
• Will the 'Collision' or 'interference' of the two particles B1,B2 at detector B have any influence of the behaviour or measurement of particle C at detector C.
There will no correlation between the axis of rotation of detector A and detector C. See also (*)
My prediction will be even stronger: any possible collision between the two particles B1 and B2 will have no influence on the behaviour on either particle A or C.
A different question is: Do you need complex numbers to describe this behaviour? No.

What amazes me the almost nothing is described at this 'collision'

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Created: 7 January 2022