## Comments about the article in Nature: Quantum theory based on real numbers can be experimental falsified

To study the full text select this link: https://www.nature.com/articles/s41586-021-04160-4
• The text in italics is copied from the article
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### Page 625

Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers.
It is an open question to what extend complex numbers are essential in mathematics.
There are two types of descriptions (explanations or theories) of physical experiments:
• In natural language. This is always required.
• By means of mathematics. This requires that the parameters can be measured.
The results of these measurements are always natural numbers.
The issue is to what extend complex numbers can be used. In physics often real space is identified with the x,y,z coordinates. A point in space is then identified with an origin and measuring the distance
Physics, however, aims to explain, rather than describe, experiments through theories.
The primary reason to explain in physics, is by means of a description of an experiment in natural language.
A theory is a description of a proposed experiment. It is more something like a thought experiment.
Although 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 raises a huge limitation because a complex Hilbert space is a pure mathematical concept and not a physical entity. As a consequence no physical measurements can be performed in complex Hilbert space.
Here we investigate whether complex numbers are actually needed in the quantum formalism.
Okay.
A more simpler question to what extend mathematics actual is needed.
We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements.
• The most important issue is to perform experiments, divided in the description of the experiment and the outcome of the experiment i.e. what is observed and what is measured.
• Secondly that the outcome cannot be explained by real Hilbert-space formulations but require the concept of complex Hilbert-space formulations.
This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.
Such an comparison is very tricky.

### "1. Main"

Whether complex numbers are needed within a theory to correctly explain experiments, or whether real numbers only are sufficient, is not straightforward.
The question is first of all is to what extend experiments can be explained in natural language, secondly when measurements are involved that mathematics can be involved and thirdly that complex numbers can inprove our understanding.
Complex numbers are sometimes introduced in electromagnetism to simplify calculations: one might, for instance, regard the electric and magnetic fields as complex vector fields to describe electromagnetic waves.
Correct.
In its Hilbert space formulation, quantum theory is defined in terms of the following four postulates.
A much more basic question has to be answered first: What is quatum theory? Or even better: What is quantum mechanics? Or what is entanglement?
(1)(1) For every physical system S, there corresponds a Hilbert space HS and its state is represented by a normalized vector Theta in HS, that is, =1.
That means there also exists a Hilbert space HU for the total Universe U. What is the relevance of that claim? What is the relevance for any sub space s'?
As originally introduced by Dirac and von Neumann, the Hilbert spaces Hs in postulate (1) are traditionally taken to be complex. We call the resulting postulate (1c)
The paramount question is: How are the parameters of the Hilbert spaces measured?
Contrary to classical physics, complex numbers (in particular, complex Hilbert spaces) are thus an essential element of the very definition of complex quantum theory
Okay.
The resulting ‘real quantum theory’, which has appeared in the literature under various names, obeys the same postulates (2)–(4) but assumes real Hilbert spaces Hs in postulate (1), a modified postulate that we denote by (1R).

### Page 626

However, we show that this is not the case: the measurement statistics generated in certain finite-dimensional quantum experiments involving causally independent measurements and state preparations do not admit a real quantum representation, even if we allow the corresponding real Hilbert spaces to be infinite dimensional.
The role of measurement statistics and causally independent measurements is discussed in Reflection 1 - Measurements in physics.
It is instructive to address our main question as a game between two players—the ‘real’ quantum physicist Regina and the ‘complex’ quantum physicist Conan.
Okay
Regina is convinced that our world is governed by real quantum theory, whereas Conan believes that only complex quantum theory can describe it.
That means that Regina is convinced that all process can be described by real numbers and Conan is convinced that complex numbers are required.
My impression is that Regina is convinced that all processes take place in 3D and that only a coordinate system in x,y and z is sufficient.
It seems logical that Conan should come up with an experiment which future state he can predict while Regina can not.
Through a well chosen quantum experiment, Conan aims to prove Regina wrong; that is, to falsify real quantum theory by exhibiting an experiment that this theory cannot explain.
...But that the complex quantum theory can explain?
At first, Conan thinks of conducting simple experiments involving a single quantum system.
An important issue is if he thinks about an experiment or actual does this experiment,
Unfortunately, for any such quantum experiment, Regina can find a real quantum explanation.
That means the explanation involves only natural language.
For instance, if rho is the complex density matrix that Conan uses to model his experiment, Regina could propose the state
rho' = equation (1)
where |±i> = 1/sqr(2) (|O>±i|1>), and the asterisk denotes complex conjugation.
The problem to what extend do we know that both rho and rho' are an actual model of the experiment?
At the same time Regina gives a very poor explanation of the proposed (?) experiment by Conan. Her whole explanation should primarly be done in natural language and or required using standard arithmatic using ordinary numbers. Anyway it is critically important that we discuss here an actual experiment, otherwise we never know if Regina understands the experiment.
Fig. 1 (left) explains how Regina can analogously define real measurement operators that, acting on rho', reproduce the statistics of any (complex) measurement conducted by Conan on rho.
Also here the same issue. Regina should try to start from the real measurements performed on an actual experiment and stay within the real world.
She should also try to understand the real measurements performed by Canon and the statistics calculated by Conan using these real measurements.

### Figure 1 Simulating single-site and multipartite quantum experiments through real quantum theory

Left: a single-site quantum experiment.
A complex quantum system in state rho is probed via the measurement {pir}r by Conan.
What is the difference between a complex-quantum system and a quantum system that is complex?
One way to reproduce the measurement statistics of this experiment using real quantum theory requires adding an extra real qubit: the state rho is then replaced by the real state rho' in equation (1),
Why does Regina has to introduce an extra qubit in the complex-quantum system to make the measurement statistics the same as by Conan?
Right: a multipartite quantum experiment. A complex Bell scenario consists of two particles (or systems) distributed between Alice and Bob, who perform local measurements, labelled by x and y, and get results a and b.
Okay.
A complex Bell scenario consists of two particles (or systems) distributed between Alice and Bob, who perform local measurements, labelled by x and y, and get results a and b.
OKay.
Conan may next consider experiments involving several distant labs, where phenomena such as entanglement and Bell non-locality can manifest.
Okay.
A source emits two particles (for example, a crystal pumped by a laser emitting two photons) in a state rhoAB, each being measured by different observers, called Alice (A) and Bob (B) (Fig. 1, right).
Okay.
Next we consider whether Conan could similarly refute real quantum theory via a (complex) quantum Bell experiment.
Okay
As pointed out by Bell, there exist quantum experiments where the observed correlations, encapsulated by the measured probabilities P(a,b|x,y), are such that they cannot be reproduced by any local deterministic model.
The question is what specific experiment he has in mind.
Such an experiment should necessarily violate some Bell inequality; otherwise, one could reproduce the measured probabilities with diagonal (and hence real) density matrices and measurement operators.

### Page 627

To find a gap between the predictions of real and complex quantum theory, Conan shall explore more complicated Bell inequalities.
The Bell inequalities are mathematical operations performed on the results of an experiment

### Reflection 1 - Measurements in physics.

We are living in a physical world which changes in time. We mean by that, that at each instant the physical world is in a certain state and at the next instant the state has changed. To be more specific

### Reflection 1 - Measurements in physics.

We are living in a physical world which changes in time. We mean by that, that at each instant the physical world is in a certain state and at the next instant the state has changed. To be more specific that the parametrs of the objects i.e. positions have changed and or the number of objects.
The general rule of these changes are that the objects in close contact can influence each other and at longer periods all objects influence each other.
In order to study the physical world we describe the physical world as a collection of (sub)processes which inturn can influence each other.
The general object of science is to study identical processes in detail in order to predict the future and or to influence the evolution of certain process to reach a predefined goal.
Generally speaking there are two ways to describe the evolution of identical processes: 1) In natural language. 2) In combination with (1) using mathematics. The overal question of this article is to what extend complex numbers have to be used

### Reflection 2

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Created: 7 January 2022