Quantum theory needs complex numbers

Date:
December 15, 2021
Source:
ICFO-The Institute of Photonic Sciences
Summary:
An international team of researchers shows through a concrete
theoretical experiment that the prediction by standard complex
quantum theory cannot be expressed by its real counterpart and
ratifies its need of complex numbers.



FULL STORY ========================================================================== Physicists construct theories to describe nature. Let us explain it
through an analogy with something that we can do in our everyday life,
like going on a hike in the mountains. To avoid getting lost, we generally
use a map. The map is a representationof the mountain, with its houses,
rivers, paths, etc. By using it, it is rather easy to find our way to the
top of the mountain. But the map is not the mountain. The map constitutes
the theory we use to represent the mountain's reality.


========================================================================== Physical theories are expressed in terms of mathematical objects,
such as equations, integrals or derivatives. During history, physics
theories evolved, making use of more elaborate mathematical concepts
to describe more complicated physics phenomena. Introduced in the early
20th century to represent the microscopic world, the advent of quantum
theory was a game changer. Among the many drastic changes it brought,
it was the first theory phrased in terms of complex numbers.

Invented by mathematicians centuries ago, complex numbers are made of
a real and imaginary part. It was Descartes, the famous philosopher
considered as the father of rational sciences, who coined the term
"imaginary," to strongly contrast it with what he called "real"
numbers. Despite their fundamental role in mathematics, complex numbers
were not expected to have a similar role in physics because of this
imaginary part. And in fact, before quantum theory, Newton's mechanics
or Maxwell's electromagnetism used real numbers to describe, say, how
objects move, as well as how electro-magnetic fields propagate. The
theories sometimes employ complex numbers to simplify some calculations,
but their axioms only make use of real numbers.

Schro"dinger's bewilderment Quantum theory radically challenged this
state of affairsbecause its building postulates were phrased in terms
of complex numbers. The new theory, even if very useful for predicting
the results of experiments, and for instance perfectly explains the
hydrogen atom energy levels, went against the intuition in favor of
real numbers. Looking for a description of electrons, Schro"dinger was
the first to introduce complex numbers in quantum theory through his
famous equation. However, he could not conceive that complex numbers
could actually be necessary in physics at that fundamental level. It
was as though he had found a map to represent the mountains but this
map was actually made out of abstract and non-intuitive drawings. Such
was his bewilderment that he wrote a letter to Lorentz on June 6, 1926,
stating "What is unpleasant here, and indeed directly to be objected
to, is the use of complex numbers. ? is surely fundamentally a real
function." Several decades later, in 1960, Prof. E.C.G. Stueckelberg,
from the University of Geneva, demonstrated that all predictions of
quantum theory for single-particle experiments could equally be derived
using only real numbers. Since then, the consensus was that complex
numbers in quantum theory were only a convenient tool.

However, in a recent study published in Nature, ICFO researchers
Marc-Olivier Renou and ICREA Prof. at ICFO Antonio Aci'n, in collaboration
with Prof.

Nicolas Gisin from the University of Geneva and the Schaffhausen
Institute of Technology, Armin Tavakoli from the Vienna University
of Technology, and David Trillo, Mirjam Weilenmann, and Thinh P. Le,
led by Prof. Miguel Navascue's, from the Institute of Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy of Sciences in Vienna
have proven that if the quantum postulates were phrased in terms of real numbers, instead of complex, then some predictions about quantum networks
would necessarily differ. Indeed, the team of researchers came up with a concrete experimental proposal involving three parties connected by two
sources of particles where the prediction by standard complex quantum
theory cannot be expressed by its real counterpart.

Two sources and three nodes To do this, they thought of a specific
scenario that involves two independent sources (S and R), placed
between three measurement nodes (A, B and C) in an elementary quantum
network. The source S emits two particles, say photons, one to A, and
the second to B. The two photons are prepared in an entangled state,
say in polarization. That is, they have correlated polarization in
a way which is allowed by (both complex and real) quantum theory but
impossible classically. The source R does exactly the same, emits two
other photons prepared in an entangled state and sends them to B and to
C, respectively. The key point in this study was to find the appropriate
way to measure these four photons in the nodes A, B, C in order to obtain predictions which cannot be explained when quantum theory is restricted
to real numbers.

As ICFO researcher Marc-Olivier Renou comments "When we found this
result, the challenge was to see if our thought experiment could be
done with current technologies. After discussing with colleagues from Shenzhen-China, we found a way to adapt our protocol to make it feasible
with their state-of-the-art devices. And, as expected, the experimental
results match the predictions!." This remarkable experiment, realized in collaboration with Zheng-Da Li,Ya-Li Mao,Hu Chen, Lixin Feng, Sheng-Jun
Yang, Jingyun Fan from the Southern University of Science and Technology,
and Zizhu Wang from the University of Electronic Science and Technology is published at the same time as the Nature paper in Physical Review Letters.

The results published in Nature can be seen as a generalization of
Bell's theorem, which provides a quantum experiment which cannot be
explained by any local physics formalism. Bell's experiment involves
one quantum source S that emits two entangled photons, one to A, and
the second to B, prepared in an entangled state. Here, in contrast,
one needs two independent sources, the assumed independence is crucial
and was carefully designed in the experiment.

The study also shows how outstanding predictions can be when combining
the concept of a quantum network with Bell's ideas. For sure, the
tools developed to obtain this first result are such that they will
allow physicists to achieve a better understanding of quantum theory,
and will one day trigger the realization and materialization of so far unfathomable applications for the quantum internet.

========================================================================== Story Source: Materials provided by
ICFO-The_Institute_of_Photonic_Sciences. Note: Content may be edited
for style and length.


========================================================================== Journal Reference:
1. Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Thinh P. Le,
Armin
Tavakoli, Nicolas Gisin, Antonio Aci'n, Miguel Navascue's. Quantum
theory based on real numbers can be experimentally
falsified. Nature, 2021; DOI: 10.1038/s41586-021-04160-4 ==========================================================================

Link to news story: https://www.sciencedaily.com/releases/2021/12/211215112812.htm

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