News

Quantum theory needs complex number

Physics thought it could do without complex numbers, which combine real and imaginary numbers. An international team of researchers is proving that they are in fact indispensable in quantum physics.

 

Gisin_Quantique_BandeauWeb.jpg

Illustration picture (Pixabay)

 

If quantum postulates were formulated only in real and not complex numbers, certain predictions about quantum networks could not be correctly formulated.  An international team, including researchers from the University of Geneva (UNIGE), SIT, IQOQI and ICFO, shows that quantum theory needs complex numbers. Their work, published in the journal Nature, opens the way to a better understanding of quantum theory and to potential developments in the field of the quantum internet.

 

Physicists develop theories to describe nature. An example: to avoid getting lost when hiking in the mountains, we usually study a map. It is a representation of the mountain, with its houses, rivers and roads, which makes it fairly easy to find your way to the top. But the map is not the mountain, it is the theory we use to represent it. Physical theories are expressed in terms of mathematical objects, such as equations, integrals or derivatives. Introduced at the beginning of the 20th century to represent the microscopic world, the advent of quantum theory has changed this. Among the many radical changes it brought, it was the first theory formulated in terms of complex numbers.

Invented by mathematicians centuries ago, complex numbers are composed of a real part and an imaginary part. Despite their fundamental role in mathematics, they were not expected to play a similar role in physics. Newton's mechanics or Maxwell's electromagnetism relied on real numbers to describe, for example, how objects move or how electromagnetic fields propagate. Physicists could sometimes use complex numbers to simplify certain calculations, but their axioms only used real numbers. Quantum theory, whose basic postulates are formulated in terms of complex numbers, has radically challenged this.

 

Schrödinger himself did not believe in it

Although very useful for predicting the results of experiments and capable, for example, of perfectly explaining the energy levels of the hydrogen atom, quantum theory clashes with our intuition, which favours real numbers. Schrödinger was the first to introduce complex numbers into quantum theory with his famous equation. However, he could not conceive that complex numbers could actually be necessary: "ψ is surely fundamentally a real function", he wrote in 1926. In 1960, Ernst Carl Gerlach Stueckelberg, a professor at the UNIGE, showed that the predictions of quantum theory for single-particle experiments could also be formulated using real numbers exclusively.

Since then, there has been a consensus that complex numbers in quantum theory are only a practical tool. But the team of researchers led by Miguel Navascués, professor at the Institute for Quantum Optics and Quantum Information (IQOQI) in Vienna, and including Nicolas Gisin, honorary professor at the UNIGE Faculty of Science and professor at the Schaffhausen Institute of Technology (SIT), has proven the opposite with a concrete experimental proposal involving three parts connected by two particle sources where the prediction of standard complex quantum theory cannot be expressed by its real counterpart.

 

Two sources for three measurement nodes

For their experiment, they devised a specific scenario involving two independent sources (S and R), placed between three measurement nodes (A, B and C) in an elementary quantum lattice. The S source emits two entangled particles, photons, one towards A, and the second towards B. Source R does exactly the same thing, emits two more entangled photons and sends them to B and C, respectively. The key point of this study was to find the appropriate way to measure these four photons in the A, B, C nodes in order to obtain predictions that cannot be explained when quantum theory is limited to real numbers.

Marc-Olivier Renou, researcher at the ICFO in Barcelona and co-author of the study, comments: "When we discovered this result, the challenge was to carry out the envisaged experiment using the latest technology. We adapted our protocol to implement it with the state-of-the-art equipment of our colleagues in Shenzen, China. And, as expected, the experimental results match the predictions! One difficulty has been to ensure the independence of the two S and R sources.

This work can be seen as a generalisation of Bell's theorem, which provides a quantum experience that cannot be explained by any formalism based solely on local quantities (propagating continuously from near to far). They reveal the predictions obtained by combining the concept of quantum lattice with Bell's ideas. The tools developed in this research pave the way for a better understanding of quantum theory and could enable the development of hitherto unthinkable applications for the quantum internet.

December 20, 2021
  News