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Quantum Theory Needs Complex Numbers

The physical elaborates theories to describe nature. An analogy would be, for example, a map with which we represent mountains, roads, rivers, etc. and that helps us to orient ourselves. The map is not the mountain, but constitutes the theory that we use to represent reality.

Similarly, theories of physics are expressed in terms of mathematical objects, such as equations, integrals, or derivatives. These include  quantum theory , introduced at the beginning of the 20th century and the first formulated in terms of complex numbers .

These numbers were created by mathematicians centuries ago, and are made up of  a real part and an imaginary part  (as square roots of negative numbers). It was Descartes, the famous philosopher considered the father of rational sciences, who coined the term “imaginary”, in order to be able to contrast it emphatically with what he called “real” numbers.

The authors demonstrate that if quantum postulates are expressed in terms of real numbers rather than complex numbers, then some predictions about quantum networks necessarily differ.

Despite their fundamental role in mathematics, complex numbers were not expected to play a similar role in physics because of this imaginary part. In fact, before quantum theory, Newtonian mechanics or Maxwell’s electromagnetism used real numbers to describe phenomena such as the movement of objects, or how electromagnetic fields propagate. In this case, the theories occasionally use complex numbers to simplify some calculations, but their axioms only use real numbers.

However, quantum theory managed to radically challenge the field because its postulates were built with complex numbers. Although it was very useful for predicting the results of experiments, such as a perfect explanation of the energy levels of the hydrogen atom, it was counterintuitive in favor of real numbers.

Schrödinger’s bewilderment

Looking for a description for electrons, the Austrian physicist Erwin Schrödinger was the first to introduce complex numbers into quantum theory through his famous equation, but he could not conceive that these could really be necessary in physics at that fundamental level. It was as if he had found a map to represent the mountains, but this map was actually created with abstract drawings and totally counterintuitive.

Such was his bewilderment that in 1926 he wrote a letter to his colleague HA Lorentz in which he said: “ What is unpleasant here, and in fact, one must directly object, is the use of complex numbers. Ψ is undoubtedly fundamentally a real function ”.

Decades later, in 1960, Professor ECG Stueckelberg of the University of Geneva (Switzerland), showed that all predictions of quantum theory for individual particle experiments could equally be derived using only real numbers. Since then, the consensus was that complex numbers in quantum theory were just a convenient tool.

The article bears a revealing title: ‘Quantum theory based on real numbers can be falsified experimentally’

However, researchers from several European centers, such as the Institute of Photonic Sciences (ICFO) in Spain and the Institute of Quantum Optics and Quantum Information ( IQOQI ) in Austria, published a study this week in Nature where they show that, if quantum postulates they are expressed in terms of real numbers rather than complex numbers, so some predictions about quantum networks necessarily differ.

The team presents a concrete experimental proposal, in which they include three interconnected parts and two particle sources, where the prediction of the standard complex quantum theory cannot be expressed by its real counterpart. In fact, the article bears a revealing title:  Quantum theory based on real numbers can be falsified experimentally .

Theoretical quantum experiment 

To perform the theoretical experiment, they devised a very specific scenario of an elemental quantum lattice that included two independent sources (S and R), placed between three measurement nodes (A, B, and C). Source S emits two particles, say photons, one to A and the second to B. The two photons are prepared in an entangled state, such as in polarization.

The key point was to find a way to measure four photons at three nodes to get predictions that cannot be explained when quantum theory is restricted to only real numbers.

That is, they correlated or prepared the  polarization of the particles  in a way that is allowed by quantum theory (both complex and real), but not by classical theory. Source R does the exact same thing, emitting two more prepared photons in an entangled state and sending them to B and C, respectively.

The key point of this study was to find the proper way to measure these four photons at nodes A, B, C to get predictions that cannot be explained when quantum theory is restricted to only real numbers.

Collaboration with China

As ICFO co-author and researcher Marc-Olivier Renou comments , “When we found this result, the challenge was to see if the experiment we had devised could be performed with current technologies. After discussing with colleagues at the Southern University of Science and Technology in Shenzhen, China , we found a way to adapt our protocol to make it feasible with their state-of-the-art devices. And, unsurprisingly, the experimental results – published in Physical Review Letter s – match the predictions! ” 

The Nature study can be seen as a generalization of Bell’s theorem , providing a quantum experiment that cannot be explained by any local quantum formalism. Bell’s experiment involves a quantum source S that emits two entangled photons, one to A and the second to B, prepared in an entangled state. Here, on the contrary, two  carefully designed independent fonts are needed .

The work also shows how excellent predictions can be when combining the concept of a quantum network with the ideas of Bel l. According to the authors, the tools developed and used to obtain this first result will allow physicists to achieve a better understanding of quantum theory, and will one day trigger the realization and materialization of applications hitherto unthinkable for the quantum internet .


Marc-Olivier Renou et al. “Quantum theory based on real numbers can be experimentally falsified.” Nature , 2021

Team formed by ICFO researchers Marc-Olivier Renou and ICREA Professor Antonio Acín, in collaboration with Professor Nicolas Gisin from the University of Geneva and the Schaffhausen Institute of Technology (Switzerland), Armin Tavakoli from the Vienna University of Technology, and David Trillo, Mirjam Weilenmann and Thinh P. Le, led by Professor Miguel Navascués from the Institute of Quantum Optics and Quantum Information (IQOQI) in Vienna (Austria).

Fountain: ICFO

Rights: Creative Commons.



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