Quantum Theory Needs Complex Numbers

THE physicist develop 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 it constitutes the theory we use to represent reality.

Likewise, theories of physics are expressed in terms of mathematical objects such as equations, integrals or derivatives. Among them is the 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 composed of a real part and an imaginary part (like square roots of negative numbers). It was Descartes, the famous philosopher considered to be the father of the rational sciences, who coined the term “imaginary” in order to emphatically contrast it 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 lattices 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, the Newtonian mechanics or Maxwell’s electromagnetism they used real numbers to describe phenomena like the movement of objects or how electromagnetic fields propagate. In this case, theories occasionally use complex numbers to simplify some calculations, but their axioms use only real numbers.

However, quantum theory managed to radically challenge the field because its postulates were built with complex numbers. While it was very useful for predicting experimental results, as a perfect explanation of the energy levels of the hydrogen atom, it went against the intuition that it favored real numbers.

Schrodinger’s Perplexity

Looking for a description for electrons, the Austrian physicist Erwin Schrodinger He was the first to introduce complex numbers into quantum theory through his famous equation, but he could not conceive that they might actually be needed in physics at that fundamental level. It was as if I had found a map to represent the mountains, but this map was actually created with abstract and totally counter-intuitive drawings.

Such was his perplexity that in 1926 he wrote a letter to his colleague HA Lorentz in which he said: “what is unpleasant here, and indeed one must object directly, is the use of complex numbers. Ψ is, without a doubt, fundamentally a real function”.

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

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

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

quantum theoretical experiment

To carry out the theoretical experiment, they devised a very specific scenario of a elementary quantum network which 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 primed in an entangled state, like polarization.

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

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

Collaboration with China

As the ICFO co-author and researcher comments Marc-Olivier Renou, “When we found this result, the challenge was to see if the experiment we had planned could be carried out with current technologies. After discussing with colleagues from Southern University of Science and Technology in Shenzhen (China), we found a way to adapt our protocol to make it workable with your next generation devices. And, unsurprisingly, the experimental results – published in Physical Review Letters – matches predictions! ”.

the study of Nature can be seen as a generalization of the Bell’s Theorem, which provides 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 independent sources carefully designed.

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

Reference:

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 Professor Antonio Acín from ICREA, 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).

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