# Quantum physics needs complex numbers

Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Le Phuc Thinh, Armin Tavakoli, Nicolas Gisin, Antonio Acin, Miguel Navascues

Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Le Phuc Thinh, Armin Tavakoli, Nicolas Gisin, Antonio Acin, Miguel Navascues

**26/1/21**Published in : arXiv:2101.10873

Complex numbers, i.e., numbers with a real and an imaginary part, are essential for mathematical analysis, while their role in other subjects, such as electromagnetism or special relativity, is far less fundamental. Quantum physics is the only physical theory where these numbers seem to play an indispensible role, as the theory is explicitly formulated in terms of operators acting on complex Hilbert spaces. The occurrence of complex numbers within the quantum formalism has nonetheless puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum physics, where states and observables are represented by real operators, seemed much more natural. In fact, previous works showed that such "real quantum physics" can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states. Thus, are complex numbers really needed for a quantum description of nature? Here, we show this to be case by proving that real and complex quantum physics make different predictions in network scenarios comprising independent quantum state sources. This allows us to devise a Bell-type quantum experiment whose input-output correlations cannot be approximated by any real quantum model. The successful realization of such an experiment would disprove real quantum physics, in the same way as standard Bell experiments disproved local physics.