Quantum Theory needs complex numbers

Abstract

While complex numbers are essential in mathematics, they are not needed to describe physical experiments, expressed in terms of probabilities, hence real numbers. Physics however aims to explain, rather than describe, experiments through theories. While most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural. Are complex numbers really needed in the quantum formalism? Here, we show this to be case by proving that real and complex quantum theory, understood in terms of operators in Hilbert spaces and tensor products to represent independent systems, make different predictions in network scenarios comprising independent states and measurements.  This allows us to devise a Bell-like experiment whose successful realization would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.