Quantum theory based on real numbers can be experimentally falsified

Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, asthose are expressed in terms of probabilities, hence real numbers. Physics, however, aimsto explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory wasthe first to be formulated in terms of operators acting on complex Hilbert spaces(1,2). This has puzzled countless physicists, including the fathers ofthe theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural(3). In fact, previous studies have shown that such areal quantum theory can reproduce the outcomes of any multipartite experiment, as long asthe parts share arbitrary real quantum states(4). Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows usto devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.