Quantum theory based on real numbers can be experimentally falsified

TLDR

Although complex numbers are mathematically essential, physical experiments are described by probabilities and can be modeled with real numbers, yet quantum theory uniquely employs complex Hilbert spaces—a choice that has long puzzled physicists who have explored real‑operator alternatives capable of reproducing multipartite experimental outcomes. The study investigates whether complex numbers are necessary in quantum theory. We demonstrate that real and complex Hilbert-space formulations predict different outcomes in network scenarios with independent states and measurements, enabling a Bell‑like experiment whose success would falsify real quantum theory.

Abstract

Abstract Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although 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 1,2 . 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 3 . In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the 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 us to 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.