Gap between quantum theory based on real and complex numbers is arbitrarily large

Quantum information theory, the formalism for representing information contained in quantum systems, is based on complex Hilbert spaces. It was recently shown that in quantum networks involving three parties with nontrivial locality constraints, this formalism predicts correlations that cannot be explained by real quantum theory, a variant of quantum mechanics based on real Hilbert spaces. In this work, we study a scenario with N + 1 parties sharing quantum systems in a star network. We then construct a multipartite Bell inequality that exhibits a gap between quantum theory and its variant based on real numbers, which grows linearly with N, and is thus arbitrarily large in the asymptotic limit. This implies, that, as the number of parties grows, Hilbert space formalism based on real numbers becomes exceedingly worse at describing complex networks of quantum systems. We also compute the tolerance of this gap to experimental errors.