Quantum theory does not need complex numbers

Abstract: The longstanding debate over whether quantum theory fundamentally requires complex numbers--or if their use is merely a convenient choice--has persisted for decades. Until recently, this question was considered open. However, in [M.-O. Renou et al, Nature 600, 625-629, 2021], a decisive argument was presented asserting that quantum theory needs complex numbers. In this work, we demonstrate that a formulation of quantum theory based solely on real numbers is indeed possible while retaining key features such as theory-representation locality (i.e. local physical operations are represented by local changes to the states) and the positive semi-definiteness of its states and effects. We observe that the standard system combination rule--the tensor product--was derived after the development of single-system complex quantum theory. By starting from a single-system quantum theory using only real numbers, we derive a combination rule that produces a real quantum theory with properties analogous to those of conventional complex quantum theory. We also prove that the conventional tensor product rule can also lead to a real and representation-local theory, albeit with a modified characterization of the state space. We thus conclude that complex numbers are a mere convenience in quantum theory.