Imaginary numbers, introduced in the 16th century as a mathematical tool, have long been viewed with suspicion. Yet they now underpin much of modern physics, particularly quantum mechanics, where complex numbers—combining real and imaginary parts—are routine. A key question has been whether complex numbers are merely a convenient mathematical device or an essential feature of quantum reality. New research from a team in France, Poland, and Spain provides a definitive answer: in sufficiently large quantum networks, complex numbers are physically required.
The researchers focused on the composition postulate, a foundational principle in quantum theory that describes how separate quantum systems combine. They constructed a star-shaped quantum network in which several outer parties each receive one part of a quantum system, while a central party, Eve, receives the other parts from independent sources. Each outer party performs one of several simple two-outcome measurements, while Eve conducts a single measurement with many possible outcomes. The team then designed a conditional Bell test—a method traditionally used to compare classical and quantum predictions—to compare standard quantum theory, which uses complex numbers, with a formulation restricted to real numbers.
Previous work had already shown that complex numbers are necessary in some scenarios, but the extent of the discrepancy remained unclear. The new study, published in Reports on Progress in Physics, reveals that the ratio between the predictions of the two theories grows proportionally to N−1, where N is the number of outer parties in the network. As the network expands, real-number quantum theory becomes increasingly inaccurate at reproducing the results of standard complex-number quantum theory. The gap can become arbitrarily large, meaning that for large enough networks, complex numbers offer a predictive advantage that cannot be ignored.
This finding has profound implications for the foundations of quantum mechanics. It suggests that complex numbers are not just a mathematical convenience but an inherent part of the theory’s structure, at least when the composition postulate holds. The result also highlights the power of network-based approaches in probing fundamental questions about quantum theory. By designing networks with many parties, researchers can amplify differences between theoretical frameworks, making subtle distinctions experimentally testable.
The study’s authors note that their work gives a clear answer to the long-standing question of whether complex numbers are necessary. In sufficiently large quantum networks, they are not optional—they are physically required. This could influence future research in quantum information and computation, where the choice of mathematical framework may affect practical applications. It also deepens our understanding of why quantum mechanics takes the form it does, suggesting that the use of complex numbers is not an accident but a fundamental feature of nature.
