Ruling out real-valued standard formalism of quantum theory

Abstract: Standard quantum theory was formulated with complex-valued Schrodinger equations, wave functions, operators, and Hilbert spaces. Previous work attempted to simulate quantum systems using only real numbers by exploiting an enlarged Hilbert space. A fundamental question arises: are complex numbers really necessary in the standard formalism of quantum theory? To answer this question, a quantum game has been developed to distinguish standard quantum theory from its real-number analog by revealing a contradiction in the maximum game scores between a high-fidelity multi-qubit quantum experiment and players using only real-number quantum theory. Here, using superconducting qubits, we faithfully experimentally implement the quantum game based on entanglement swapping with a state-of-the-art fidelity of 0.952(1), which beats the real-number bound of 7.66 by 43 standard deviations. Our results disprove the real-number formulation and establish the indispensable role of complex numbers in the standard quantum theory.