  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.
