Ruling out real-number description of quantum mechanics

Standard quantum mechanics has been formulated with complex-valued Schrödinger equations, wave functions, operators, and Hilbert spaces. However, previous work has shown possible to simulate quantum systems using only real numbers by adding extra qubits and exploiting an enlarged Hilbert space. A fundamental question arises: are the complex numbers really necessary for the quantum mechanical description of nature? To answer this question, a non-local game has been developed to reveal a contradiction between a multiqubit quantum experiment and a player using only real numbers. Here, based on deterministic and high-fidelity entanglement swapping with superconducting qubits, we experimentally implement the Bell-like game and observe a quantum score of 8.09(1), which beats the real number bound of 7.66 by 43 standard deviations. Our results disprove the real-number description of nature and establish the indispensable role of complex number in the quantum mechanics. Physicists use mathematics to describe nature. In classical physics, real number appears complete to describe the physical reality in all classical phenomenon, whereas complex number is only sometimes employed as a convenient mathematical tool. In quantum mechanics, the complex number was introduced as the first principle in Schrödinger's equation and Heisenberg’s commutation relation. The complex wavefunction has been shown to represent physical reality of quantum objects. Experimentally, the real and imaginary parts of the wavefunction has been directly measured. Today, the quantum mechanics with complex wavefunction seems the most successful theory to describe nature. However, whether the complex number is necessary to represent the theory of quantum mechanics is still an open question. Starting with von Neumann in 1936, previous works have shown possible to simulate quantum systems using only real numbers by adding extra qubits and exploiting an enlarged Hilbert space. For example, by adding an extra qubit ( 0 1 ) / 2 i i    , a single-quantum system with a complex density matrix  and Hermitian operator H can be simulated through ( ) ( ) tr H tr H    , where  and H are real and of the form: * ( ) / 2 i i i i            , * . H H i i H i i         Therefore, in this case one cannot distinguish between the realand complex-number representations. It has also been shown that real-number quantum state is universal for quantum computing, which again challenged the fundamental role of complexnumber quantum mechanics. Very recently, Renou et al. developed an elegant scheme to provide an observable effect in quantum experiments to distinguish between the two representations. The scheme is a Bell-like three-party game based on deterministic entanglement swapping. With the assumption that the quantum states produced by two independent sources is a tensor product of the states produced by each of the sources, the real-number quantum mechanics cannot obtain maximal violation of a Bell CHSH-like inequality, thus being falsified. Figure 1a shows an illustration of the non-local game for Alice, Bob, and Charlie. First, two pairs of Einstein-Podolsky-Rosen (EPR) entangled qubits are distributed between Alice and Bob, and between Bob and Charlie, respectively. Bob then performs a joint Bell-state measurement (BSM) on his two received qubits, which randomly projects them into one of the four Bell states: =( 0 0 1 1 ) / 2, =( 0 1 1 0 ) / 2, =( 0 0 1 1 ) / 2, =( 0 1 1 0 ) / 2.     