Proposals for ruling out the real quantum theories in an entanglement-swapping quantum network with causally independent sources

The question of whether complex numbers play a fundamental role in quantum theory has been debated since the inception of quantum mechanics. Recently, a feasible proposal to differentiate between real and complex quantum theories based on the technique of testing Bell nonlocalities has emerged [Nature 600, 625-629 (2021)]. Based on this method, the real quantum theory has been falsified experimentally in both photonic and superconducting quantum systems [Phys. Rev. Lett. 128, 040402 (2022), Phys. Rev. Lett. 128, 040403 (2022)]. The quantum networks with multiple independent sources which are not causally connected have gained significant interest as they offer new perspective on studying the nonlocalities. The independence of these sources imposes additional constraints on observable covariances and leads to new bounds for classical and quantum correlations. In this study, we examine the discrimination between the real and complex quantum theories with an entanglement swapping scenario under a stronger assumption that the two sources are causally independent, which wasn't made in previous works. Using a revised Navascu\'es-Pironio-Ac\'in method and Bayesian optimization, we find a proposal with optimal coefficients of the correlation function which could give a larger discrimination between the real and quantum theories comparing with the existing proposals. This work opens up avenues for further exploration of the discrimination between real and complex quantum theories within intricate quantum networks featuring causally independent parties.