On the Theory of the Multiplication of Causal Functions

Tamás Gál

The formula for differentiation of functional $A[\ensuremath{\rho}]$ with respect to $\ensuremath{\rho}(\mathbf{r})$ while keeping the normalization $\ensuremath{\int}\ensuremath{\rho}(\mathbf{r})d\mathbf{r}$ of $\ensuremath{\rho}(\mathbf{r})$ fixed is derived, and basic properties arising from it are concerned. The results are then generalized for time-dependent theories. One of the essential consequences of normalization conservation for functional differentiation, namely, that it ruins the symmetry in r of multiple derivatives, is shown to give the resolution of the causality paradox of response functions in time-dependent density-functional theory. The formula for the differentiation of functionals $A[\ensuremath{\rho}]$ that conserves the shape of $\ensuremath{\rho}(\mathbf{r})$ is also presented.

Steven P. Sourbron

Yahohao Deng, Jianwei Wang

We report strict verification of multiple types of indefinite causal orders and demonstrate distillation of indefinite causal order utilizing a programmable silicon-photonic quantum chip. This experimental implementation contributes to the development of a resource theory for indefinite causal order.

Jian Hua Yao, Hu Chen, Ya-Li Mao +2 more

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 [Renou et al., Nature (London) 600, 625 (2021)]. Based on this method, the real quantum theory has been falsified experimentally in both photonic and superconducting quantum systems [Li et al., Phys. Rev. Lett. 128, 040402 (2022); Chen, et al., 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 a 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 was not made in previous works. Using a revised Navasc\'ues-Pironio-Ac\'{\i}n 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 compared 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.

Zygmunt Zawirski