Paper detail

Internal nonlocality in generally dilated Hermiticity

According to von Neumann, the global Hamiltonian of whole universe must be Hermitian in order to keep the eigenvalues real and to construct a self-consistent quantum theory. In addition to the open system approach by introducing environmental degrees of freedom to a small system, a global Hermitian Hamiltonian can also be generated through the dilation from a small Hilbert space. For example, a local non-Hermitian $\cal PT$-symmetric system can be simulated with a global Hermitian one by the Naimark dilation. When shared by Alice and Bob, the internal nonlocality in such dilated Hermitian systems is revealed recently, but only with a two-fold structure. In this paper, we extend such a discussion to the generalized case when the two-fold structure breaks. The internal nonlocality is discussed with different correlation pictures and the corresponding correlation bounds. Our results provide a device-independent test on the reliability of the simulation in the global Hermiticity.