The Naimark dilated PT-symmetric brachistochrone

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 syst…