No-signaling principle and Bell inequality in ${\mathcal{PT}}$ -symmetric quantum mechanics

-symmetric quantum mechanics, the extension of conventional quantum mechanics to the non-Hermitian Hamiltonian invariant under the combined parity () and time reversal () symmetry, has been successfully applied to a variety of fields?such as solid state physics, mathematical physics, optics, quantum field theory. Recently, the extension of -symmetrical theory to entangled quantum systems has been challenged, in that formulation within the conventional Hilbert space violates the no-signaling principle. Here, we revisit the derivation of non-signaling principle in the framework of inner product prescription. Our results preserve the no-signaling principle for a two-qubit system, reaffirm the invariance of the entanglement, and reproduce the Clauser?Horne?Shimony?Holt (CHSH) inequality. We conclude that -symmetric quantum mechanics satisfies the requirements for a fundamental theory and provides a consistent description of quantum systems.