Remarkable extension of Noether's theorem in PT-symmetric systems

Noether’s theorem[1],relating the symmetry of a physical system to its constant of motion,is a fundamental physical law. It has been shown that the expectation value of a time-independent operator is a constant of motion if it commutes with the Hermitian Hamiltonian. However,applying Noether’s theorem to open systems and obtaining the corresponding conserved quantities are still tricky.