Extension of Noether's theorem in PT-symmetry systems and its experimental demonstration in an optical setup

Noether's theorem is one of the fundamental laws in physics, relating the symmetry of a physical system to its constant of motion and conservation law. On the other hand, there exist a variety of non-Hermitian parity-time (PT)-symmetric systems, which exhibit novel quantum properties and have attracted increasing interest. In this work, we extend Noether's theorem to a class of significant PT-symmetry systems for which the eigenvalues of the PT-symmetry Hamiltonian circumflex expressionccent HPTchange from purely real numbers to purely imaginary numbers, and introduce a generalized expectation value of an operator based on biorthogonal quantum mechanics. We find that the generalized expectation value of a time-independent operator is a constant of motion when the operator presents a standard symmetry in the PT-symmetry unbroken regime, or a chiral symmetry in the PT-symmetry broken regime. In addition, we experimentally investigate the extended Noether's theorem in PT-symmetry single-qubit and two-qubit systems using an optical setup. Our experiment demonstrates the existence of the constant of motion and reveals how this constant of motion can be used to judge whether the PT-symmetry of a system is broken. Furthermore, a novel phenomenon of masking quantum information is first observed in a PT-symmetry two-qubit system. This study not only contributes to full understanding of the relation between symmetry and conservation law in PT-symmetry physics, but also has potential applications in quantum information theory and quantum communication protocols.