Depiction of Hamiltonian PT-symmetry

The theory of PT-symmetry describes the non-hermitian Hamiltonian with real energy levels, which means that the Hamiltonian H is invariant neither under parity operator P, nor under time reversal operator T, PTH = H. Whether the Hamiltonian is real and symmetric is not a necessary condition for ensuring the fundamental axioms of quantum mechanics: real energy levels and unitary time evolution. The theory of PT-symmetry plays a significant role in studying quantum physics and quantum information science, Researchers have paid much attention to how to describe PT-symmetry of Hamiltonian. In the paper, we define operator F according to the PT-symmetry theory and the normalized eigenfunction of Hamiltonian. Then we first describe the PT-symmetry of Hamiltonian in dimensionless cases after finding the features of commutator and anti-commutator of operator CPT and operator F. Furthermore, we find that this method can also quantify the PT-symmetry of Hamiltonian in dimensionless case. I(CPT, F) = ||[CPT, F]|| CPT represents the part of PT-symmetry broken, and J(CPT, F) = ||[CPT, F]|| CPT represents the part of PT-symmetry. If I(CPT, F) = ||[CPT, F]|| CPT = 0, Hamiltonian H is globally PT-symmetric. Once I(CPT, F) = ||[CPT, F]|| CPT not equal 0, Hamiltonian H is PT-symmetrically broken. In addition, we propose another method to describe PT-symmetry of Hamiltonian based on real and imaginary parts of eigenvalues of Hamiltonian, to judge whether the Hamiltonian is PT symmetric. ReF = 1/4||(CPTF+F)||CPT represents the sum of squares of real part of the eigenvalue E-n of Hamiltonian H, ImF = 1/4||(CPTF-F)||CPT is the sum of imaginary part of the eigenvalue E-n of a Hamiltonian H. If ImF = 0, Hamiltonian H is globally PT-symmetric. Once ImF not equal 0, Hamiltonian H is PT-symmetrically broken. ReF = 0 implies that Hamiltonian H is PT-asymmetric, but it is a sufficient condition, not necessary condition. The later is easier to realize in the experiment, but the studying conditions are tighter, and it further requires that CPT phi(n) (x) = phi(n) (x). If we only pay attention to whether PT-symmetry is broken, it is simpler to use the latter method. The former method is perhaps better to quantify the PT-symmetrically broken part and the part of local PT-symmetry.