Density-matrix formalism for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math> -symmetric non-Hermitian Hamiltonians with the Lindblad equation

In the presence of Lindblad decoherence, i.e. dissipative effects in an open quantum system due to interaction with environment, we examine transition probabilities between eigenstates two-level described by non-Hermitian Hamiltonians equation, for which parity-time-reversal (PT) symmetry is conserved. First, density matrix formalism PT-symmetric Hamiltonian systems developed. It shown that operators $L^{}_j$ are pseudo-Hermitian, namely, $\eta L^{}_j \eta^{-1} = L^\dagger_j$ $\eta$ being a linear and positive-definite metric, respect PT as well. We demonstrate generalized $\rho^{}_{\rm G}(t) \equiv \rho(t) \eta$, instead normalized N}(t) \rho(t)/{\rm tr}\left[\rho(t)\right]$, should be implemented calculation accordance linearity requirement. Second, used derive general cases Hamiltonians. some concrete examples, calculate compact analytical formulas explore their main features numerical illustrations. also make comparison our present results previous ones using state vectors absence decoherence.