Time reversal transformations, star-conjugation symmetry, and microscopic reversibility in the matrix formulation of quantum liouvillian dynamics

We analyze the symmetries of open dissipative models for quantum Liouvillian dynamical systems. Our analysis yields two representation independent symmetry relations. One of the symmetry relations, star-conjugation symmetry, is shown to represent a necessary and sufficient condition for ensuring the reality of observables and the Hermiticity of the system density operator. The other symmetry relation represents a necessary and sufficient condition for the realization of microscopic reversibility. These symmetry relations and the structure of the global dynamics are shown to give rise to a number of generalized symmetry relations for quantities expressible as matrix elements of the retarded, advanced, and global propagators. It is shown that the generalized symmetry relations may be used to facilitate the solution of spectral and temporal problems. This is demonstrated in a concrete way in an application entailing the use of dual Lanczos transformation theory to obtain an exact solution of a prototype model for excited state dynamics. Apart from elucidating the symmetries of open dissipative systems, the reported analysis reveals that there are two different and equally legitimate schemes for performing time-reversal transformations in the matrix formulation of quantum Liouvillian dynamics. The properties and ramifications of these time-reversal transformation schemes are discussed in detail.