Unitary-Scaling Decomposition and Dissipative Behaviour in Finite-Dimensional Linblad Dynamics

We investigate a decomposition of a unital Lindblad dynamical map of an open quantum system into two distinct types of mapping on the Hilbert-Schmidt space of quantum states. One component of the decomposed map corresponds to reversible behaviours, while the other to irreversible characteristics. For a finite dimensional system, we employ real vectors or Bloch representations and express a dynamical map on the state space as a real matrix acting on the representation. It is found that rotation and scaling transformations on the real vector space, obtained from the real-polar decomposition, form building blocks for the dynamical map. Consequently, the change of the linear entropy or purity, which indicates dissipative behaviours, depends only on the scaling part of the dynamical matrix. The rate of change of the entropy depends on the structure of the scaling part of the dynamical matrix, such as eigensubspace partitioning, and its relationship with the initial state. In particular, the linear entropy is expressed as a weighted sum of the exponential-decay functions in each scaling component, where the weight is equal to $\vert\vec{x}_k(ρ)\vert^2$ of the initial state $ρ$ in the subspace. The dissipative behaviours and the partition of eigensubspaces in the decomposition are discussed and illustrated for qubit systems.