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

We investigate the problem of a decomposition of a dissipative dynamical map, which is used in quantum dynamics of a finite open quantum system, into two distinct types of mapping on a Hilbert-Schmidt space of quantum states represented by density matrices. One type corresponds to reversible behaviours, while the other to irreversible characteristics. For a finite dimensional system, in which one can equip the state-observable system by a complex matrix space, we employ real vectors or Bloch representations and express a 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 orthogonalsymmetric or the real-polar decomposition, behave as building blocks for a dynamical map. We introduce an additional parameter of the dynamics together with the time parameter. This assertion is reduced to a scaling parameter in the scaling part of the dynamical matrix as a function in time t, making the dynamical map be a one-parameter map. Specifically, for the Lindblad-type dynamical map, in which the dynamical map forms a one-parameter semigroup satisfying the complete positivity condition, normal condition and Makovianity, we interpret the conditions on the Lindblad map in our framework. As results, we find that the change of linear entropy or purity, which we use it as an indicator of dissipative behaviours, is increasing in time and possesses an asymptote expected in thermodynamics. The rate of change of the linear entropy depends on the structure of the scaling part of the dynamical matrix. Moreover, the initial state plays an important role in this change. The issues on initial dependence of dissipative behaviours and the role of eigensubspace partitioning of the dynamical matrix are discussed, and an example on a qubit system is illustrated. PACS numbers: 05.30.-d