Finding the Kraus decomposition from a master equation and vice versa

Abstract

For any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, a general procedure is reviewed for constructing the corresponding linear map from the initial state to the state at time t, including its Kraus-type representations. Formally, this is equivalent to solving the master equation. For an N-dimensional Hilbert space it requires (i) solving a first order N^2 x N^2 matrix time evolution (to obtain the completely positive map), and (ii) diagonalising a related N^2 x N^2 matrix (to obtain a Kraus-type representation). Conversely, for a given time-dependent linear map, a necessary and sufficient condition is given for the existence of a corresponding master equation, where the (not necessarily unique) form of this equation is explicitly determined. It is shown that a `best possible' master equation may always be defined, for approximating the evolution in the case that no exact master equation exists. Examples involving qubits are given.Comment: 16 pages, no figures. Appeared in special issue for conference QEP-16, Manchester 4-7 Sep 200

Investigating the generality of time-local master equations

“…A simple motivation for this is given in Ref. [16]. One uses the fact that any physical time evolution in quantum mechanics is described by a completely positive (CP) map [17],…”

Local-in-time master equations with memory effects: applicability and interpretation

“…However, the form of a master equation is not in general unique, i.e., the dynamics can be described equivalently by another type of master equation, which is local in time [28]. More precisely, it was shown in [11,12], that under fairly general assumptions, the master equation of the form of equation (2) can always be cast into a local in time form…”

“…One can proof that the dynamical maps for different times t given by equation (13) are completely positive if and only if t 0 γ(s)ds ≥ 0 for all times t. Thus in order for the master equation (12) to describe a physical process, the decay rate γ(t) needs not to be positive but only it's integral has to be. Therefore, in general, master equations with negative decay rates do produce physical processes, when certain additional constraints are met.…”

“…For simplicity we take the system Hamiltonian H = 0. Equation (12) can be easily solved to give the dynamical maps describing the process for time t ρ(t) = κ(t)ρ ee (0) |e e| + (1 − κ(t)ρ ee (0)) |g g|…”

“…Since merely all realistic quantum systems are open, i.e., they are influenced by their surroundings, the description of such dynamics is crucial for our understanding of the nature. Therefore, the fundamental issues of open quantum systems as well as their mathematical description have gained a large amount of interest in the recent years [1,2,3,4,5,6,7,8,9,10,11,12,13].…”