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.
Cite
@article{arxiv.0801.4100,
title = {Finding the Kraus decomposition from a master equation and vice versa},
author = {Erika Andersson and Jim D. Cresser and Michael J. W. Hall},
journal= {arXiv preprint arXiv:0801.4100},
year = {2008}
}
Comments
16 pages, no figures. Appeared in special issue for conference QEP-16, Manchester 4-7 Sep 2006