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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2008
- DOI:
- 10.48550/arXiv.0801.4100
- arXiv:
- arXiv:0801.4100
- Bibcode:
- 2008arXiv0801.4100A
- Keywords:
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- Quantum Physics
- E-Print:
- 16 pages, no figures. Appeared in special issue for conference QEP-16, Manchester 4-7 Sep 2006