Finding the Kraus decomposition from a master equation and vice versa
By Erika Andersson, J. D. Cresser, Michael J. W. Hall. 2007
155 citations recorded by the source corpus.
For any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, a general procedure is given 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×N 2 matrix time evolution (to obtain the completely positive map), and (ii) diagonalizing a related N 2×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.
Subfields: Artificial Intelligence, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/87f2c33c377598b369c070511463ec8d4a850e70
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Citations in the corpus
- Quantum coherence and geometric quantum discord (2017, background): “…equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, one can always construct a linear map which gives ρ(t) = E(ρ(0)) (the opposite case may not always be true), and the linear map can be expressed in the Kraustype representations (Andersson et al., 2007).”
- Concepts of quantum non-Markovianity: A hierarchy (2017, background): “Here G2 t1 is a two-parameter family of linear TP (but not necessarily CP) maps such that E2 t0 ρ = G t2 t1 E1 t0 ρ, for all ρ ∈ B(Hs) (one may take G t2 t1 = E2 t0 E ′t1 t0 , where here φ′ denotes the map corresponding to the Moore-Penrose pseudo-inverse of map φ [64]).”