### Abstract

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 x N-2 matrix time evolution ( to obtain the completely positive map), and ( ii) diagonalizing 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.

Original language | English |
---|---|

Pages (from-to) | 1695-1716 |

Number of pages | 22 |

Journal | Journal of Modern Optics |

Volume | 54 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2007 |

### Cite this

*Journal of Modern Optics*,

*54*(12), 1695-1716. https://doi.org/10.1080/09500340701352581

}

*Journal of Modern Optics*, vol. 54, no. 12, pp. 1695-1716. https://doi.org/10.1080/09500340701352581

**Finding the Kraus decomposition from a master equation and vice versa.** / Andersson, Erika; Cresser, James D.; Hall, Michael J. W.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Finding the Kraus decomposition from a master equation and vice versa

AU - Andersson, Erika

AU - Cresser, James D.

AU - Hall, Michael J. W.

PY - 2007

Y1 - 2007

N2 - 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 x N-2 matrix time evolution ( to obtain the completely positive map), and ( ii) diagonalizing 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.

AB - 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 x N-2 matrix time evolution ( to obtain the completely positive map), and ( ii) diagonalizing 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.

U2 - 10.1080/09500340701352581

DO - 10.1080/09500340701352581

M3 - Article

VL - 54

SP - 1695

EP - 1716

JO - Journal of Modern Optics

JF - Journal of Modern Optics

SN - 0950-0340

IS - 12

ER -