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Finding the Kraus decomposition from a master equation and vice versa

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Andersson, Erika, Cresser, James D., Hall, Michael J. W.·2007·Journal of Modern Optics
DOI 10.1080/09500340701352581
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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×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.

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  • The proposed measures are suitable for providing information on deviations from Markovianity in experimental implementations of effective spin models using controllable systems, such as trapped ions.

    “would be suitable for providing information on the deviations from Markovianity in the experimental implementation of effective spin models using controllable systems, as for instance, trapped ions [17].”

    From Entanglement and Non-Markovianity of Quantum Evolutions (2010) · cites this paper as [17]

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Research works printed but not linked: 4

  1. [6]H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  2. [11]H. J. Carmichael, An open systems approach to quantum optics (Springer-Verlag, Berlin, 1993).
  3. [21]K. Kraus, States, effects and operations (Springer-Verlag, Berlin, 1983).
  4. [24]C. M. Caves, J. Supercond. 12 (6) 707 (1999); also published as quant-ph/9811082. · arXiv

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Authors
Andersson, E., Cresser, J. D., & Hall, M. J. W.
Affiliations
  • University of Strathclyde (GB)
  • Macquarie University (AU)
  • Australian National University (AU)
Corresponding author
Andersson, Erika
ORCID
Venue
Journal of Modern Optics
Publisher
Informa UK Limited
Published
2007-08-15
Volume
54
Issue
12
Pages
1695-1716
ISSN
  • 0950-0340 (ISSN-L)
  • 1362-3044
DOI
10.1080/09500340701352581
OpenAlex
W2140406475
Cited by
155
References
28
Citations by year
2026: 2 · 2025: 12 · 2024: 14 · 2023: 13 · 2022: 11 · 2021: 9 · 2020: 13 · 2019: 8 · 2018: 14 · 2017: 13 · 2016: 8 · 2015: 9 · 2014: 7 · 2013: 6 · 2012: 6
FWCI
1.63
Type
journal article
Language
en
Open access
Locations
  • Journal of Modern Optics (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • ANU Open Research (Australian National University) (submittedVersion)
  • Griffith Research Online (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Royal Society
  • Macquarie University
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
Keywords
Kraus decomposition, master equation, completely positive maps, non-Markovian dynamics, quantum dynamical maps, time-local master equation, qubits, quantum state evolution

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