Canonical form of master equations and characterization of non-Markovianity

Michael J W Hall, James D. Cresser, Li Li, Erika Andersson

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Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalization procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented which reflect, to varying degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose using the negative decoherence rates themselves, as they appear in the canonical form of the master equation, to completely characterize non-Markovianity. The advantages of this are especially apparent when more than one decoherence channel is present. We show that a measure proposed by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)PRLTAO0031-900710.1103/PhysRevLett.105. 050403] is a surprisingly simple function of the canonical decoherence rates, and give an example of a master equation that is non-Markovian for all times t>0, but to which nearly all proposed measures are blind. We also give necessary and sufficient conditions for trace distance and volume measures to witness non-Markovianity, in terms of the Bloch damping matrix.

Original languageEnglish
Article number042120
Pages (from-to)042120-1-042120-11
Number of pages11
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
Issue number4
Publication statusPublished - 28 Apr 2014

Bibliographical note

Hall, M. J., Cresser, J. D., Li, L., & Andersson, E. (2014). Canonical form of master equations and characterization of non-Markovianity. Physical Review A, 89(4), 042120. Copyright (2014) by the American Physical Society. The original article can be found at


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