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Cited by 448Quantum Mechanics and ApplicationsQuantum Information and CryptographyAdvanced Thermodynamics and Statistical Mechanics

Symmetries and conserved quantities in Lindblad master equations

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Albert, Victor V., Jiang, Liang·2014·Physical Review A
DOI 10.1103/physreva.89.022118
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Abstract

This work is concerned with determination of the steady-state structure of time-independent Lindblad master equations, especially those possessing more than one steady state. The approach here is to treat Lindblad systems as generalizations of unitary quantum mechanics, extending the intuition of symmetries and conserved quantities to the dissipative case. We combine and apply various results to obtain an exhaustive characterization of the infinite-time behavior of Lindblad evolution, including both the structure of the infinite-time density matrix and its dependence on initial conditions. The effect of the environment in the infinite-time limit can therefore be tracked exactly for arbitrary state initialization and without knowledge of dynamics at intermediate time. As a consequence, sufficient criteria for determining the steady state of a Lindblad master equation are obtained. These criteria are knowledge of the initial state, a basis for the steady-state subspace, and all conserved quantities. We give examples of two-qubit dissipation and single-mode $d$-photon absorption where all quantities are determined analytically. Applications of these techniques to quantum information, computation, and feedback control are discussed.

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  • The Liouville space formalism is very effective for finding solutions or mathematically characterizing many complex problems in quantum mechanics.

    “For many complex problems in quantum mechanics, the Liouville space formalism turns out to be very effective in finding solutions or mathematically characterizing the problem.”

    From Fundamentals of quantum mechanics in Liouville space (2020) · cites this paper as [2]

    Section I (p. 2) explicitly adopts Liouville space to analyze unitary and dissipative systems, and the introduction (p. 2) describes an exhaustive characterization of the asymptotic states of Lindblad master equations. This supports its effectiveness for that class of problems, but the paper does not establish the claim across many complex problems in quantum mechanics generally. (LLM, conf=0.82)

    Experimental — accuracy not yet measured

  • |ρ(∞)〉〉 is a linear combination of those {|ζ_k〉〉} with Re[λ_k]=0.

    “This means that | ρ ( ∞ ) 〉 〉 is a linear combination of those {| ζ k 〉 〉} with Re[ λ k ] = 0 [2, 5, 48].”

    From Fundamentals of quantum mechanics in Liouville space (2020) · cites this paper as [2]

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Research works printed but not linked: 5

  1. [24]M. M¨ uller, S. Diehl, G. Pupillo, and P. Zoller, in Adv. Atom. Mol. Opt. Phy. 61 (Academic Press, 2012) pp. 1-80.
  2. [32]F. Ticozzi and L. Viola, IEEE T. Automat. Contr. 53 , 2048 (2008).
  3. [38]F. Fagnola and R. Rebolledo, Infin. Dimens. Anal. Qu. 11 , 467 (2008); A. Dhahri, F. Fagnola, and R. Rebolledo, Infin. Dimens. Anal. Qu. 13 , 413 (2010); P. Rouchon and A. Sarlette, e-print arXiv:1302.6899. · arXiv
  4. Approach to Quantum Optics (Wiley, Weinheim, 2009).
  5. [62]S. G. Schirmer, in IEEE Decis. Contr. P. , Vol. 1 (IEEE, 2001) p. 298.

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Authors
Albert, V. V., & Jiang, L.
Affiliations
Yale University (US)
ORCID
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2014-02-21
Volume
89
Issue
2
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.89.022118
OpenAlex
W2044042033
Cited by
448
References
117
Citations by year
2026: 56 · 2025: 72 · 2024: 60 · 2023: 44 · 2022: 43 · 2021: 39 · 2020: 49 · 2019: 29 · 2018: 19 · 2017: 11 · 2016: 12 · 2015: 8 · 2014: 6
FWCI
6.14
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell 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 › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
Keywords
Lindblad master equation, steady state, first integrals, quantum symmetries, dissipative quantum systems, quantum feedback control, quantum information

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