Symmetries and conserved quantities in Lindblad master equations

By Victor V. Albert, Liang Jiang. 2013

437 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/56dc55d31af749d6e844261f533e37538ed2e358

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Citations in the corpus

Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps (2019, background): “Since the Liouvillian is a non-Hermitian matrix, it too can exhibit EPs [37–44].”
Performance and structure of single-mode bosonic codes (2017, background): “Expressing the normalization factor in terms of such overlaps ([83], Eq.”
Hybrid-Liouvillian formalism connecting exceptional points of non-Hermitian Hamiltonians and Liouvillians via postselection of quantum trajectories (2020, background): “To every Lindblad master equation corresponds a nonHermitian Liouvillian superoperator L which can display Liouvillian exceptional points (LEPs) [12, 23, 46–52].”
Geometry and Response of Lindbladians (2015, background): “, symmetry [50,51] or topology [11], the asymptotic subspace canbe degenerate yet not support a qubit [e.”
Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe (2022, background): “Already open quantum systems in finite dimensions are more complicated with recent advances in reservoir engineering [12, 564, 565] as well as fix-point and symmetry analysis [11, 14].”
A short introduction to the Lindblad master equation (2019, background): “The spectrum of Liouvillian superoperators has been analysed in several recent papers [33, 37].”
Dynamically protected cat-qubits: a new paradigm for universal quantum computation (2013, neutral): “ fjC(0mod4) i;jC (2mod4) igcomprising our logical qubit’s basis. The corresponding conserved quantity for the populations of jC(0mod4) iand jC (2mod4) i is once again identical to the non-driven case [31], J 00 = P 1 n=0 j4nih4nj. While an analytical expression for the other conserved quantity J 02 ”
Open quantum systems with local and collective incoherent processes: Efficient numerical simulations using permutational invariance (2018, background): “Recent studies have highlighted the rich physics that can be inferred from the properties of the Lindbladian superoperators and their spectra [36, 37, 218].”