Symmetries and conserved quantities in Lindblad master equations

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.