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The continuous-time differential Lyapunov equation is widely used in linearoptimal control theory, a branch of mathematics and engineering. In quantumphysics, it is known to appear in Markovian descriptions of linear (quadraticHamiltonian, linear equations of motion) open quantum systems, typically fromquantum master equations. Despite this, the Lyapunov equation is seldomconsidered a fundamental formalism for linear open quantum systems. In thiswork we aim to change that. We establish the Lyapunov equation as a fundamentaland efficient formalism for linear open quantum systems that can go beyond thelimitations of various standard quantum master equation descriptions, whileremaining of much less complexity than general exact formalisms. This alsoprovides valuable insights for non-Hermitian quantum physics. In particular, wederive the Lyapunov equation for the most general number conserving linearsystem in a lattice of arbitrary dimension and geometry, connected to anarbitrary number of baths at different temperatures and chemical potentials.Three slightly different forms of the Lyapunov equation are derived via anequation of motion approach, by making increasing levels of controlledapproximations, without reference to any quantum master equation. Then wediscuss their relation with quantum master equations, positivity, accuracy andadditivity issues, the possibility of describing dark states, generalperturbative solutions in terms of single-particle eigenvectors and eigenvaluesof the system, and quantum regression formulas. Our derivation gives a clearunderstanding of the origin of the non-Hermitian Hamiltonian describing thedynamics and separates it from the effects of quantum and thermal fluctuations.Many of these results would have been hard to obtain via standard quantummaster equation approaches.

Lyapunov equation in open quantum systems and non-Hermitian physics