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For discrete-time systems, governed by Kraus maps, the work of D. Petz hascharacterized the set of universal contraction metrics. In the present paper,we use this characterization to derive a set of quadratic Lyapunov functionsfor continuous-time systems, governed by Lindblad differential equations, thathave a steady-state with full rank. An extremity of this set is given by theBures metric, for which the quadratic Lyapunov function is obtained byinverting a Sylvester equation. We illustrate the method by providing a strictLyapunov function for a Lindblad equation designed to stabilize a quantumelectrodynamic "cat" state by reservoir engineering. In fact we prove that anyLindblad equation on the Hilbert space of the (truncated) harmonic oscillator,which has a full-rank equilibrium and which has, among its decoherencechannels, a channel corresponding to the photon loss operator, globallyconverges to that equilibrium.

Contraction and stability analysis of steady-states for open quantum systems described by Lindblad differential equations