  For discrete-time systems, governed by Kraus maps, the work of D. Petz has
characterized the set of universal contraction metrics. In the present paper,
we use this characterization to derive a set of quadratic Lyapunov functions
for continuous-time systems, governed by Lindblad differential equations, that
have a steady-state with full rank. An extremity of this set is given by the
Bures metric, for which the quadratic Lyapunov function is obtained by
inverting a Sylvester equation. We illustrate the method by providing a strict
Lyapunov function for a Lindblad equation designed to stabilize a quantum
electrodynamic "cat" state by reservoir engineering. In fact we prove that any
Lindblad equation on the Hilbert space of the (truncated) harmonic oscillator,
which has a full-rank equilibrium and which has, among its decoherence
channels, a channel corresponding to the photon loss operator, globally
converges to that equilibrium.

Optimization and Control;Quantum Physics