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

TL;DR

This paper develops quadratic Lyapunov functions for continuous-time open quantum systems described by Lindblad equations, demonstrating stability and convergence to steady states, including quantum cat states, using contraction metrics and reservoir engineering.

Contribution

It introduces a method to derive quadratic Lyapunov functions for Lindblad systems with full-rank steady states, extending contraction metric concepts from discrete to continuous quantum dynamics.

Findings

Quadratic Lyapunov functions are derived using contraction metrics.

The Bures metric provides an extremal Lyapunov function via Sylvester equation.

Lindblad equations with photon loss channels converge to full-rank equilibrium states.

Abstract

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.