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

Abstract : For discrete-time quantum 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.
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Submitted on : Wednesday, April 2, 2014 - 11:18:16 PM
Last modification on : Thursday, September 24, 2020 - 5:04:18 PM

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Pierre Rouchon, Alain Sarlette. Contraction and stability analysis of steady-states for open quantum systems described by Lindblad differential equations. Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, Dec 2013, Firenze, Italy. pp.6568 - 6573, ⟨10.1109/CDC.2013.6760928⟩. ⟨hal-00971495⟩

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