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

Authors: P. Rouchon, A. Sarlette, Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, pp. 6568 - 6573, 10-13 Dec. 2013, Firenze DOI: 10.1109/CDC.2013.6760928

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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BibTeX:

@Proceedings{,

author = {P. Rouchon, A. Sarlette},

editor = {},

title = {Contraction and stability analysis of steady-states for open quantum systems described by Lindblad differential equations},

booktitle = {Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on},

volume = {},

publisher = {},

address = {Firenze},

pages = {6568 - 6573},

year = {2013},

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.},

keywords = {}}

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.

Download PDF

BibTeX:

@Proceedings{,

author = {P. Rouchon, A. Sarlette},

editor = {},

title = {Contraction and stability analysis of steady-states for open quantum systems described by Lindblad differential equations},

booktitle = {Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on},

volume = {},

publisher = {},

address = {Firenze},

pages = {6568 - 6573},

year = {2013},

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.},

keywords = {}}