Lindblad微分方程描述的开放量子系统定态的压缩性和稳定性分析
Contraction and stability analysis of steady-states for open quantum systems described by Lindblad differential equations
摘要：
对于离散时间系统，由Kraus映射，D。Petz描述了一组通用的收缩度量。在本文中，我们使用这个特性来推导出一组二次李雅普诺夫函数的连续时间系统，由Lindblad微分方程，具有一个稳态满秩。这个集合的一个极值由Bures度量给出，其中二次李雅普诺夫函数通过反演Sylvester方程获得。我们说明了该方法通过提供一个严格的李雅普诺夫函数的Lindblad方程设计稳定的量子电动力学的“猫”状态水库工程。事实上，我们证明了任何Lindblad方程的（截断）谐振子的希尔伯特空间，其中有一个满秩平衡，并有，在其退相干通道，对应于光子损失运营商的通道，全球收敛到该平衡。
英文摘要：
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.
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