Reachability in Controlled Markovian Quantum Systems: An Operator-Theoretic Approach

F. V. Ende
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引用次数: 5

Abstract

In quantum systems theory one of the fundamental problems boils down to: Given an initial state, which final states can be reached by the dynamic system in question? Formulated in the framework of bilinear control systems, the evolution shall be governed by an inevitable Hamiltonian drift term, finitely many control Hamiltonians allowing for (at least) piecewise constant control amplitudes, plus a (possibly bang-bang switchable) noise term in Kossakowski-Lindblad form. Now assuming switchable coupling of finite-dimensional systems to a thermal bath of arbitrary temperature, the core problem of reachability boils down to studying points in the standard simplex amenable to two types of controls that can be used interleaved: Permutations within the simplex, and contractions by a dissipative one-parameter semigroup. We illustrate how the solutions of the core problem pertain to the reachable set of the original controlled Markovian quantum system. This allows us to show that for global as well as local switchable coupling to a temperature-zero bath one can generate every quantum state from every initial state up to arbitrary precision. Moreover we present an inclusion for non-zero temperatures as a consequence of our results on d-majorization. Then we consider infinite-dimensional open quantum-dynamical systems following a unital Kossakowski-Lindblad master equation extended by controls. Here the drift Hamiltonian can be arbitrary, the finitely many control Hamiltonians are bounded, and the switchable noise term is generated by a single compact normal operator. Via new majorization results of ours, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one, as up to now only has been known in finite-dimensional analogues.
受控马尔可夫量子系统的可达性:一种算子论方法
在量子系统理论中,一个基本问题可以归结为:给定一个初始状态,所讨论的动态系统可以达到哪些最终状态?在双线性控制系统的框架中,演化应由一个不可避免的哈密顿漂移项,有限个控制哈密顿量允许(至少)分段恒定的控制幅度,加上一个(可能是bang-bang可切换的)Kossakowski-Lindblad形式的噪声项来控制。现在,假设有限维系统与任意温度的热浴的可切换耦合,可达性的核心问题归结为研究标准单纯形中的点,这些点适用于两种可交错使用的控制:单纯形内的置换和耗散单参数半群的收缩。我们说明了核心问题的解如何与原始受控马尔可夫量子系统的可达集有关。这使我们能够证明,对于全局和局部可切换耦合到零温度浴,可以从每个初始状态生成任意精度的每个量子态。此外,我们提出了一个包含非零温度作为我们的结果在d-多数化。然后,我们考虑无限维开放量子动力系统遵循由控制扩展的一元Kossakowski-Lindblad主方程。其中,漂移哈密顿量可以是任意的,有限个控制哈密顿量是有界的,可切换噪声项由单个紧正规算子产生。通过我们的新多数化结果,我们表明这种双线性量子控制系统允许近似达到任何被初始状态多数化的目标状态,因为到目前为止只在有限维类似物中已知。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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