开放量子系统中量子复杂性的几何结构
The Geometry of Quantum Complexity in Open Systems
中文总结 AI 辅助
研究将量子复杂性几何方法从封闭系统扩展到开放系统，通过混合态空间最优控制问题定义复杂性，分析多个例子，表明改变惩罚因子会改几何性质，为量化耗散量子系统复杂性提供几何框架。
中文摘要 AI 辅助
我们将尼尔森用于量子复杂性的几何方法从封闭量子系统扩展到开放量子系统，其动力学由林德布拉德演化支配。在此框架下，复杂性通过混合态空间上的最优控制问题定义，酉和非酉生成元都有成本赋值。结果表明，所得几何结构与酉演化情形下出现的黎曼几何有根本不同。在开放系统设置中，自然几何通常是次芬斯勒几何。耗散使测地线不可逆，同时可允许的切向方向受物理允许控制的限制。我们分析了几个有物理动机的例子，包括受去极化和振幅阻尼通道作用的单个量子比特以及阻尼谐振子。我们表明，与酉情形类似，改变成本泛函中的惩罚因子会通过标志曲率（截面曲率的芬斯勒类似物）的变化来修改几何性质。我们的结果为量化耗散量子系统中复杂性的抽象概念提供了一个几何框架，与可实验实现的设置有潜在联系。
英文摘要
We extend Nielsen's geometric approach for quantum complexity from closed to open quantum systems, whose dynamics is governed by Lindbladian evolution. In this framework, complexity is defined through an optimal-control problem on the space of mixed states, with a cost assigned to both unitary and non-unitary generators. We show that the resulting geometric structure differs fundamentally from the Riemannian geometry that emerges in the case of unitary evolution. In the open-system setting, the natural geometry is typically sub-Finslerian. Dissipation makes the geodesics non-reversible, while the admissible tangent directions are restricted by the physically allowed controls. We analyze several physically motivated examples, including a single qubit subject to depolarizing and amplitude-damping channels, as well as the damped harmonic oscillator. We show that, similarly to the unitary case, varying the penalty factors in the cost functional modifies the geometric properties through changes in the flag curvature, the Finslerian analog of sectional curvature. Our results provide a geometric framework for quantifying the abstract notion of complexity in dissipative quantum systems, with potential connections to experimentally realizable setups.
