量子信道、复Stiefel流形与优化
Quantum channels, complex Stiefel manifolds, and optimization
AI总结：
本文建立了量子信道空间与复Stiefel流形商空间之间的同胚，并应用于量子控制目标泛函的极值分析，同时定义了推广Bures角的信道度量。
摘要：
开放量子系统最一般的动力学通常由量子信道表示，它是一个完全正迹保持映射（CPTP或Kraus映射）。量子信道的著名表示由Choi矩阵和Kraus算子和表示（OSR）描述。如前所述，可以利用Kraus OSR通过某个复Stiefel流形在酉群作用下的适当商空间中的点来参数化量子信道。在本工作中，我们建立了量子信道拓扑空间与复Stiefel流形商空间之间的同胚。该同胚可应用于各种量子优化问题。作为示例，我们将其应用于分析定义在复Stiefel流形上的多种量子控制目标泛函的极值点，包括平均值、量子门生成、涉及熵的热力学量等。最后，我们定义了由Stiefel流形上的黎曼度量诱导的量子信道空间上的度量，并证明它是密度矩阵之间Bures角的推广。
英文摘要：
Most general dynamics of an open quantum system is commonly represented by a quantum channel, which is a completely positive trace-preserving map (CPTP or Kraus map). Well-known representations of quantum channels are described by Choi matrices and by Kraus operator-sum representation (OSR). As was shown before, one can use Kraus OSR to parameterize quantum channels by points of a suitable quotient of some complex Stiefel manifold by the action of the unitary group. In this work, we establish a homeomorphism between the topological space of quantum channels and the quotient of the complex Stiefel manifold. This homeomorphism can be applied to various quantum optimization problems. As an example, we apply it to the analysis of extrema points for a wide variety of quantum control objective functionals defined on the complex Stiefel manifolds, including mean value, generation of quantum gates, thermodynamic quantities involving entropy, etc. Finally, a metric on the space of quantum channels induced by the Riemannian metric on the Stiefel manifold is defined, and we show that it is a generalization of the Bures angle between density matrices.
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