$C$ -数值域和局域$C$ -数值域在量子控制和量子信息中的意义
The Significance of the $C$-Numerical Range and the Local $C$-Numerical Range in Quantum Control and Quantum Information
摘要：
本文展示了C-数值范围相关的新结构如何从量子控制的实际问题中产生-反之亦然，对这些结构的理解如何有助于解决量子信息中的热点问题。我们首先概述了C-数值范围在量子理论当前研究问题中的作用：最大化初始状态的幺正轨道上的点到目标状态C的投影的量子力学任务通过最大化迹函数与A的C-数值半径相关|\tr \{C^ \dagger UAU^ \dagger \}|.在n个量子比特的量子控制中，人们可能感兴趣的是（i）对于整个动力学具有U \in SU（2 ^n），或者（ii）将动力学限制为每个量子比特上的{ \em局部}操作，即限制为n重张量积SU（2）\otimes SU（2）\otimes. \otimes SU（2）.有趣的是，后者则导致一个新的实体，{ \em本地} C-数值范围W_{\rm loc}（C，A），其复杂的几何形状既不是星形的，也不是简单地连接在传统的C-数值范围。这在随附的论文（math-ph/0702005）中有所说明。我们提出了新的应用程序的C-数值范围在量子控制的梯度流的本地幺正群：（1）他们作为强大的工具，决定是否可以逆转的量子相互作用的时间（在某种意义上概括哈恩著名的自旋回波）;（2）他们允许优化证人的量子纠缠。我们的结论是相关的C-数值范围的约束量子优化问题，我们也给拉格朗日型梯度流算法。
英文摘要：
This paper shows how C-numerical-range related new strucures may arise from practical problems in quantum control--and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of C-numerical ranges in current research problems in quantum theory: the quantum mechanical task of maximising the projection of a point on the unitary orbit of an initial state onto a target state C relates to the C-numerical radius of A via maximising the trace function |\tr \{C^\dagger UAU^\dagger\}|. In quantum control of n qubits one may be interested (i) in having U\in SU(2^n) for the entire dynamics, or (ii) in restricting the dynamics to {\em local} operations on each qubit, i.e. to the n-fold tensor product SU(2)\otimes SU(2)\otimes >...\otimes SU(2). Interestingly, the latter then leads to a novel entity, the {\em local} C-numerical range W_{\rm loc}(C,A), whose intricate geometry is neither star-shaped nor simply connected in contrast to the conventional C-numerical range. This is shown in the accompanying paper (math-ph/0702005). We present novel applications of the C-numerical range in quantum control assisted by gradient flows on the local unitary group: (1) they serve as powerful tools for deciding whether a quantum interaction can be inverted in time (in a sense generalising Hahn's famous spin echo); (2) they allow for optimising witnesses of quantum entanglement. We conclude by relating the relative C-numerical range to problems of constrained quantum optimisation, for which we also give Lagrange-type gradient flow algorithms.
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