标题： 相对 $C$"-数值范围在量子控制和量子信息中的应用 显示英文标题

标题： Relative $C$"-Numerical Ranges for Applications in Quantum Control and Quantum Information

摘要： 受量子信息和量子控制应用的推动，一种新的$C$"数值范围"——相对$C$"数值范围"，记作$W_K(C,A)$——被引入。 它通过在经典$C$"数值范围"定义中的酉群 U(N) 替换为其任意紧致连通子群$K \subset U(N)$而产生。 相对$C$"数值范围"的几何性质被详细分析。 反例证明其几何结构比经典情况更为复杂：例如，$W_K(C,A)$既不是星形的也不是单连通的。 然而，经典的$C$"数值范围的旋转对称性的一个已知结果扩展到了$W_K(C,A)$，这通过基于李理论的新方法得以证明。 此外，我们集中研究子群$SU_{\rm loc}(2^n) := SU(2)\otimes ... \otimes SU(2)$，即 SU(2) 的$n$阶张量积，在应用中对此特别感兴趣。 在这种情况下，得到了$W_{K}(C,A)$是复平面原点处圆形圆盘的充分条件。 最后，详细说明了前文结果在$SU(2) \otimes SU(2)$上的应用。 显示英文摘要

摘要： Motivated by applications in quantum information and quantum control, a new type of $C$"-numerical range, the relative $C$"-numerical range denoted $W_K(C,A)$, is introduced. It arises upon replacing the unitary group U(N) in the definition of the classical $C$"-numerical range by any of its compact and connected subgroups $K \subset U(N)$. The geometric properties of the relative $C$"-numerical range are analysed in detail. Counterexamples prove its geometry is more intricate than in the classical case: e.g. $W_K(C,A)$ is neither star-shaped nor simply-connected. Yet, a well-known result on the rotational symmetry of the classical $C$"-numerical range extends to $W_K(C,A)$, as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup $SU_{\rm loc}(2^n) := SU(2)\otimes ... \otimes SU(2)$, i.e. the $n$-fold tensor product of SU(2), which is of particular interest in applications. In this case, sufficient conditions are derived for $W_{K}(C,A)$ being a circular disc centered at origin of the complex plane. Finally, the previous results are illustrated in detail for $SU(2) \otimes SU(2)$.