The theory of symmetry of quantum mechanical systems is applied to study the structure and properties of several classes of relevant maps in quantum information theory: CPTP, PPT and Schwarz maps. First, we develop the general structure that equivariant maps $Φ:\mathcal A \to \mathcal B$ between $C^\ast$-algebras satisfy. Then, we undertake a systematic study of unital, Hermiticity-preserving maps that are equivariant under natural unitary group actions. Schwarz maps satisfy Kadison's inequality $Φ(X^\ast X) \geq Φ(X)^\ast Φ(X)$ and form an intermediate class between positive and completely positive maps. We completely classify $U(n)$-equivariant on $M_n(\mathbb C)$ and determine those that are completely positive and Schwarz. Partial classifications are then obtained for the weaker $DU(n)$-equivariance (diagonal unitary symmetry) and for tensor-product symmetries $U(n_1) \otimes U(n_2)$. In each case, the parameter regions where $Φ$ is Schwarz or completely positive are described by explicit algebraic inequalities, and their geometry is illustrated. Finally, we further show that the $U(n)$-equivariant family satisfies $\mathrm{PPT} \iff \mathrm{EB}$, while the $DU(2)$, symmetric $DU(3)$, $U(2) \otimes U(2)$ and $U(2) \otimes U(3)$, families obey the $\mathrm{PPT}^2$ conjecture through a direct symmetry argument. These results reveal how group symmetry controls the structure of non-completely positive maps and provide new concrete examples where the $\mathrm{PPT}^2$ property holds.

翻译：本文将量子力学系统的对称性理论应用于研究量子信息论中几类相关映射的结构与性质：CPTP、PPT 与施瓦茨映射。首先，我们建立了 $C^\ast$-代数间等变映射 $Φ:\mathcal A \to \mathcal B$ 所满足的一般结构。随后，我们对在自然酉群作用下保持等变的保单位、保厄米性映射进行了系统研究。施瓦茨映射满足卡迪逊不等式 $Φ(X^\ast X) \geq Φ(X)^\ast Φ(X)$，构成了正映射与完全正映射之间的一个中间类别。我们完全分类了作用于 $M_n(\mathbb C)$ 上的 $U(n)$-等变映射，并确定了其中为完全正映射和施瓦茨映射的类别。接着，我们针对较弱的 $DU(n)$-等变性（对角酉对称性）以及张量积对称性 $U(n_1) \otimes U(n_2)$ 获得了部分分类结果。在每种情形下，$Φ$ 为施瓦茨映射或完全正映射的参数区域均由显式的代数不等式描述，并图示了其几何结构。最后，我们进一步证明 $U(n)$-等变映射族满足 $\mathrm{PPT} \iff \mathrm{EB}$，而 $DU(2)$、对称 $DU(3)$、$U(2) \otimes U(2)$ 以及 $U(2) \otimes U(3)$ 映射族则通过直接的对称性论证服从 $\mathrm{PPT}^2$ 猜想。这些结果揭示了群对称性如何控制非完全正映射的结构，并为 $\mathrm{PPT}^2$ 性质成立的情形提供了新的具体示例。