We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.

中文翻译：

---

量子理论中的对角幺正和正交对称

我们分析了矩阵代数之间的二部矩阵和线性映射，在对角酉群和正交群的作用下，它们分别是不变和协变的。通过展示来自文献的大量示例，其中包括对角对称状态和 Choi 型映射等值得注意的条目，我们表明这类矩阵（和映射）包含各种各样的场景，从而统一了他们的研究。我们检查它们的线性代数结构，并通过它们的凸圆锥表现来研究不同的正性概念。特别是，我们将众所周知的完全正矩阵的锥推广到三重完全正矩阵的锥，并将其与相关不变态的可分离性（或相应量子通道的纠缠破坏特性）联系起来。对于线性映射，我们根据它们的 Kraus、Stinespring 和 Choi 表示提供了所述协方差的明确表征，并系统地分析了正性、可分解性、完全正性等的通常属性。我们还描述了这些映射的不变子空间，并使用它们的结构为相关的不变二分状态的可分离性提供必要和充分条件。和 Choi 表示，并系统地分析了正性、可分解性、完全正性等的通常属性。我们还描述了这些映射的不变子空间，并使用它们的结构为相关的不变二分状态的可分离性提供必要和充分条件。和 Choi 表示，并系统地分析了正性、可分解性、完全正性等的通常属性。我们还描述了这些映射的不变子空间，并使用它们的结构为相关的不变二分状态的可分离性提供必要和充分条件。