We study equivariant linear maps between finite-dimensional matrix algebras, as introduced in Collins et al. (Linear Algebra Appl 555:398–411, 2018). These maps satisfy an algebraic property which makes it easy to study their positivity or k-positivity. They are therefore particularly suitable for applications to entanglement detection in quantum information theory. We characterize their Choi matrices. In particular, we focus on a subfamily that we call (a, b)-unitarily equivariant. They can be seen as both a generalization of maps invariant under unitary conjugation as studied by Bhat (Banach J Math Anal 5(2):1–5, 2011) and as a generalization of the equivariant maps studied in Collins et al. (2018). Using representation theory, we fully compute them and study their graphical representation and show that they are basically enough to study all equivariant maps. We finally apply them to the problem of entanglement detection and prove that they form a sufficient (infinite) family of positive maps to detect all k-entangled density matrices.

我们研究了Collins等人介绍的有限维矩阵代数之间的等变线性映射。（Linear Algebra Appl 555：398–411，2018）。这些图满足代数性质，这使得研究其正性或k正性变得容易。因此，它们特别适用于量子信息理论中的纠缠检测。我们表征他们的Choi矩阵。特别是，我们专注于我们称为的一个亚科（a，  b）-单等变。它们既可以看作是Bhat研究的在单一共轭下不变映射的推广（Banach J Math Anal 5（2）：1-5，2011年），也可以看作是Collins等人研究的等价映射的推广。（2018）。使用表示理论，我们充分地计算了它们并研究了它们的图形表示，并表明它们基本上足以研究所有等变图。我们最终将它们应用于纠缠检测问题，并证明它们形成了足够的（无限）正图族，可以检测所有k个纠缠的密度矩阵。