The decompositions of separable Werner states and isotropic states are well-known tough issues in quantum information theory. In this work, we investigate them in the Bloch vector representation, exploring the symmetric informationally complete positive operator-valued measure (SIC-POVM) in the Hilbert space. In terms of regular simplexes, we successfully get the decomposition for arbitrary Werner state in \({\mathbb {C}}^N\otimes {\mathbb {C}}^N\), and the explicit separable decompositions are constructed based on the SIC-POVM. Meanwhile, the decomposition of isotropic states is found related to the decomposition of Werner states via partial transposition. It is interesting to note that when dimension N approaches to infinity, the Werner states are either separable or non-steerably entangled, and most of the isotropic states tend to be steerable.

可分离的维尔纳态和各向同性态的分解是量子信息理论中众所周知的难题。在这项工作中，我们在 Bloch 向量表示中研究它们，探索 Hilbert 空间中的对称信息完全正算子值度量 (SIC-POVM)。在正则单纯形方面，我们成功地得到了\({\mathbb {C}}^N\otimes {\mathbb {C}}^N\) 中任意 Werner 状态的分解，并且显式可分离分解是基于SIC-POVM。同时，发现各向同性态的分解与通过部分转置的Werner态的分解有关。有趣的是，当维度N 接近无穷大时，维尔纳态要么是可分离的，要么是不可操纵的纠缠，而且大多数各向同性态往往是可操纵的。