The Decompositions of Werner and Isotropic States

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.