Higher dimensional bipartite composite systems with the same density matrix: separable, free entangled, or PPT entangled?

Given the density matrix of a bipartite quantum state, could we decide whether it is separable, free entangled, or PPT entangled? Here, we give a negative answer to this question by providing a lot of concrete examples of $$16 \times 16$$ density matrices, some of which are well known. We find that both separability and distillability are dependent on the decomposition of the density matrix. To be more specific, we show that if a given matrix is considered as the density operators of different composite systems, their entanglement properties might be different. In the case of $$16 \times 16$$ density matrices, we can look them as both $$2 \otimes 8$$ and $$4 \otimes 4$$ bipartite quantum states and show that their entanglement properties (i.e., separable, free entangled, or PPT entangled) are completely irrelevant to each other. [ABSTRACT FROM AUTHOR]