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A bipartite quantum state $\rho$ is called a CPB(Complete Product Basis)-state if its eigenvectors can be expressed as product state (a pure state of the form of $|\phi\rangle \otimes |\psi\rangle$). (Groisman et al., (2007)) Although this class of separable states seems to have useful physical meanings, (in the sense that these states are 'more separable' than the states with entangled eigenbasis) I can't find any necessary or sufficient conditions for a given separable state to be a CPB-state. Is there any research done on this topic?

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