Suppose one has a tripartite system A,B,C with density matrix $\rho$ , and with reduced density matrix $\rho_{BC}=\text{Tr}_A\ \rho$. Suppose $\rho_{BC}$ is a diagonal matrix.

As the partial transpose of a diagonal matrix corresponds to the same matrix, and since all the matrix elements are positive, the negativity $\mathcal{N}(\rho_{BC})=\sum_i \frac{\vert \lambda_i\vert-\lambda_i}{2}$, where $\lambda_i$ are the eigenvalues of the partial transpose $\rho_{BC}^{T_B}$ is zero.

How it is possible to say whether the state is entangled or not?


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A state with a diagonal density matrix is always separable (=not entangled): If $$ \rho= \sum p_{ij} |ij\rangle\langle ij|\ , $$ then a separable decomposition is given by $$ \rho = \sum p_{ij} \sigma_{ij}^A\otimes\sigma_{ij}^B\ , $$ with $\sigma_{ij}^A=|i\rangle\langle i|$ and $\sigma_{ij}^B=|j\rangle\langle j|$.


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