Negativity for a diagonal reduced density matrix

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?

1 Answer

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 $$\sigma_{ij}^B=|j\rangle\langle j|$$.