Bell-diagonal states are 2-qubit states that are diagonal in the Bell basis. Since those states lie in $\mathbb{C}^{2} \otimes \mathbb{C}^{2}$, the Peres-Horodecki criterion is a sufficient condition to show separability and it's also pretty easy to check: $\rho = \sum_{i \in [0,3]}\lambda_i|\psi_{i}\rangle\langle\psi_{i}| $ is PPT (or separable) if and only if $Tr(\rho) \geq 2\lambda_i \geq 0$ for every $i$. (Here {$|\psi_{i}\rangle$} are the Bell states)
In my research I am dealing with a generalization of those states. In particular, my question is about states in $\mathbb{C}^{2^d} \otimes \mathbb{C}^{2^d}$ that are diagonal in the basis given by the $d$-fold tensor product of Bell states.
For example, for $d=2$, the states I am considering are diagonal in the basis: $$ |\psi_{0}\rangle\otimes|\psi_{0}\rangle,|\psi_{0}\rangle\otimes|\psi_{1}\rangle, \ldots,|\psi_{3}\rangle\otimes|\psi_{3}\rangle. $$
I am wondering the following:
are there some nice criteria already known to check when these states are PPT or separable?
Notice that those states are in general different from the states diagonal in what is called the generalized Bell basis in literature.
