Suppose we have a mixed state $\rho$ over $\mathcal H_A\otimes \mathcal H_B$, where $\mathcal H_A=\mathcal H_B=\mathbb{C}^d$ is a finite dimensional Hilbert space. Can $\rho$ be entangled if it is symmetric ($\mathsf{Swap}_{AB}\rho=\rho$) and invariant to partial transposition ($\rho^{T_A}=\rho^{T_B}=\rho$)?
To answer the question, I have been looking throught the literature on the more general set of symmetric entangled states that have positive partial transpose. So far, however, I have not made progress in finding these states, so I'm wondering if anything is known about them.
