A reduction of the separability problem to SPC states in the filter normal form (pp0057-0070) 
         Daniel Cariello
           doi: https://doi.org/10.26421/QIC24.1-2-3
Abstracts: It was recently suggested  that a solution to the separability problem for states that remain positive under partial transpose composed with realignment (the so-called symmetric with positive coefficients states or simply SPC states) could  shed light on entanglement in general. Here we show that such a solution would solve the problem completely.Given a state in $ \mathcal{M}_k\otimes\mathcal{M}_m$, we build a SPC state in  $ \mathcal{M}_{k+m}\otimes\mathcal{M}_{k+m}$ with the same Schmidt number.   It is known  that this type of state can be put in the filter normal form retaining its type. A solution to the separability problem in $\mathcal{M}_k\otimes\mathcal{M}_m$ could be obtained by solving the same problem for SPC states in the filter normal form within $\mathcal{M}_{k+m}\otimes\mathcal{M}_{k+m}$.  This SPC state can be built arbitrarily close to the orthogonal projection on the symmetric subspace of  $ C^{k+m}\otimes C^{k+m}$. All the information required to understand entanglement in $ \mathcal{M}_s\otimes\mathcal{M}_t$ $(s+t\leq k+m)$ lies inside an arbitrarily small ball around that projection.  We also show that the Schmidt number of any state $\gamma\in\mathcal{M}_n\otimes\mathcal{M}_n$ which commutes with the flip operator and lies inside a small ball around that projection cannot exceed $\lfloor\frac{n}{2}\rfloor$.
Key Words: Separability Problem; Schmidt Number; Symmetric Subspace