Sanpera–Bruß–Lewenstein conjecture on Schmidt number of $3\times3$ PPT bound entangled states

Let $\mathcal{H}_A$ and $\mathcal{H}_B$ be Hilbert spaces of dimension $3$, and let a bipartite density matrix represent a quantum state on $\mathcal{H}_A\otimes\mathcal{H}_B\cong\mathbb{C}^3\otimes\mathbb{C}^3$. Its Schmidt number is the minimum, over all convex decompositions into pure states, of the largest Schmidt rank among the pure states in the decomposition. A state has positive partial transpose if its partial transpose is positive, and it is bound entangled if it is entangled but cannot be distilled into pure entanglement by local operations and classical communication.

Sanpera–Bruß–Lewenstein conjecture. All bound entangled states with positive partial transpose in $\mathbb{C}^3\otimes\mathbb{C}^3$ have Schmidt number $2$.

The conjecture concerns the structure of the simplest positive-partial-transpose bound entangled states. The source reports strong evidence for special cases and later states that the inclusion $\mathbb{T}\subset\mathbb{V}_2$ gives an affirmative answer, so the conjecture is solved in the paper's setting.