exist NPT bound entangled states [@biyog]. Note also that both separable as well as PPT states form convex sets. Figure \[fig:phase\] depicts the structure of the state space with respect to the partial transposition criteria and distillability.

![The structure of the state space in light of the partial transposition criteria and distillability. Separable and as well as PPT states form convex sets (while NPT states do not). It also shows the conjectured NPT bound entangled states. It is not clear whether the set of non-distillable states is convex.[]{data-label="fig:phase"}](phase.pdf)

Theorem 2 is a necessary condition of separability in any arbitrary dimension. However, for some special cases, the partial transposition criterion is both a necessary and a sufficient condition for separability:\
**Theorem 3** [@HorodeckiPPT]\
*In ${\mathbb{C}}^2 \otimes {\mathbb{C}}^2$ or ${\mathbb{C}}^2 \otimes {\mathbb{C}}^3$, a state $\rho$ is separable if and only if $\rho^{T_{A}}\ \ge\ 0$*.

As mentioned above, PPT bound entangled states exist. However, as the Theorem 3 shows, they can exist only in dimensions higher than $\mathbb{C}^2 \otimes \mathbb{C}^2$ and $\mathbb{C}^2 \otimes \mathbb{C}^3$.

Majorization
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The partial transposition criterion, although powerful, is not able to detect entanglement in a finite volume of states. It is, therefore, interesting to discuss other independent criteria. 