# For 2x2 and 2x3 systems, is the partial transpose the only positive but not CP operation?

Question: For 2x2 and 2x3 systems, is the partial transpose the only positive but not completely positive operation that is possible?

Why this came up: The criteria for detecting if a state $$\rho$$ is entangled is:

$$\forall \Lambda$$ such that $$\Lambda$$ is a positive but not completely positive operator, if $$(I \otimes \Lambda ) \rho \geq 0$$ then the state is separable, otherwise it is entangled.

However, we know the Peres-Horodecki criteria for 2x2 and 2x3 systems says it is enough to check only for one positive operation, that is, the partial transpose. Hence I was wondering if this was true.

every such map $$\Lambda$$ can be written as
$$\Lambda=\Lambda_1+\Lambda_2\circ T$$
where $$\Lambda _{1}$$ and $$\Lambda _{2}$$ are completely positive and T is the transposition map