# 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.

The partial transpose is not the only positive but not completely positive operation that is possible on 2x2 and 2x3 systems. Trivially, any completely positive operation (such as a local unitary) combined with the partial transpose is a different positive operation.

The point is that, as wikipedia puts it

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

so, somehow, the partial transpose is the only element that you need to describe all possible positive maps on 2x2 and 2x3 systems. That doesn't mean it's the only map.