  We have reexamined the moments of positive maps and the criterion based on
these moments to detect entanglement. For two qubits, we observed that
reduction map is equivalent to partial transpose map as the resulting matrices
have the same set of eigenvalues although both matrices look different in same
computational basis. Consequently, the detection power of both maps is same.
For $2 \otimes 4$ systems, we find that moments of reduction map are capable to
detect a family of bound entangled states. For qutrit-qutrit systems, we show
that moments of reduction map can detect two well known families of bound
entangled states. The moments of another positive map can detect the complete
range of entanglement for a specific family of quantum states, whereas the
earlier criterion fails to detect a small range of entangled states. For three
qubits system, we find that applying reduction map to one of the qubit is
equivalent to partial transpose operation. In particularly, for GHZ state and W
state mixed with white noise, all the moments of a reduction map are exactly
the same as the moments of partial transpose map.
