  Many of the properties of the partial transposition are not clear so far.
Here the number of the negative eigenvalues of K(T)(the partial transposition
of K) is considered carefully when K is a two-partite state. There are strong
evidences to show that the number of negative eigenvalues of K(T) is N(N-1)/2
at most when K is a state in Hilbert space N*N. For the special case, 2*2
system(two qubits), we use this result to give a partial proof of the
conjecture sqrt(K(T))(T)>=0. We find that this conjecture is strongly connected
with the entanglement of the state corresponding to the negative eigenvalue of
K(T) or the negative entropy of K.
