Partial Transposition on Bipartite System

Many properties of partial transposition are unclear as yet. Here we carefully consider the number of the negative eigenvalues of ρ^T (ρ's partial transposition) when ρ is a two-partite state. There is strong evidence to show that the number of negative eigenvalues of ρT is N(N-1)/2 at most when ρ is a state in Hilbert space C^N×C^N. For the special case, the 2×2 system,
we use this result to give a partial proof of the conjecture |ρ^T|^T≥0. We find that this conjecture is strongly connected with the entanglement of the state corresponding to the negative eigenvalue of ρ^^T or the negative entropy of ρ.