Inertia of two-qutrit entanglement witnesses

Inertia of partial transpose of positive semidefinite matrices

“…This is presented in theorem 22. Combining with the results in [15], we can conclude the list of inertia of partial transpose of all two-qutrit NPT states. Then we apply our results to study the inertia of partial transpose of 3 × 4 NPT states.…”

“…(i) If a = b * , then M 13 = M † 13 . By ( 14) and ( 15) and lemma 14, we have rank (14) and lemma 14, we have i − (M Γ − M) ⩽ 2, which is a contradiction with (15). So the first column of M † 23 − M 23 are not all zeros.…”

“…Hence, one can straightforwardly show that ρ Γ is an EW. The inertia of EWs has been investigated in [14,15]. For example, it has been shown that the number of positive eigenvalues of an EW is at least three.…”

“…The inertia is also related to the entanglement measure such as the negativity [12], and the inertia of partial transpose of 2 × n and 3 × 3 NPT states has been partially investigated in [13][14][15]. Due to the mathematical difficulties of characterizing 3 × 3 systems, the existence of two inertias (4, 1, 4) and (3,2,4) of two-qutrit states remains unknown in [15]. In this paper, we develop novel techniques and exclude their existence.…”