Inertia of decomposable entanglement witnesses

We investigate the inertia (i.e., the array of numbers of negative, zero and positive eigenvalues of an Hermitian matrix) of decomposable entanglement witnesses (EWs). We show that the 2 × n and two-qutrit decomposable EWs have the same inertias as those of non-positive-transpose (NPT) EWs. We also show that if an m × n EW W has inertia (p, a_p, mn − p − a_p) with p≥1, then for every integer b ∈ [0, a_p], then we can find an EW W_b such that InW_b = (p, b, mn − p − b). If W is a decomposable (resp. NPT) EW, then we can choose W_b as also a decomposable (resp. NPT) EW. We further show that the m × n decomposable EW with the maximum number of negative eigenvalues can be chosen as an NPT EW. Then we explicitly characterize the 2 × 3 EWs, and decomposable EWs P^Γ + Q with positive semidefinite matrices P of rank one and Q. We also show that a 2 × 4 non-decomposable EW has no inertia (3, 2, 3). Then we show some properties of a 2 × 4 non-decomposable EW of inertia (2, 3, 3), if it exists.