Spectral characterization of entanglement witnesses in 2 × n and 3 × n systems

In this paper, we investigate the inertia (i.e. the eigenvalue triplet counting negative, zero, and positive eigenvalues) of entanglement witnesses arising from the partial transposes of non-positive partial transpose (NPT) states across bipartite (2 × n and 3 × n) and tripartite (three-qubit) systems. For any 2 × n NPT state ρ whose partial transpose ρ^Γ has inertia (1,2n−4,3), we prove that ρ can be converted into a 2 × 2 NPT subsystem via stochastic local operations and classical communication (SLOCC). For any 3 × n NPT state ρ whose partial transpose ρ^Γ has inertia (1,3n−4,3), we prove that ρ can always be projected into a 2 × 2 NPT subsystem via local operations. Through extensive numerical simulations generating random sparse positive semidefinite matrices, we catalog all possible inertia trios (In((Formula presented)),In((Formula presented)),In((Formula presented))) for three-qubit NPT states. We identify 72 distinct inertia trios, revealing asymmetric distributions and co-occurrence constraints. These findings establish fundamental links between inertia patterns, distillability properties, and the geometric structure of entangled states, providing new spectral tools for characterizing entanglement in quantum information processing.