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 ([Formula: see text] and [Formula: see text]) and tripartite (three-qubit) systems. For any [Formula: see text] NPT state [Formula: see text] whose partial transpose [Formula: see text] has inertia [Formula: see text], we prove that [Formula: see text] can be converted into a [Formula: see text] NPT subsystem via stochastic local operations and classical communication (SLOCC). For any [Formula: see text] NPT state [Formula: see text] whose partial transpose [Formula: see text] has inertia [Formula: see text], we prove that [Formula: see text] can always be projected into a [Formula: see text] NPT subsystem via local operations. Through extensive numerical simulations generating random sparse positive semidefinite matrices, we catalog all possible inertia trios [Formula: see text] 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.