k-Positive Maps: New Characterizations and a Generation Method

We study [Formula: see text]-positive linear maps on matrix algebras and address two problems, (i) characterizations of [Formula: see text]-positivity and (ii) generation of nondecomposable [Formula: see text]-positive maps. On the characterization side, we derive optimization-based conditions equivalent to [Formula: see text]-positivity that (a) reduce to a simple check when [Formula: see text], (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for [Formula: see text]), and (c) recast [Formula: see text]-positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single [Formula: see text]-positive map, produces one-parameter families that remain [Formula: see text]-positive and nondecomposable for small enough times. We illustrate this by generating such families for [Formula: see text] and [Formula: see text]. We also formulate a semidefinite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying [Formula: see text]-positivity and a systematic way to sample [Formula: see text]-positive nondecomposable maps.