Positive maps and extendibility hierarchies from copositive matrices

This work introduces and systematically studies a new convex cone of PCOP (pairwise copositive). We establish that this cone is dual to the cone of PCP (pairwise completely positive) and, critically, provides a complete characterization for the positivity of the broad class of covariant maps. We provide a way to lift matrices from the cone of COP to PCOP, thereby creating a powerful bridge between the theory of copositive forms and the positive maps. We develop an analogous framework for decomposable maps, introducing the cone PDEC. As a primary application of this framework, we define a novel family of linear maps $\Phi_t^G$ parameterized by a graph $G$ and a real parameter $t$. We derive exact thresholds on $t$ that determine when these maps are positive or decomposable, linking these properties to fundamental graph-theoretic parameters. This construction yields vast new families of positive indecomposable maps, for which we provide explicit examples derived from infinite classes of graphs, most notably rank 3 strongly regular graphs such as Paley graphs. On the dual side, we investigate the entanglement properties of large classes of (symmetric) states. We prove that the SOS hierarchies used in polynomial optimization to approximate the cone of copositive matrices correspond precisely to dual cones of witnesses for different levels of the PPT bosonic extendibility hierarchy}-. In the setting of the DPS hierarchy for separability, we construct a large family of optimal entanglement witnesses that are not certifiable by any level of the PPT bosonic extendibility hierarchy, answering a long standing open question from [DPS04]. Leveraging the duality, we also provide an explicit construction of (mixture of) bipartite Dicke states that are simultaneously entangled and $K_r$-PPT bosonic extendible for any desired hierarchy level $r \geq 2$ and local dimension $n \geq 5$.