Sang-Jun Park

The characterization of positive, non-completely positive linear maps is a central problem in operator algebras and quantum information theory, where such maps serve as entanglement witnesses. This work introduces and systematically studies a new convex cone of pairwise copositive matrices, denoted $COPCP_n$. We establish that this cone is dual to the cone of pairwise completely positive matrices and, critically, provides a complete characterization for the positivity of the broad and physically relevant class of covariant maps. We provide a way to systematically lift matrices from the classical cone of copositive matrices, $COP_n$, to the new pairwise cone $COPCP_n$, thereby creating a powerful bridge between the well-studied theory of copositive forms and the structure of positive maps. We develop an analogous framework for decomposable maps, introducing the cone $PDEC_n$ of pairwise decomposable matrices. For several families of linear maps having diagonal unitary symmetry such as generalized Choi maps, we characterize membership in these cones using simple properties of the parameters of the maps. 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, decomposable, or completely positive, 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, such as the (mixture of) Dicke states. We prove that the sum-of-squares (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 boundary 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 bipartite (mixture of) Dicke states that are simultaneously entangled and $K_r$-PPT bosonic extendible for any desired hierarchy level $r \geq 2$ and local dimension $n \geq 5$.

We investigate the structure of $k$-positivity and Schmidt numbers for classes of linear maps and bipartite quantum states possessing symplectic group symmetry. Specifically, we consider (1) linear maps on $M_d(\mathbb{C})$ which are covariant under conjugation by unitary symplectic matrices $S$, and (2) $d\otimes d$ bipartite states which are invariant under $S\otimes S$ or $S\otimes \overline{S}$ actions, each parametrized by two real variables. We provide a complete characterization of all $k$-positivity and decomposability conditions for these maps and explicitly compute the Schmidt numbers for the corresponding bipartite states. In particular, our analysis yields a broad class of PPT states with Schmidt number $d/2$ and the first explicit constructions of $(d/2-1)$-positive indecomposable linear maps, achieving the best-known bounds. These constructions also allow us to exhibit PPT states with an arbitrarily large gap between the Schmidt number of a state and its partial transpose, improving on previously known examples. Our results thus provide a natural and analytically tractable setting in which both highly positive indecomposability and a high degree of PPT entanglement can be studied in full generality. Finally, we show that the PPT squared conjecture holds true within the classes of PPT linear maps which are either symplectic covariant or conjugate-symplectic covariant.