Aabhas Gulati

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 $n \geq 5$.

In this paper, we provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors. This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d ≥ 3 and n ≥ 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions. We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.