Positive maps and extendibility hierarchies from copositive matrices

TL;DR

This work develops a unified framework linking pairwise matrix cones to positive maps and quantum entanglement. By introducing the pairwise copositive cone $\mathsf{PCOP}_n$ and the pairwise decomposable cone $\mathsf{PDEC}_n$, the authors establish dualities with $\mathsf{PCP}_n$ and $\mathsf{SPN}_n$, and provide constructive lifting theorems from COP to pairwise COP that generate large families of positive indecomposable maps. They apply these tools to graph-based maps $\Phi_t^G$, deriving exact positivity/decomposability thresholds in terms of graph invariants like $\lambda(G)$, $\omega(G)$, and a new parameter $\sigma(G)$, yielding infinite families of indecomposable maps from rank-3 strongly regular and Paley graphs. The dual perspective connects SOS hierarchies for copositive matrices with PPT bosonic extendibility, enabling the explicit construction of entanglement witnesses without any finite SOS certificate and demonstrating PPT entangled Dicke states for arbitrarily high extendibility. Overall, the paper offers a powerful bridge between convex geometry, graph theory, and quantum entanglement, with broad implications for separability criteria and entanglement detection.

Abstract

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 $Φ_t^G$ 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 $n \geq 5$.

Figures (6)

• Figure 1: The positivity properties of the linear maps $\Phi^G_t$ from \ref{['eq:def-Phi-G-t']} as the parameter $t$ varies. The blue region corresponds to the important case of positive indecomposable maps.
• Figure 2: Graphs on 6 vertices leading to positive indecomposable maps. From left to right, the square + path graph, the tadpole graph, and the wheel graph. A maximal clique of each graph is highlighted in red. Note that the wheel graph is the first graph not covered by the Proposition \ref{['prop:triangle-free-bipartite']} as it is not triangle-free.
• Figure 3: The Petersen (left) and the Shrikhande (right) are SRG with respective parameters $(10, 3, 0, 1)$ and $(16,6,2,2)$. Maximal cliques are highlighted in red.
• Figure 4: On the left, we denote the state a bipartite $\rho$, and on the right, the the notion of $r$-extension that we consider in Definition \ref{['def:star-graph-extendibility']}. The marginal over any edges of the graph should be equal to $\rho$. Moreover, we demand that the systems corresponding to the nodes of the star graph together live in the bosonic subspace $\mathsf{BExt}^{(r)}_n$, and for PPT extendible states $\mathsf{PPTBExt}^{(r)}_n$, we impose the PPT condition across all bipartitions.
• Figure 5: On the left, we denote a bipartite bosonic state $\rho$, and on the right, the the notion of $r$-extension that we consider in Definition \ref{['def:star-graph-extendibility']}. The marginal over every edge of the graph should be equal to $\rho$. Moreover, we demand that the global state corresponding to the complete graph should live in the bosonic subspace. For PPT extendible states $\mathsf{PPTBExt}^{(\mathcal{K}_r)}_n$, we impose the PPT condition across all bipartitions.

Theorems & Definitions (114)

• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Remark 2.6
• Lemma 2.7: Choi-Jamiołkowski Isomorphism choi1975completely
• Definition 2.8
• Proposition 2.9
• Remark 2.10