Rank-based entanglement detection for PPT entangled states

TL;DR

This work introduces a rank-1 generated-cone framework to detect PPT entanglement by mapping bipartite density operators into constrained PSD cones and exploiting a fundamental gap: separable states map into the rank-1 generated subcone $R_1[\mathcal{M}^+_n(\mathcal{L})]$, while certain PPT entangled states map outside this subcone. The authors develop a general theorem using maps $Z_K$ and $Z_K^\Gamma$ that translate separability and PPT entanglement into a rank-1 generation test on the image cones, enabling construction of PPT entangled states and edge states across several faces of the PSD cone. They instantiate the framework with concrete families—sparse PSD matrices, restricted rank-1 states via diagonal-proportional correlation cones, scaled correlation matrices, and bosonic symmetric states—each yielding new PPT entangled examples and practical witnesses. By weaving convex analysis, graph theory, and invariant-state structures, the approach provides a flexible, multi-parameter toolkit for entanglement detection and expansion of known PPT entangled families with potential applications in quantum information tasks.

Abstract

We develop a new method for entanglement detection in bipartite quantum states by using the violation of the rank-1-generated property of matrices. The positive-semidefinite matrices form a convex cone that has extremal elements of rank 1. But, convex conic subsets resulting from the presence of linear constraints allow extremal elements of rank >= 2. The problem of deciding when a matrix is rank-1 generated, i.e, a sum of rank-1 positive-semidefinite (PSD) matrices, has been studied extensively in optimization theory. This rank-1 generated property acts as an entanglement criterion, and we use this property to find novel classes of PPT (Positive under partial transposition) entangled states. We do this by mapping some faces of PPT density matrices to convex cones that are not rank-1 generated. We show that all separable states map to a rank-1 generated matrix. In general, the same is not true for the corresponding mapping of PPT entangled states. We also extend this approach to construct PPT entangled edge states. Finally, we provide witnesses that detect violations of the rank-1 generated property.

Figures (4)

• Figure 1: We plot the slice of the (normalised) Bloch sphere with $r_z = 0$ and represent the constraints $r_x \geq 0$ and $r_y \geq 0$ in (light purple and light green, respectively). The extremal points (points on the boundary of the sphere) are the rank-1 extremal elements. The region in pink is the set $\mathcal{M}^+_d (\mathcal{L})$ and in orange is the $\operatorname{R}_1[\mathcal{M}^+_d (\mathcal{L})]$.
• Figure 2: The figure represents a cycle of the graph $v_1 \rightarrow v_2 \rightarrow v_3 \rightarrow v_4 \rightarrow v_1$ and $v_1 \rightarrow v_3$ is the called the chord of the graph.
• Figure 3: The graph $\mathcal{C}_4$ is the smallest non-chordal graph. It is also triangle-free, hence for any state with $G(A) \subseteq \mathcal{C}_4$, we can apply the theorem \ref{['thm:triangle-free-graphs']} to detect entanglement
• Figure 4: The graph in the figure is not triangle-free, but it has a triangle-free induced subgraph (the subgraph in red), allowing the application of \ref{['cor:induced-graph']}

Theorems & Definitions (49)

• Definition 2.1: Convex cones
• Definition 2.2
• Theorem 2.3
• Definition 2.4: Rank-1 generated cones
• Lemma 2.5
• proof
• Definition 2.6: Separability and Entanglement
• Definition 2.7: Separability Problem (Informal)
• Definition 2.8: Range criterion horodecki1997separability
• Definition 2.9: lewenstein2000optimization