Sufficient criteria for absolute separability in arbitrary dimensions via linear map inverses

TL;DR

The paper tackles the problem of characterizing absolute separability (AS) and absolute PPT (AP) across arbitrary dimensions by leveraging inverses of non-CP maps. It develops eigenvalue-based sufficient criteria derived from unitary-covariant, reduction-like maps Λ_α and their inverses, and then unifies these criteria using convex geometry to yield stronger, spectrum-only tests expressible as convex programs. The main contributions include a compact inequality describing the convex hull of AS criteria, an SDP-formulation to combine multiple criteria, and robust extensions to multipartite and symmetric subspaces with explicit eigenvalue bounds for SAS and SAP. These results provide practically verifiable, dimension-agnostic tests that rely only on spectral data, with potential applications in resource theories and invariant-set analyses. The framework also suggests pathways to extend similar convex-geometry methods to other invariant convex sets in quantum information, such as steering and nonlocality criteria.

Abstract

Quantum states that remain separable (i.e., not entangled) under any global unitary transformation are known as absolutely separable and form a convex set. Despite extensive efforts, the complete characterization of this set remains largely unknown. In this work, we employ linear maps and their inverses to derive new sufficient analytical conditions for absolute separability in arbitrary dimensions, providing extremal points of this set and improving its characterization. Additionally, we employ convex geometry optimization to refine the characterization of the set when multiple non-comparable criteria for absolute separability are available. We also address the closely related problem of characterizing the absolute PPT (positive partial transposition) set, which consists of quantum states that remain positive under partial transposition across all unitary transformations. Finally, we extend our results to multipartite states.

Figures (3)

• Figure 1: Schematic representation of quantum states and the AS set for $D=3$ in barycentric coordinates. Normalized states fill the area enclosed by the blue vertices. In red (dashed) the simplex corresponding to $\lambda_{\min}\geq 1/(D+2)$, in orange(dots) the one corresponding to $\lambda_{\max}\leq 1/(D-1)$ (see Theorem \ref{['Prop:Lewenstein']}). In green the convex hull of both simplexes. The light shaded region fulfills the ordering $\lambda_{0}\leq\lambda_{1}\leq\lambda_{2}$ we consider. The dark shadow polytope highlights the intersection of the convex hull with the ordered zone. The figure is illustrative as $D=3$ does not correspond to any bipartite splitting.
• Figure 2: Detected absolutely separable (AS) states. Left panel: Schematic two-dimensional representation of the convex hull (orange) of the Gurvits-Barnum ball (yellow) and our simplex (blue) in barycentric coordinates (cf. Figure \ref{['fig:Simplex']}). Right panel: Purity $\mathrm{Tr}(\rho^2)$ and minimal eigenvalue $\lambda_{\min}(\rho)$ of the detected states for a $2$-qutrit system. The points in the striped region do not correspond to physical states.
• Figure 3: Left panel: Comparison of the states detected for a symmetric $2-$qubit system with the convex hull (green line) of the sufficient conditions through the inverse of the linear map with $\alpha=1$ (dashed red line) and $\alpha=-3/4$ (dotted orange line) with the complete characterization $\sqrt{\lambda_{0}}+\sqrt{\lambda_{1}}\geq 1$ (black solid line). Right panel: Comparison of the states detected for a symmetric $3-$qubit system with the convex hull (green line) of the sufficient conditions through the inverse of the linear map with $\alpha=2/3$ (dashed red line) and $\alpha=-2/3$ (dotted orange line) with the best known analytical criterion serrano-ensastiga_absolute-separability_2024 (black sphere). The representation shows a $2D$ cut of the non-increasingly ordered region of eigenvalues $\lambda_{0}\leq\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}$.

Theorems & Definitions (36)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Definition 2
• Lemma 1
• proof
• Theorem 4
• proof