$k$-Positivity and high-dimensional bound entanglement under symplectic group symmetry

TL;DR

This work identifies and analyzes a rich class of quantum objects with unitary symplectic symmetry, enabling complete analytic control over $k$-positivity and Schmidt numbers. By introducing two-parameter families $\mathcal{L}_{p,q}^{(d)}$ and $\rho_{a,b}^{(d)}$, the authors obtain explicit, geometry-driven criteria for $k$-positivity, decomposability, and PPT entanglement, and they construct optimal $k$-positive indecomposable maps for all $k=1,\dots,d/2-1$ (the $k$-Breuer–Hall maps). They also demonstrate PPT-squared conjecture validity within these symmetry classes and resolve a Sindici–Piani SDP bound for PPT entanglement, highlighting the interplay between symmetry, high-dimensional bound entanglement, and positive maps. Collectively, the results provide a natural and tractable framework to study strong forms of positivity and high-dimensional PPT entanglement.

Abstract

We investigate the structure of $k$-positivity and Schmidt numbers for classes of linear maps and bipartite quantum states exhibiting 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 (optimal) $k$-positive indecomposable linear maps for arbitrary $k=1,\ldots, d/2-1$, achieving the best-known bounds. Overall, our results offer a natural and analytically tractable framework in which both strong forms of positive indecomposability and high degrees of PPT entanglement can be studied systematically. We present two further applications of symplectic group symmetry. First, we show that the PPT-squared conjecture holds within the class of PPT linear maps that are either symplectic-covariant or conjugate-symplectic-covariant. Second, we resolve a conjecture of Pal and Vertesi concerning the optimal lower bound of the Sindici-Piani semidefinite program for PPT entanglement.

Figures (3)

• Figure 1: $k$-Positivity (left) and Schmidt number (right) regions for $d=4$. The red dashed hyperbola (left) and ellipse (right) partially determine the boundaries for $k=3$. Furthermore, the dark blue quadrilateral (left) and dark green quadrilateral (right) indicate the decomposability and PPT regions, resp.
• Figure 2: The complete positivity (top left), complete copositivity (top right), positivity (bottom left), and decomposability (bottom right) regions of $\mathcal{L}_{p,q}$
• Figure 3: The $k$-Positivity regions $\mathbb{P}_k$ of $\mathcal{L}_{p,q}$ in the case: (1) top left, (2) top right, (3) bottom left, and (4) bottom right.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2: Theorem \ref{['thm-Symp-opt']}
• Theorem 2.1
• Theorem 2.2
• Corollary 2.3
• proof
• Proposition 2.4: VW01PJPY24
• Proposition 2.5: PJPY24PY24
• Proposition 2.6: NP25
• Theorem 2.7: PJPY24PY24