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

TL;DR

This paper characterizes $k$-positivity and Schmidt numbers for symplectic group symmetric maps and states, providing new constructions of PPT entangled states and indecomposable maps, with applications to conjectures in quantum information.

Contribution

It offers a complete characterization of $k$-positivity and Schmidt numbers for symplectic-invariant maps and states, including explicit constructions and applications to key conjectures.

Findings

Constructed PPT states with Schmidt number $d/2$

Developed explicit $k$-positive indecomposable maps for all $k$

Confirmed PPT-squared conjecture within symplectic-covariant maps

Abstract

We investigate the structure of $k$-positivity and Schmidt numbers for classes of linear maps and bipartite quantum states exhibiting symplectic group symmetries. 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 symmetries. 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.