Sharp Asymptotics for $k$-Positivity Shifts under Singular Perturbations

Abstract

We show that, as the Choi matrix of a completely positive perturbation direction approaches a singular limit, the shift required for $k$-positivity can diverge more slowly than that required for complete positivity. For every $k\ge 2$, we construct a $(k-1)$-positive map $\Phi$ and a completely positive map $\Gamma$ on ${\mathbb{M}}_{k+1}$ with singular Choi matrix ${\mathbf{C}}_{\Gamma }$. Regularizing $\Gamma$ by ${\Gamma }_{\epsilon }=\Gamma +\epsilon \Delta$, where $\Delta (X)=\mathrm{Tr}(X)I$, we determine the least $t\ge 0$ for which $\Phi +t{\Gamma }_{\epsilon }$ is $j$-positive. More precisely, $s_{j,{\Gamma }_{\epsilon }}(\Phi )=0$ for $j<k$, while $s_{k,{\Gamma }_{\epsilon }}(\Phi )\sim c_k{\epsilon }^{-(2k+1)/(2k+2)}$ and $s_{k+1,{\Gamma }_{\epsilon }}(\Phi )={\epsilon }^{-1}$. The leading constant $c_k>0$ is determined explicitly. The fractional exponent is obtained through singular-value estimates and an explicit construction of rank-$k$ perturbations of a nilpotent Jordan block.