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 $k\geq 2$, 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 $\Gamma_\varepsilon=\Gamma+\varepsilon\Delta$, where $\Delta(X)=\mathrm{Tr}(X)I$, we determine the least $t\geq 0$ for which $\Phi+t\Gamma_\varepsilon$ is $j$-positive. More precisely, $s_{j,\Gamma_\varepsilon}(\Phi)=0$ for $j<k$, while $s_{k,\Gamma_\varepsilon}(\Phi)\sim c_k\varepsilon^{-(2k+1)/(2k+2)}$ and $s_{k+1,\Gamma_\varepsilon}(\Phi)=\varepsilon^{-1}$. 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.

Quantum Physics (quant-ph); Functional Analysis (math.FA)

arXiv:2506.03773 [quant-ph]

(or arXiv:2506.03773v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2506.03773 Focus to learn more arXiv-issued DOI via DataCite

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