A Hierarchy of Deviation from Complete Positivity and Optimal Entanglement Witnesses
Mohsen Kian
TL;DR
The paper develops a spectral framework to quantify how far a Hermitian map is from complete positivity by introducing the CP-distance $d_{\mathrm{CP}}(\Phi)$ and its directional variant $d_\Gamma(\Phi)$, with explicit formulas in terms of the Choi matrix. It then generalizes this idea to a hierarchy $d_k(\Phi)$ based on the cones of $k$-positive maps, deriving a sharp spectral characterization via Schmidt-rank-$k$ vectors and the Choi matrix, which yields a rigorous threshold for certifying Schmidt numbers. The results provide a universal construction of dimension-sensitive entanglement witnesses: from any Hermitian map, one obtains optimal witnesses for Schmidt-number detection by shifting the Choi matrix by $d_k(\Phi)I_{mn}$. The framework links cone geometry, robustification against depolarizing noise, and entanglement certification, enabling scalable witness construction and stability analyses across dimensions.
Abstract
We introduce the \emph{CP-distance} to quantify the deviation of Hermitian linear maps from complete positivity, defined as the minimal depolarizing noise required to render a map completely positive. We derive a closed spectral formula for this distance and extend the framework to \emph{directional robustness} against arbitrary completely positive maps, establishing stability and tensor-product properties. Expanding this to the intermediate cones of $k$-positive maps, we introduce a \emph{hierarchy of deviation}, $d_k(Φ)$. We derive a spectral formula for $d_k$ based on entanglement depth and demonstrate that it serves as an optimal threshold for certifying Schmidt numbers, allowing for the universal construction of dimension-sensitive entanglement witnesses.