A Hierarchy of Deviation from Complete Positivity and Optimal Entanglement Witnesses
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908745616850944 |
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| author | Kian, Mohsen |
| author_facet | Kian, Mohsen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Hierarchy of Deviation from Complete Positivity and Optimal Entanglement Witnesses Kian, Mohsen Quantum Physics Functional Analysis 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. |
| title | A Hierarchy of Deviation from Complete Positivity and Optimal Entanglement Witnesses |
| topic | Quantum Physics Functional Analysis |
| url | https://arxiv.org/abs/2506.03773 |