A Hierarchy of Deviation from Complete Positivity and Optimal Entanglement Witnesses

2025/06/04 by Mohsen Kian, Kian, Mohsen · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Quantum Information and Cryptography #Random Matrices and Applications #Quantum Mechanics and Applications

paper · pdf · doi:10.48550/arxiv.2506.03773

Abstract

We introduce the 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 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 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.

Citations

• Completely positive linear maps on complex matrices

Cited by

• Hermitian Maps: Approximations and Completely Positive Extensions

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