Sharp Asymptotics for k-Positivity Shifts under Singular Perturbations

The gist The CP-distance to quantify the deviation of Hermitian linear maps from complete positivity and its extension to a hierarchy of deviation, dk(Φ), serves as an optimal threshold for certifying Schmidt numbers and constructing dimension-sensitive entanglement witnesses.

CP-Distance Definition and Properties

The CP-distance dCP(Φ) is defined as the smallest scalar k ≥ 0 for which the map Ψ = Φ + ktr(·)In is completely positive, given by dCP(Φ) = max[0, −λmin(CΦ)] <ref:2506.037731,five or more words copied word for word from that page. This distance measures the minimal “shift” needed to make all eigenvalues non-negative when adding a simple completely positive map <ref:2506.037734,five or more words copied word for word from that page. The basic properties of the CP-distance include subadditivity, homogeneity, convexity, and invariance under unitary conjugation <ref:2506.037738,five or more words copied word for word from that page.

Directional Robustness and dΓ(Φ)

The paper introduces a robustness functional along an arbitrary completely positive direction called the directional robustness dΓ(Φ), defined as dΓ(Φ):= inf t ≥ 0: Φ + tΓ is completely positive <ref:2506.037731,five or more words copied word for word from that page. Theorem 3.3 establishes a spectral characterization for this quantity when the reference map Γ has a positive Choi matrix CΓ ≻ 0, showing dΓ(Φ) = max[0, −λmin(H)] where H:= B−1/2AB−1/2 <ref:2506.037737,five or more words copied word for word from that page. This framework is shown to be Lipschitz stable with respect to the map Φ <ref:2506.037749,five or more words copied word for word from that page.

Hierarchy of Deviation and Entanglement Certification

The paper extends the concept to a hierarchy of deviation dk(Φ), defined as dk(Φ):= inf [t ≥ 0: Φ + t∆ ∈ Pk] <ref:2506.037741,five or more words copied word for word from that page. Theorem 4.2 provides the spectral formula for dk(Φ) as dk(Φ) = max[0, − min v⟩∈Sk ⟨vCΦv⟩] <ref:2506.037746,five or more words copied word for word from that page. This formula shows that dk is determined by the minimum expectation value over the subspace of states with bounded entanglement depth <ref:2506.037748,five or more words copied word for word from that page.

Optimal Dimensionality Certification

The quantity dk(Φ) serves as a universal benchmark for entanglement certification, as Theorem 4.3 proves that if tr(CΦρ) < −dk(Φ), then the Schmidt number of ρ is strictly greater than k (i.e., SN(ρ) ≥ k + 1) <ref:2506.037751,five or more words copied word for word from that page. Furthermore, the threshold −dk(Φ) is sharp because there exists a quantum state σ with SN(σ) ≤ k such that tr(CΦσ) = −dk(Φ) <ref:2506.037754,five or more words copied word for word from that page. This allows for the universal construction of high-dimensional entanglement witnesses using the shifted Choi matrix WΦ,k:= CΦ + dk(Φ)Imn <ref:2506.037756,five or more words copied word for word from that page.

Tensor Product Properties

The directional robustness is invariant under tensor products with a full-rank completely positive map, meaning dΓ⊗Ξ(Φ ⊗ Ξ) = dΓ(Φ) <ref:2506.037759,five or more words copied word for word from that page. Additionally, the CP-distance scales linearly with the identity map when tensored with an identity map on Mr, resulting in dCP(Φ ⊗ idr) = r dCP(Φ) <ref:2506.037761,five or more words copied word for word from that page.

The paper establishes a hierarchical framework of deviation measures—dCP and dk—which quantify how far a Hermitian map is from complete positivity or k-positivity, respectively, providing rigorous spectral formulas for these distances and demonstrating their operational significance in certifying entanglement through optimal thresholds <ref:2506.037742,five or more words copied word for word from that page.

1. textbfThe CP-distance as an operational feasibility metric: dCP(Φ) = max(0, −λmin(CΦ)). This allows AI systems to quantify how close a learned or generated quantum operation is to being physically realizable by adding minimal noise, which is crucial for designing noise-robust quantum circuits or channels that can withstand realistic hardware imperfections.

2. textbfDirectional robustness against arbitrary CP maps: dΓ(Φ) = inf t ≥ 0: CΦ + t CΓ ≥ 0. This enables AI to develop methods for certifying the robustness of a map against a specific, known physical process (represented by the CP map Γ), moving beyond simple complete positivity checks to assess stability along relevant operational directions.

3. textbfUniversal Schmidt number certification via dk(Φ) = max 0, − min v⟩∈Sk ⟨vCΦv⟩. This provides a rigorous threshold for certifying high-dimensional entanglement; specifically, if tr(CΦρ) < −dk(Φ), the AI system can definitively conclude that the state ρ has a Schmidt number SN(ρ) ≥ k + 1, enabling automated detection of complex entanglement structures in generated quantum states.

4. textbfWitness construction via dk+1(Φ). For any Hermitian map Φ such that dk+1(Φ) > dk(Φ), the shifted Choi matrix WΦ,k:= CΦ + dk(Φ)Imn is a non-trivial entanglement witness of class k. This allows the AI to automatically generate optimal witnesses for detecting states that violate a specific Schmidt rank constraint k, ensuring the detection threshold is sharp and cannot be improved upon for that rank.

5. textbfAmplification by identity map: dCP(Φ ⊗ idr) = r dCP(Φ). This property enables the AI to scale entanglement detection capabilities across tensor product systems; if an operation is detectable with a certain robustness, the system can predict its performance in higher-dimensional composite systems by simply scaling the distance metric.