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The map Φ2\Phi_2Φ2​ of the dimension-ten pair is PPT

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OAI.DimensionTen.phiTwo_ppt

by elem · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

completely-positivepptquantum-information

The explicit map Φ2:M10(C)→M10(C)\Phi_2 : M_{10}(\mathbb C) \to M_{10}(\mathbb C)Φ2​:M10​(C)→M10​(C) of the source, Φ2(B)=R†(E∗BE)\Phi_2(B) = R^{\dagger}(E^* B E)Φ2​(B)=R†(E∗BE) where R(A)=K S(A)TK∗R(A) = K\,S(A)^T K^*R(A)=KS(A)TK∗ is the complementary map built from SSS and the Hodge-type signed permutation KKK on C6\mathbb C^6C6, and R†R^\daggerR† is its Hilbert–Schmidt adjoint, is PPT: it is complex linear, completely positive, and its output transpose B↦Φ2(B)TB \mapsto \Phi_2(B)^TB↦Φ2​(B)T is completely positive, with complete positivity in the amplification sense of the mission.

Preamble
import Mathlib
import Definitions.Def_DimensionTenPair

open Matrix Complex
open scoped Matrix ComplexOrder
Formal statement
namespace OAI.DimensionTen

theorem phiTwo_ppt : PPT phiTwo := by sorry

end OAI.DimensionTen
Source
OpenAI, Entanglement with zero distillable secret key in local dimension ten (September 27, 2026), https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf, Section 6, equations (6.4)–(6.5) and the paragraph following (6.5) (adjoint and composition clauses of Lemma 2.1 make Φ₂ PPT).

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