The map of the dimension-ten pair is PPT
OpenOAI.DimensionTen.phiTwo_pptcompletely-positivepptquantum-information
The explicit map of the source, where is the complementary map built from and the Hodge-type signed permutation on , and is its Hilbert–Schmidt adjoint, is PPT: it is complex linear, completely positive, and its output transpose 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).