The PPT² conjecture holds for spin-factor maps

The PPT2 conjecture asserts that the composition of two completely positive and completely copositive maps is entanglement breaking. A counterexample in dimension ten has recently been announced, and for maps on M4 the conjecture is open. Let 𝒮 ⊂ M4 be the real span of the identity γ0 and five pairwise anticommuting Hermitian unitaries γ1, …, γ5. It is a spin factor whose positive cone is a Lorentz cone. We prove the conjecture for all compositions Ψ∘Φ of completely positive maps Φ: M4 → Mk and Ψ: Mk → M4, with k arbitrary, such that the ranges of Ψ and of the adjoint Φ* lie in invertible congruences of the complex span of 𝒮. Such maps are automatically completely copositive, and the class contains maps whose Choi matrices are entangled. A theorem of Hildebrand on separability for pairs of Lorentz cones reduces the claim to one trace inequality between the coefficient matrices of the two Choi matrices. We derive this inequality from the operator inequality Σj=15FjF0−1Fj ⪯ F0, valid whenever Σμ=05Fμ ⊗ γμ ⪰ 0 and F0 ≻ 0, which in turn follows from the Kadison–Schwarz inequality. As a corollary, Ψ∘Φ is entanglement breaking whenever Ψ is such a map and Φ is a completely positive map with Kraus operators of rank at most two. Mathematics Subject Classification (2020): 81P40; 47L07, 15A69, 46L07.