The PPT² conjecture holds for spin-factor maps

Abstract

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 → M_k and Ψ: M_k → 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 Σ_j=1⁵F_jF₀⁻¹F_j ⪯ F₀, valid whenever Σ_μ=0⁵F_μ ⊗ γ_μ ⪰ 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.