Diagonal-unitary twirls of composites of PPT maps through a qutrit

Let Φ: Md → M3 and Ψ: M3 → Md be completely positive and completely copositive (PPT) maps, and let W be the Choi matrix of Ψ ∘ Φ. The PPT2 conjecture asserts that W is separable. We study the twirl of W by the group of local diagonal unitaries D ⊗ D̄, whose separability is equivalent to the statement that no diagonal-unitary-covariant witness detects W. The twirl is determined by Aij = ⟨ij|W|ij⟩ and the coherence matrix Bik = ⟨ii|W|kk⟩. We prove that for every d the twirl is separable whenever rank B ≤ 2. The proof factors W through Gram data Ta = Ga*Ha, classifies the two-dimensional spaces span{Ta} into five types, and combines a peeling lemma with known separability criteria for low-rank PPT states and, for two of the five types, with the theorem that composites of PPT maps through a qubit are entanglement breaking. A peel at an index c writes Ψ (or Φ) as the sum of an entanglement-breaking part and a PPT remainder whose coherence vanishes at c. Chains of peels, interleaved with the removal of zero rows of B, close further cases at every coherence rank, and for d = 4 a single peel suffices. For every d ≥ 3 the twirl plus the diagonal operator ((d − 3)/2) ∑i Bii|ii⟩⟨ii| is separable. At d = 4, if the compression of the Choi matrix of Φ or of Ψ* to a set of at least two outer indices is a sum of product states with linearly independent outer factors, then Φ, respectively Ψ, is entanglement breaking. At d = 4 and coherence rank 3 with a linear relation ∑a λaTa = 0 in which some λa vanishes, the twirl is separable, and Ψ ∘ Φ is entanglement breaking when exactly one λa vanishes. A cycle criterion for an arbitrary factorization B = FF* yields explicit product-state certificates. Numerically, this criterion certifies the twirl at each of the 42 generic configurations without peels produced by our random semidefinite searches at d = 4. At some pairs of extreme PPT maps it fails for every factorization of B. In particular, for every choice of Kraus operators, it fails when the columns of F are the diagonals of the products of a Kraus operator of Ψ with one of Φ. The twirl is nevertheless separable at these pairs: the Choi matrices of Φ and of Ψ* are supported on S ⊗ ℂ3 and S′ ⊗ ℂ3 for three-dimensional subspaces S, S′ ⊆ ℂd, and the known case M3 → M3 → M3 of the PPT2 conjecture applies. For d ≤ 4 and every coherence rank, the twirl is separable for every composite Φ* ∘ Ξ ∘ Φ in which Ξ is a sum of conjugations by positive semidefinite matrices. This follows from a criterion of Johnston and MacLean for completely positive coherence matrices. For d ≤ 4 we extend their criterion to coherence matrices that become entrywise nonnegative under conjugation by a diagonal unitary, provided that, around each cycle of the support graph of B, the product of the entries of A is the same in both directions. We also show that the number of product terms in a separable decomposition of the twirl can exceed the coherence rank, even for entanglement-breaking pairs through a qutrit. Finally, given the theorem that composites of PPT maps through a qubit are entanglement breaking, we show that W + (1/5) Φ*(I)⊤ ⊗ Ψ(I) is separable for every d, that for d ≥ 4 no pair-independent average of certificates from this theorem lowers the constant 1/5, even with PPT2 for maps on M3 and the separability of PPT states on ℂ2 ⊗ ℂ3, and that certificates adapted to the pair go beyond it: exact rational certificates show that W − (1/20) Φ*(I)⊤ ⊗ Ψ(I) is separable for an explicit pair at d = 4, and that W + (1/10) Φ*(I)⊤ ⊗ Ψ(I) is separable for an explicit Φ: M4 → M3 and every PPT map Ψ. Mathematics Subject Classification (2020): 81P40; 15A69, 47L07.