We prove that the composition of two PPT maps through an intermediate qubit system is entanglement breaking. Specifically, for arbitrary finite dimensions m and n, if Φ: Mₘ → M₂ and Ψ: M₂ → Mₙ are completely positive and completely copositive, then Ψ ∘ Φ is entanglement breaking. The proof uses the special 2 × 2 block structure imposed by the qubit intermediate system, Schur-complement inequalities, an operator interpolation argument, and an explicit separability decomposition. As a consequence, no counterexample to the PPT-squared conjecture can occur through the 4 → 2 → 4 composition pattern.
We prove that the composition of two PPT maps through an intermediate qubit system is entanglement breaking. Specifically, for arbitrary finite dimensions m and n, if Φ: Mₘ → M₂ and Ψ: M₂ → Mₙ are completely positive and completely copositive, then Ψ ∘ Φ is entanglement breaking. The proof uses the special 2 × 2 block structure imposed by the qubit intermediate system, Schur-complement inequalities, an operator interpolation argument, and an explicit separability decomposition. As a consequence, no counterexample to the PPT-squared conjecture can occur through the 4 → 2 → 4 composition pattern.
The PPT2 conjecture asserts that composites of PPT (completely positive and completely copositive) maps are entanglement breaking. For PPT maps Φ: Md1 → M3 and Ψ: M3 → Md2 let W be the Choi matrix of Ψ ∘ Φ. With the theorem that composites of PPT maps through a qubit are entanglement breaking (the qubit theorem), it is known that W + (1/5) Φ*(I)⊤ ⊗ Ψ(I) is separable, while PPT2 through a qutrit is the same statement with 1/5 replaced by 0. We ask how far 1/5 can be lowered. Certificates built from the qubit theorem independently of the pair cannot lower it, and for d1, d2 ≥ 3 with max(d1, d2) ≥ 4 neither can averages of certificates that apply maps separately to the input, middle and output spaces, even with PPT2 on M3 added. The least λ for which J(Φ) + λ Φ*(I)⊤ ⊗ I is a sum of PPT operators supported on ℂd1 ⊗ V for planes V is the one-sided constant λQD(Φ) ≤ 1/5; by the qubit theorem it is valid for every partner. It is below 1/5 when one compression of the middle qutrit to a qubit is strictly PPT of full rank, and its largest value Λ(d) over d1 = d is below 1/5 unless some extreme PPT channel is rigid, that is, has λQD = 1/5. We prove, without the qubit theorem, that no PPT channel M3 → M3 is rigid, so that some c < 1/5 makes ρ + c I ⊗ I separable for the Choi matrix ρ of every PPT channel M3 → M3. With the qubit theorem, W + c Φ*(I)⊤ ⊗ Ψ(I) is then separable for all PPT maps Φ: M3 → M3 and Ψ: M3 → Md2, and exact rational certificates beat 1/5 at explicit factors with d1 = 4. The exact computations used in the proofs are also checked in Lean 4. This paper develops Section 10, Appendix A and supplement S12-S13 of version 3 of "Diagonal-unitary twirls of composites of PPT maps through a qutrit" (doi:10.5281/zenodo.23675449) with a new introduction and preliminaries; its supplement S1-S2 is S12-S13 there. The statements and proofs are those of that version, renumbered, with references to the companion paper, version 4 of that record (doi:10.5281/zenodo.23680469). Programs, certificates and Lean checks are in the deposit, version 4 (doi:10.5281/zenodo.23680464). Dagger k of this paper is dagger k+24 in version 3 and in the deposit's folders, map files, replay record and Lean targets; the deposit's README uses this paper's numbering. Mathematics Subject Classification (2020): 81P40; 15A69, 47L07.
Entanglement swapping is a fundamental operation in quantum repeaters for establishing entanglement between distant parties. The positive partial transpose (PPT) squared conjecture asks whether two PPT entangled links can generate terminal entanglement through entanglement swapping, or equivalently, whether the composition of two PPT maps is always entanglement breaking. Motivated by this conjecture, we investigate the map-composition problem beyond the PPT setting. For qutrit completely positive (CP) maps, we prove that the composition of any CP map whose Choi matrix is 1-undistillable with any CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in either order. Moreover, we show that the cone of 1-undistillable CP maps is exactly the largest qutrit cone of CP maps whose composition with every CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in both orders. Finally, although map composition captures only the standard maximally entangled outcome in entanglement swapping, we prove that any 1-undistillable two-qutrit state and any state of Schmidt number at most two cannot generate terminal entanglement under an arbitrary selective measurement on the intermediate systems.
We prove that every PPT linear map has finite entanglement breaking index, thereby establishing the eventual entanglement breaking property of PPT channels in full generality. Furthermore, by utilizing completely positive maps with low entanglement dimensionality, we show that a large family of PPT maps, which strictly contains the class of 2-superpositive maps, has entanglement breaking index bounded above by 3, uniformly in the dimension. In particular, these results provide evidence that the PPT-cubed conjecture may hold in full generality.
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.
M. Christandl conjectured that the composition of any trace preserving PPT map with itself is entanglement breaking. We prove that Christandl's conjecture holds asymptotically by showing that the distance between the iterates of any unital or trace preserving PPT map and the set of entanglement breaking maps tends to zero. Finally, for every graph we define a one-parameter family of maps on matrices and determine the least value of the parameter such that the map is variously, positive, completely positive, PPT and entanglement breaking in terms of properties of the graph. Our estimates are sharp enough to conclude that Christandl's conjecture holds for these families.
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. Given the same theorem, for Φ: M3 → M3 a constant below 1/5 works uniformly: there is c < 1/5 such that W + c Φ*(I)⊤ ⊗ Ψ(I) is separable for every such Φ and every PPT map Ψ: M3 → Mn. Version 2 adds Theorem 10.19: no PPT channel M₃ → M₃ is rigid, so Λ(3) < 1/5. Given the theorem that composites of PPT maps through a qubit are entanglement breaking, it follows that a single constant below 1/5 applies to every pair of PPT maps M₃ → M₃ → Mₙ (Corollary 10.20). Exact certificates, checked in Lean, show that the constant of certificates adapted to one factor is at most 1/10 at two explicit points built from the Tiles unextendible product basis (Remark 10.21). The accompanying Lean code now contains 2688 theorems, 1833 of which are checked by Lean's kernel alone. Mathematics Subject Classification (2020): 81P40; 15A69, 47L07.