The PPT² conjecture asserts that the composition of two completely positive and completely copositive (PPT) maps is entanglement breaking. A recent counterexample uses two maps through a six-dimensional middle algebra between ten-dimensional outer algebras, and certifies entanglement by the range criterion: the range of the composite Choi matrix contains no product vector. We ask how small such a certificate can be. For PPT maps Φ: Mₘ → Mₖ and Ψ: Mₖ → Mₙ with Ψ ∘ Φ ≠ 0 we prove, with computer-assisted steps, that the range of the Choi matrix of Ψ ∘ Φ contains a product vector whenever max(m, k, n) ≤ 6 and either k ≤ 5 or an outer dimension is at most five. If both outer dimensions are four, this holds for every middle dimension k. In dimension six only the reduced type (6, 6, 6) remains, and we bound its Choi ranks. The proofs use that, after a reduction, a product-free composite range forbids Kraus operators of rank at most two, that the rank-loss loci of a PPT map are finite unions of linear subspaces with generically constant range on each subspace, a rank inequality for PPT maps at product vectors in the kernel of their Choi matrices, top Chern classes at middle dimensions four and five, an excess intersection on Gr(3, 6) at middle dimension six, and, for the type (5, 6, 5), the Euler characteristic of a sheaf on a multiple line in ℙ². For the known example we show that the quadratic Veronese images of the rank-loss directions of its base map and of their kernel vectors are Gale dual by the Cayley–Bacharach theorem, and that the kernel conditions determine the Kraus spaces of both factors. If the rank-loss scheme of the base map is reduced and of the expected dimension zero, then among the lifts through arbitrary Schur functors with nonzero composite only those through exterior powers give an unextendible product set of kernel product vectors, and these only when one of the three dimensions is at least ten. For a general base map, the member of the family with outer dimension six is one product vector short of the range criterion, and its factors are rigid. For such base maps on M₄ that are not entanglement breaking, a compression of the lift to outer dimensions 4 ≤ D ≤ 9 keeps at most 2D − 4 of the twenty kernel product vectors. Finally, symmetry under a compact connected group acting irreducibly on both outer spaces, or under SU(2) acting without multiplicities on one of them, puts a product vector into the range. Version 2 splits the paper into a main paper (ppt2-paper2-v2-main.pdf) and a supplement (ppt2-paper2-v2-supplement.pdf); the mathematics is the same as in version 1. Mathematics Subject Classification (2020): 81P40; 15A69, 14M15, 14C17, 47L07.
The PPT² conjecture asserts that the composition of two completely positive and completely copositive (PPT) maps is entanglement breaking. A recent counterexample uses two maps through a six-dimensional middle algebra between ten-dimensional outer algebras, and certifies entanglement by the range criterion: the range of the composite Choi matrix contains no product vector. We ask how small such a certificate can be. For PPT maps Φ: Mₘ → Mₖ and Ψ: Mₖ → Mₙ with Ψ ∘ Φ ≠ 0 we prove, with computer-assisted steps, that the range of the Choi matrix of Ψ ∘ Φ contains a product vector whenever max(m, k, n) ≤ 6 and either k ≤ 5 or an outer dimension is at most five. If both outer dimensions are four, this holds for every middle dimension k. In dimension six only the reduced type (6, 6, 6) remains, and we bound its Choi ranks. The proofs use that, after a reduction, a product-free composite range forbids Kraus operators of rank at most two, that the rank-loss loci of a PPT map are finite unions of linear subspaces with generically constant range on each subspace, a rank inequality for PPT maps at product vectors in the kernel of their Choi matrices, top Chern classes at middle dimensions four and five, an excess intersection on Gr(3, 6) at middle dimension six, and, for the type (5, 6, 5), the Euler characteristic of a sheaf on a multiple line in ℙ². For the known example we show that the quadratic Veronese images of the rank-loss directions of its base map and of their kernel vectors are Gale dual by the Cayley–Bacharach theorem, and that the kernel conditions determine the Kraus spaces of both factors. If the rank-loss scheme of the base map is reduced and of the expected dimension zero, then among the lifts through arbitrary Schur functors with nonzero composite only those through exterior powers give an unextendible product set of kernel product vectors, and these only when one of the three dimensions is at least ten. For a general base map, the member of the family with outer dimension six is one product vector short of the range criterion, and its factors are rigid. For such base maps on M₄ that are not entanglement breaking, a compression of the lift to outer dimensions 4 ≤ D ≤ 9 keeps at most 2D − 4 of the twenty kernel product vectors. Finally, symmetry under a compact connected group acting irreducibly on both outer spaces, or under SU(2) acting without multiplicities on one of them, puts a product vector into the range. Version 2 splits the paper into a main paper (ppt2-paper2-v2-main.pdf) and a supplement (ppt2-paper2-v2-supplement.pdf); the mathematics is the same as in version 1. Mathematics Subject Classification (2020): 81P40; 15A69, 14M15, 14C17, 47L07.
The PPT² conjecture asserts that the composition of two completely positive and completely copositive (PPT) maps is entanglement breaking. A recent counterexample uses two maps through a six-dimensional middle algebra between ten-dimensional outer algebras, and certifies entanglement by the range criterion: the range of the composite Choi matrix contains no product vector. We ask how small such a certificate can be. For PPT maps Φ: Mₘ → Mₖ and Ψ: Mₖ → Mₙ with Ψ ∘ Φ ≠ 0 we prove, with computer-assisted steps, that the range of the Choi matrix of Ψ ∘ Φ contains a product vector whenever max(m, k, n) ≤ 6 and either k ≤ 5 or an outer dimension is at most five. If both outer dimensions are four, this holds for every middle dimension k. In dimension six only the reduced type (6, 6, 6) remains, and we bound its Choi ranks. The proofs use that, after a reduction, a product-free composite range forbids Kraus operators of rank at most two, that the rank-loss loci of a PPT map are finite unions of linear subspaces with generically constant range on each subspace, a rank inequality for PPT maps at product vectors in the kernel of their Choi matrices, top Chern classes at middle dimensions four and five, an excess intersection on Gr(3, 6) at middle dimension six, and, for the type (5, 6, 5), the Euler characteristic of a sheaf on a multiple line in ℙ². For the known example we show that the quadratic Veronese images of the rank-loss directions of its base map and of their kernel vectors are Gale dual by the Cayley–Bacharach theorem, and that the kernel conditions determine the Kraus spaces of both factors. If the rank-loss scheme of the base map is reduced and of the expected dimension zero, then among the lifts through arbitrary Schur functors with nonzero composite only those through exterior powers give an unextendible product set of kernel product vectors, and these only when one of the three dimensions is at least ten. For a general base map, the member of the family with outer dimension six is one product vector short of the range criterion, and its factors are rigid. For such base maps on M₄ that are not entanglement breaking, a compression of the lift to outer dimensions 4 ≤ D ≤ 9 keeps at most 2D − 4 of the twenty kernel product vectors. Finally, symmetry under a compact connected group acting irreducibly on both outer spaces, or under SU(2) acting without multiplicities on one of them, puts a product vector into the range. Version 3 adds a description of a Lean formalization that derives the main theorem from nine stated algebro-geometric hypotheses (code: doi:10.5281/zenodo.23643757); the mathematics is otherwise the same as in version 2. Mathematics Subject Classification (2020): 81P40; 15A69, 14M15, 14C17, 47L07.
The PPT² conjecture asserts that the composition of two completely positive and completely copositive (PPT) maps is entanglement breaking. A recent counterexample uses two maps through a six-dimensional middle algebra between ten-dimensional outer algebras, and certifies entanglement by the range criterion: the range of the composite Choi matrix contains no product vector. We ask how small such a certificate can be. For PPT maps Φ: Mₘ → Mₖ and Ψ: Mₖ → Mₙ with Ψ ∘ Φ ≠ 0 we prove, with computer-assisted steps, that the range of the Choi matrix of Ψ ∘ Φ contains a product vector whenever max(m, k, n) ≤ 6 and either k ≤ 5 or an outer dimension is at most five. If both outer dimensions are four, this holds for every middle dimension k. In dimension six only the reduced type (6, 6, 6) remains, and we bound its Choi ranks. The proofs use that, after a reduction, a product-free composite range forbids Kraus operators of rank at most two, that the rank-loss loci of a PPT map are finite unions of linear subspaces with generically constant range on each subspace, a rank inequality for PPT maps at product vectors in the kernel of their Choi matrices, top Chern classes at middle dimensions four and five, an excess intersection on Gr(3, 6) at middle dimension six, and, for the type (5, 6, 5), the Euler characteristic of a sheaf on a multiple line in ℙ². For the known example we show that the quadratic Veronese images of the rank-loss directions of its base map and of their kernel vectors are Gale dual by the Cayley–Bacharach theorem, and that the kernel conditions determine the Kraus spaces of both factors. If the rank-loss scheme of the base map is reduced and of the expected dimension zero, then among the lifts through arbitrary Schur functors with nonzero composite only those through exterior powers give an unextendible product set of kernel product vectors, and these only when one of the three dimensions is at least ten. For a general base map, the member of the family with outer dimension six is one product vector short of the range criterion, and its factors are rigid. For such base maps on M₄ that are not entanglement breaking, a compression of the lift to outer dimensions 4 ≤ D ≤ 9 keeps at most 2D − 4 of the twenty kernel product vectors. Finally, symmetry under a compact connected group acting irreducibly on both outer spaces, or under SU(2) acting without multiplicities on one of them, puts a product vector into the range. Mathematics Subject Classification (2020): 81P40; 15A69, 14M15, 14C17, 47L07.
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.
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.
The positive-partial-transpose-squared (PPT-squared) conjecture asks whether the composition of two positive-partial-transpose (PPT) quantum channels must be entanglement breaking (EB). The problem remains open in general and, within the diagonal orthogonal covariant (DOC) class, the first unresolved deterministic dimension is four. A sufficient condition for PPT-squared composition is derived for DOC maps in arbitrary finite dimension. The proof isolates the quantum coherence data into two correlation matrices, mixes each with the identity until it lies in the convex hull of rank-one correlation matrices, and converts the resulting objects into an explicit triplewise completely positive (TCP) core. The remaining contribution is purely classical and is TCP whenever a simple entrywise mixing inequality is satisfied. For PPT DOC maps ΦA,B,C and ΦD,E,F on Md, the entrywise bound (AD)ij≥2⌊d⌋AiiAjjDiiDjj for all i,j guarantees that ΦA,B,C∘ΦD,E,F is EB. In dimension four the universal coefficient becomes four. A continuous family of bistochastic PPT channels is then constructed; every member with a=1 is not EB, while every pairwise composition within the family is proved to be EB. A rank-sensitive refinement shows explicitly how low coherence rank can improve the universal coefficient. The resulting certificate is sufficient rather than necessary and is complementary to factor-width and semidefinite-hierarchy approaches.
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.