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.
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.
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 (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 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.
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 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.