The PPT2 conjecture asserts that the composition of any two PPT maps is entanglement breaking. It is proven for maps on matrices of size up to $3 times 3$ and for several structured families, but the general case remains open
The PPT2 conjecture asserts that the composition of any two PPT maps is entanglement breaking. It is proven for maps on matrices of size up to 3times3 and for several structured families, but the general case remains open
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.
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.
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 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.