We present an explicit rational certificate for a positive-partial-transpose (PPT) state on C^15625 tensor C^15625 outside the cone C2 + partial_transpose(C2), where C2 denotes the cone of positive semidefinite operators of Schmidt number at most two. A 5x5 grid-state seed has realignment norm bounded below by 1028/909, proved by a 25x25 integer matrix certificate. Six tensor copies amplify this to (1028/909)^6 > 2, while the realignment norm is at most twice the trace on C2 + partial_transpose(C2). An explicit Choi witness consequently defines a linear map that is simultaneously 2-positive and 2-copositive but nondecomposable. The accompanying archive contains the mathematical preprint, LaTeX source, exact-arithmetic verification programs, and computational checks. This is an unrefereed proof manuscript. Independent expert verification and exhaustive prior-art assessment remain outstanding. The construction does not resolve the PPT-cubed conjecture or establish an improved quantum teleportation protocol.
We present an explicit rational certificate for a positive-partial-transpose (PPT) state on C^15625 tensor C^15625 outside the cone C2 + partial_transpose(C2), where C2 denotes the cone of positive semidefinite operators of Schmidt number at most two. A 5x5 grid-state seed has realignment norm bounded below by 1028/909, proved by a 25x25 integer matrix certificate. Six tensor copies amplify this to (1028/909)^6 > 2, while the realignment norm is at most twice the trace on C2 + partial_transpose(C2). An explicit Choi witness consequently defines a linear map that is simultaneously 2-positive and 2-copositive but nondecomposable. The accompanying archive contains the mathematical preprint, LaTeX source, exact-arithmetic verification programs, and computational checks. This is an unrefereed proof manuscript. Independent expert verification and exhaustive prior-art assessment remain outstanding. The construction does not resolve the PPT-cubed conjecture or establish an improved quantum teleportation protocol.
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.
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.
This preprint constructs finite rational proof objects for both sides of the quantum value of I3322 Bell inequality. Let 𝑄t denote the supremum over finite-dimensional tensor-product binary measurements and𝑄c the commuting-operator value. On the complete word space of total length at most four, we construct a rational Gram certificate. The full signed setting-relabeling stabilizer is the dihedral group of order eight, reducing the certificate to five rational blocks. Ex-act affine reconstruction and a uniform diagonal shift of certified positive interior prove: 𝑄c (𝐼3322) ≤ 62718846253/250000000000 = 0.250875385012. Independently, a local-dimension-499 strategy specified by rational projectors and a nonzero unnormalized rational Schmidt vector has exact normalized value 𝐿499 = 0.25087538451397653548 . . .. Con-sequently, 𝐿499 ≤ 𝑄t(𝐼3322) ≤ 𝑄c (𝐼3322) ≤ 0.250875385012, with endpoint difference below 5 × 10−10. The lower certificate also proves 𝑄t(𝐼3322) > 1/4. Every theorem is checked using exact rational or integer arithmetic; floating-point computation is used only to discover candidates. Complete certificates, source code, tests, and reproducibility materials are archived at: doi:10.5281/zenodo.21462892.
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.
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.
This paper characterizes the fixed finite positive semidefinite composition partners whose tensor powers test strong positivity of every finite Hermitian matrix: a partner is universal exactly when it has a nonreal entry in the specified history basis. Constructive proofs give explicit copy bounds, finite-depth spectral and tolerance certificates, a finite-family obstruction, and a quantum-history realization. For a noisy two-history family, all-event lower bounds and explicit exposing events establish sharp weak-imaginary resource scaling. Further results include an exact pure-phase onset and conditional finite-composition CHSH certificates. The accompanying ZIP contains the LaTeX source, six exact standard-library Python checkers and their results, a package verifier and file hashes. Priority remains unestablished. No derivation of a physical probability law is claimed.Robert Barritz directed the research and its preparation for publication. OpenAI’s ChatGPT and Codex systems contributed substantially to mathematical formulation, proof development and examination, examples, exact computational verification, literature comparison, and manuscript preparation. Anthropic’s Claude contributed mathematical, computational, and literature review that informed revisions. The AI contribution extended substantially beyond language editing. AI-assisted review does not constitute human expert or journal peer review.