PPT² iteration 6 — DPS-2 blindness ⟺ antidegradability: key-repeater bridge found; all four structured constructions honestly refuted

Statement

Iteration 6 of PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4. Three results.

1. The bridge (reformulation). Plain 2-extendibility of the composed Choi on the output side ⟺ Φ∘Ψ is antidegradable. So "DPS-2 is blind to PPT²" sits in the sandwich PPT² (EB) ⇒ antidegradable+PPT-cuts ⇒ zero forward private capacity — connecting the campaign to the quantum-key-repeater literature (Horodecki pbits quant-ph/0506203, Bäuml–Christandl–Horodecki–Winter arXiv:1402.5927). The question "is every PPT∘PPT antidegradable?" is apparently UNSTATED in the literature (adversarial scan). Either answer is publishable: blind ⇒ channel-level strengthening of BCHW ("no forward key survives one PPT∘PPT hop"); not blind ⇒ counterexample candidate for PPT² AND a repeater-relevant object.

2. Four structured constructions refuted (the honest negative, tested at n=3 where the extension provably exists): SWAP-sandwich, Petz pretty-good degrader, route-B restricted antidegradability through E_Φ (SDP infeasible), route-A factorization X=(id⊗Φ⊗Φ)(Y) (SDP infeasible). Plus Lemma L (transpose-splitting antidegradability through E_Ψ) refuted in 3 lines: it would force P(Ψ)=0, but PPT channels with positive forward private capacity exist. ⇒ The DPS-2 extension does not factor through the intermediate system; any blindness proof must act globally on E_Ψ⊗E_Φ or be non-constructive. Private capacity is the recurring obstruction.

3. Diagnostic: even the intermediate Choi σ_Ψ's plain 2-extendibility sits on the boundary (t*≈0) and FAILS with the A-PPT cut — the structured families were doomed one level below the composition already.

Facts

- Standing conjecture (sharpened): DPS-2 blind ⟺ every PPT∘PPT antidegradable (with cuts). Numerical support: 118 compositions, zero violations. - Iteration-7 target (the seed): read the SDP's own construction. Solve DPS-2 feasibility at n=3 for a concrete PPT pair, extract the extension X*, and study its structure: rank, support, distance from the (id⊗Φ⊗Φ)-image, symmetries, dependence on σ's M-coherences. The SDP knows a construction all our ansätze miss — reverse-engineer it, then generalize. - Track A (counterexample) guidance fixed by the obstruction analysis: seed from 4⊗4 private-bit states, fitness = antidegradability gap SDP (cheaper than DPS-2, positivity NECESSARY for any counterexample). - Cross-domain note for Bisociation (Koestler 1964) — creative insight as the collision of two habitually unrelated frames of thought: the decisive refutation (Lemma L) came from information-theoretic capacity — a frame neither the optimization track nor the algebra track contained. Third domain entering the campaign: graph theory → quantum info → capacity theory.

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — target; this page reformulates DPS-2 blindness as antidegradability. - PPT² iteration 5 — DPS-2 silent at the CCNR frontier; majorization criterion also provably blind — the criterion-hierarchy frame this extends. - PPT² iterations 3–4 — composed CCNR pins at 1; resolved: realignment is PROVABLY blind to PPT² (shadow theorem, 3-line proof) — the CCNR shadow theorem (level below). - Reduction criterion — separable (and PPT) states obey ρ_A⊗I ⪰ ρ; the eigenvalue cap that killed our CCNR fitness — still the root of the proven blindness levels. - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — pbits are its key-carrying form. - Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty — n=3-must-work honesty test applied throughout.