PPT² iteration 2 — first n=4 ghost channel; two no-gos (padding, adjoint diagonal); the hunt becomes transfer-matrix alignment

Statement

Iteration 2 of the PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 loop. Four verified results.

1. First bound-entangled PPT channel at n=4 (the open dimension). Direct sum of the tiles ghost channel (3-block) with a pure pass-through |3⟩→|3⟩: CPTP ✓, PPT ✓, Choi CCNR = 1.0865 (= ¾·1.115 + ¼·1 exactly, orthogonal R-supports add), DPS-2 = entangled ✓. Note it sits ON the PPT boundary (min eig of C^{T_B} ≈ 0) — where ghosts must live.

2. No-go: padding can never certify. For block channels ghost⊕pass-through, composed CCNR = (1−1/n)·(ghost-block value) + (1/n)·1: the pass-through is pinned at exactly 1, the ghost block decays ⇒ the padded family is structurally ≤ 1 after composition. Measured: Φ∘Φ = 0.7864 (=¾·0.715+¼), anneal over Φ∘(UΦU†), U∈U(4), never beat the block structure (U=I local max). Positive signal: n=4 decay factor 0.724 vs 0.64 at n=3 — more room in the open dimension.

3. Composition is matrix multiplication in the realignment picture. R(C_{Φ∘Ψ}) = R(C_Ψ)·R(C_Φ) — machine-precision verified. (R(C) is the channel's transfer matrix; textbook fact, newly weaponized here.) Hence composed CCNR = ||T_Ψ·T_Φ||₁/n: a purely algebraic objective. Hölder: ||T_Ψ T_Φ||₁ ≤ σ_max(T_Ψ)·||T_Φ||₁. For the tiles ghost σ_max = 1.082, giving ceiling 1.21 > 1 even at n=3 — where the 2018 theorem forbids the value ever being reached. The n=3 obstruction is INVISIBLE to norm bounds: whatever kills alignment there is the conjecture's true mechanism.

4. No-go: the adjoint diagonal. For UNITAL channels, CCNR(Φ*∘Φ) = Σσ_i(T_Φ)²/n = n·Tr(ρ_Choi²) ≤ 1 always, since any channel Choi has A-marginal I/n forcing purity ≤ 1/n. Equality needs purity exactly 1/n (unitaries — never PPT for n≥2 — or flat-spectrum EB channels). So Ψ ≈ Φ* is a dead direction; the tiles ghost is additionally NOT unital (caught by the validator when Φ* failed TP).

Facts

- The hunt is now sharply posed: can two PPT transfer matrices be singular- structure ALIGNED enough that ||T_Ψ·T_Φ||₁ > n? At n=3 provably no (theorem), at n≥4 open — this is a reformulation of (the CCNR-certifiable part of) PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4. - Design guidance derived: (a) maximize alignment of top singular subspaces of T_Φ, T_Ψ subject to both PPT; (b) spiky R-spectrum beats flat (tiles state spectrum: σ = 0.361, 0.195, 0.177, ... — fairly flat); (c) stay on the PPT boundary; (d) avoid Ψ≈Φ* and padding (results 2,4). - Iteration-3 target: structured optimization over PAIRS at n=4 with objective ||T_Ψ T_Φ||₁, PPT+CPTP constraints (projection or parametrization), seeded from the n=4 ghost — Simulated-annealing extremal search — Metropolis local search for rare, structured witness/counterexample objects with an algebraic fitness instead of blind moves.

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — the target; this page reformulates its CCNR-visible part. - PPT² iteration 1 — WH-mixture exclusion at n=4; first bound-entangled CHANNEL built and composed (n=3 landscape) — the seeds and the decay landscape. - Choi–Jamiołkowski isomorphism — state–channel duality: a map is CP iff its Choi matrix is PSD — the reshuffle/transfer correspondence. - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — the n=4 ghost is its channel form at the open dimension. - Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty — caught the unitality error (author-error #4). - Simulated-annealing extremal search — Metropolis local search for rare, structured witness/counterexample objects — now armed with ||T_Ψ T_Φ||₁.